\section{Introduction}


In \cite{Storchak_2019, Storchak_2020} and \cite{Storchak_2021}, the method  developed earlier in \cite{Storchak_2, Storchak_2009, Storchak_1, Storchak_98}, by which it is possible to perform the reduction in the path integrals that are used for quantization of the mechanical systems with a symmetry was exteded to the the systems with interaction. In all these papers, the Wiener-type path integrals defined by Daletskii and Belopolskaya in \cite{Dalecky_1,Dalecky_2} were used as path integrals, whose measures are generated by stochastic processes on a manifold. According to their papers, the path integral is a global evolution semigroup, which is the limit of the local evolution semigroups determined by the local stochastic processes--the solutions of the stochastic differential equations.

The reason why we are interested in these finite-dimensional mechanical systems is because of their geometric properties, which are similar to those we meet  in field systems with gauge symmetry. In particular, the mechanical system from [21 ] has geometric properties similar to those of the field system describing the interaction of Yang-Mills fields with scalar fields.




In all cases that were considered, it was shown that the measure of the path integral is not invariant under reduction.
As a consequence, this leads to the appearance of a reduction Jacobian, which gives an additional potential term to the reduced Hamiltonian. Of particular interest is the study of the geometric representation of the resulting reduction Jacobian, since its geometric structure can be the same as in the corresponding gauge systems. 

We also note that such a structure is difficult to recognize, being within the framework of the Faddeev-Popov formalism, intended mainly for use in perturbation theory  and in which the measure in functional integrals is not defined as a true measure of functional analysis.

The appearance of an additional term to the potential in the pure Yang-Mills theory was demonstrated by J. Lott in \cite{Lott} when projecting the (functional) Schrödinger equation into an equation given on the manifold of the orbit space. On this space, he used independent variables to describe reduced evolution. Note that the geometric structure of the additional term coincides with the reduction Jacobian obtained  in \cite{Storchak_2021}, if we neglect the motion in the vector space. 

A similar approach, but using dependent variables, was considered by K. Gawedzki in \cite{Gawedzki}, where the geometric structure of the additional term was also disclosed. Its additional term looks different than in \cite{Lott}. But in fact, their geometric structures must coincide, since it can be shown that one representation (or the Jacobian of the Lott reduction) can be expressed in terms of another (obtained by Gawedzki).



Since in gauge theories the dynamics on the space of orbits of the gauge group is described by dependent variables (gauge fields with additional restrictions, or ``gauges''), and local evolution\footnote{Such a quantum evolution, described in terms of dependent variables, can be regarded as global if we are dealing only with a trivial principal bundle.}
occurs on the gauge surface, this was the reason for using dependent coordinates in path integrals
when considering reduction in finite-dimensional mechanical systems with symmetry. It is from this point of view that in our last work \cite{Storchak_2021} we studied an interacting mechanical system consisting of two interacting scalar particles moving on a product manifold (a smooth compact finite-dimensional Riemannian manifold and a vector space) on which a free isometric action of a compact semisimple Lie group is given.

The purpose of the present paper is to find a geometric representation of the reduction Jacobian obtained there for such an interacting system. To do this, we calculate the scalar curvature of the original manifold, which is the total space of the principal bundle associated with the system under consideration. In the case when dependent coordinates are used as local coordinates on the total space of this bundle (together with coordinates defined on the vector space, as well as group coordinates), the calculation is carried out in a special nonholonomic coordinate basis.

The work is organized as follows. Section 2 introduces the necessary definitions and briefly describes the result obtained in our previous paper. Section 3 with subsections is devoted to calculating the scalar curvature of the original manifold, where first the Christoffel symbols are calculated, then the components of the Ricci tensor, and finally the scalar curvature. In Section 4, we deal with the geometric representation of the reduction Jacobian. Appendix A contains definitions of projectors with their properties, Killing relations for the horizontal metric, and some identities that are used in the main text. Appendix B gives a representation of the scalar curvature terms. Appendix C contains a derivation of the main identity, which simplifies the calculations.

\section{Definitions}
An original diffusion of two scalar particles on a smooth compact Riemannian manifold $ \tilde {\mathcal P} = \mathcal P \times \mathcal V $ is described by the backward Kolmogorov equation 
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle
\left(
\frac \partial {\partial t_a}+\frac 12\mu ^2\kappa \bigl[\triangle
_{\cal P}(p_a)+\triangle
_{\cal V}(v_a)\bigr]+\frac
1{\mu ^2\kappa m}V(p_a,v_a)\right){\psi}_{t_b} (p_a,t_a)=0,\\
{\psi}_{t_b} (p_b,v_b,t_b)=\phi _0(p_b,v_b),
\qquad\qquad\qquad\qquad\qquad (t_{b}>t_{a}),
\end{array}\right.
\label{1}
\end{equation}
in which $\mu ^2=\frac \hbar m$ , $\kappa $  is a real positive
parameter,  $V(p,f)$ is the group-invariant potential term:
 $V(pg,g^{-1}v)=V(p,v)$, $g\in \mathcal G$, 
$\triangle _{\cal P}$ is the Laplace--Beltrami operator on the Riemannian
manifold $\cal P$, the scalar Laplacian $\triangle
_{\cal V}$ acts on the space of functions defined on  the vector space $\cal V$. In a chart $(U_{\cal P}\times U_{\cal V},\varphi )$, $\varphi=(\varphi^A,\varphi^a)$, from the atlas of the manifold $\tilde{\mathcal P}$, where the point $(p,v)$ has the  coordinates $(Q^A,f^a)$, $\triangle _{\cal P}$ is represented as\footnote{In  our formulas  we  assume that there is 
sum over the repeated indices. The indices denoted by the capital
letters  run from 1 to $n_{\cal P}=\rm{dim} \cal P$, and the indices of small Latin letters, except $i,j,k,l$, run from 1 to $n^{\cal V}=\dim \cal V$.}:
\begin{equation}
\triangle _{\cal P}(Q)=G^{-1/2}(Q)\frac \partial {\partial
Q^A}G^{AB}(Q)
G^{1/2}(Q)\frac\partial {\partial Q^B},
\label{2}
\end{equation}
where $G^{AB}(Q)$ is an inverse metric tensor of the
 Riemannian
metric on $\cal P$ in coordinate basis  $\{\frac{\partial}{\partial
Q^A}\}$, $G=det (G_{AB})$.

The coordinate expression of  the operator $\triangle_{\cal V}$ is  
\[
 \triangle_{\cal V}(f)=G^{ab}\frac{\partial}{\partial f^a\partial f^b},
\]
where  $G^{ab}$ is an  inverse metric tensor of the metric on $\mathcal V$ with the metric tensor $(G_{ab})$
 in the coordinate basis $\{\frac{\partial}{\partial f^a}\}$.
We assume that  the matrix $(G_{ab})$ consists of fixed constant elements. 

The Schr\"odinger equation is obtained  from  the forward Kolmogorov equation associated with the equation (\ref{1}) in which  one must set $\kappa =i$.


With the definition of the path integral from \cite{Dalecky_1}, the solution of the equation (\ref{1})
 can be written as follows
\begin{eqnarray}
{\psi}_{t_b} (p_a,v_a,t_a)&=&{\rm E}\Bigl[\phi _0(\eta_1 (t_b),\eta_2(t_b))\exp \{\frac
1{\mu
^2\kappa m}\int_{t_a}^{t_b}V(\eta_1(u),\eta_2(u))du\}\Bigr]\nonumber\\
&=&\int_{\Omega _{-}}d\mu ^\eta (\omega )\phi _0(\eta (t_b))\exp
\{\ldots 
\},
\label{orig_path_int}
\end{eqnarray}
where ${\eta}(t)=(\eta_1(t),\eta_2(t))$ is a global stochastic process on a manifold 
$\tilde{\cal P}=\cal P\times \cal V$, 
the measure  ${\mu}^{\eta}$ in 
the path space $\Omega _{-}=\{\omega (t)=(\omega^1(t),\omega^2(t)): \omega^{1,2} (t_a)=0,
\eta_1
(t)=p_a+\omega^1 (t), \eta_2(t)=v_a+\omega^2(t)\}$ is determined by the probability distribution of the stochastic  process $\eta(t)$.

 (\ref{orig_path_int}) is  a symbolical notation of the global semigroup defined in \cite{Dalecky_1, Dalecky_2} as a limit of the superposition of the local semigroups 
\begin{equation}
\!\psi _{t_b}(p_a,v_a,t_a)=U(t_a,t_b)\phi _0(p_a,v_a)=
{\lim}_q {\tilde U}_{\eta}(t_a,t_1)\cdot\ldots\cdot
{\tilde U}_{\eta}(t_{n-1},t_b)
\phi _0(p_a,v_a),
\label{6}
\end{equation}
where  each 
${\tilde U}_{\eta}$ is given by
\begin{equation}
 {\tilde U}_{\eta}(s,t) \phi (p',v')={\rm E}_{s,p',v'}\phi (\eta_1
 (t),\eta_2(t))\,\,\,\,\,\,
 s\leq t\,\,\,\,\,\,\eta_1 (s)=p',\;\eta_2 (s)=v'.
 \label{local_semigroup}
 \end{equation}
These local semigroups are also given by the corresponding path integrals.
 The local processes     $\{\eta_1^A(t),\eta_2^a(t)\}$  are  solutions of two  stochasic differential equations in which the stochastic differentials are taken in the sence of It\^o:
\begin{equation}
d\eta_1^A(t)=\frac12\mu ^2\kappa G^{-1/2}\frac \partial {\partial
Q^B}(G^{1/2}G^{AB})dt+\mu \sqrt{\kappa }{\mathscr X}_{\bar{M}}^A(\eta_1
(t))dw^{
\bar{M}}(t),\\
\label{eta_1}
\end{equation}
and
\begin{equation}
 d\eta_2^a(t)=\mu \sqrt{\kappa }{\mathscr X}_{\bar{b}}^a
dw^{
\bar{b}}(t),\\
\label{eta_2}
\end{equation}
where  ${\mathscr X}_{\bar{M}}^A$ and ${\mathscr X}_{\bar{b}}^a$ are  defined  by the local equalities
$\sum^{n_{\mathcal P}}_{\bar{\scriptscriptstyle K}\scriptscriptstyle =1}{\mathscr X
}_{\bar{K}}^A{\mathscr X}_{\bar{K}}^B=G^{AB}$ and  $\sum^{n_{\mathcal V}}_{\bar{\scriptscriptstyle b}\scriptscriptstyle =1}{\mathscr X
}_{\bar{b}}^a{\mathscr X}_{\bar{b}}^c=G^{ac}$, $dw^{\bar{M}}(t)$ and $dw^{\bar{b}}(t)$ are the independent Wiener processes.
(We  denote the Euclidean indices by over-barred indices.) 

 
The left-hand side of (\ref{orig_path_int}) can also be  represented as follows
\[
\psi_{t_b} (p_a,v_a,t_a)=\int G_{\tilde{\cal P}}(p_b,v_b,t_b;p_a,v_a,t_a)\phi _0(p_b,v_b)dv_{\tilde{\cal P}}(p_b,v_b),
\]
where $G_{\tilde{\cal P}}$ is the  Green's function -- the kernel of the  semigroup $U(t_a,t_b)$,
$dv_{\tilde{\cal P}}(p,v)$ is a volume element on the  manifold $\tilde{\cal P}$.
In the path integral for the Green's function $ G_{\tilde{\cal P}}$, the integration is carried out in the space of paths with fixed both ends.



The configuration space of our original mechanical system is the Riemannian manifold $ \tilde{\mathcal P} = \mathcal P \times \mathcal V $ endowed with a smooth isometric free and proper action of the compact semisimple Lie group $\mathcal G$. 

 The  manifold $\tilde{\mathcal P}$  is equipped with the  Riemannian metric which in coordinate basis $(\frac{\partial}{\partial Q^A},\frac{\partial}{\partial f^a}$)) is given by
\begin{equation}
 ds^2=G_{AB}(Q)dQ^AdQ^B+G_{ab}\,df^adf^b.
\label{metr_orig}
\end{equation}
This metric is invariant under the right action of the group  which acts as
 \[
 {\tilde Q}^A=F^A(Q,g),\;\;\;\;{\tilde f}^b=\bar D^b_a(g)f^a,
\]
where $\bar D^b_a(g)\equiv D^b_a(g^{-1})$,
 ($D^b_a(g)$ is the matrix of  the finite-dimensional representation of the group $\mathcal G$
acting in the vector space $\mathcal V$). 


The  action of the group $ \mathcal G $ on the manifold $ \tilde {\mathcal P} $ leads to appearance of the principal fiber bundle $ \pi ': \mathcal P \times \mathcal V \to \mathcal P \times_{\mathcal G} \mathcal V $, and ${\tilde{\mathcal P}}$ can be viewed as the total space of this bundle. From this it follows that  the local coordinates $(Q^A, f^a)$ of the point $(p,v)\in \tilde{\mathcal P}$ can be expressed in terms of the  coordinates defined in the principal fiber bundle.





In gauge field theories, the calculations are performed in a special local coordinates known as an adapted coordinates.
They are formed from two groups of coordinates. Some are connected with the base of the principal bundle, while others are connected with the group manifold. To determine them, it is necessary to choose local submanifolds (local surfaces) of the total space of the principal bundle that intersect transversally with each of the orbits.





In our case, such local surfaces $\tilde{\Sigma}_i$ in the principal bundle $\rm P(\tilde {\mathcal M}, \mathcal G)$ are determined by local surfaces $\Sigma_i$ chosen in the total space $\mathcal P$ of the principal fiber bundle $\rm P(\mathcal M, \mathcal G)$. For this, the local submanifold $\Sigma$ in $\mathcal P$ is given by the system of equations
$\chi^{\alpha}(Q)=0,\,\alpha=1,...,n^{\mathcal G}$, which means that the point with coordinates $Q^{\ast A}$ $\{ \chi^{\alpha}(Q^{\ast A})=0\}$ belongs to $\Sigma $. For this reason, the coordinates $Q^{\ast A}$ are called dependent coordinates.

The group coordinates $a^{\alpha}(Q)$ of a point $p\in \mathcal P$ are defined by the solution of the following equation:
\[
 \chi^{\alpha}(F^A(Q,a^{-1}(Q)))=0.
\]
It means that
\[
 Q^{\ast A}=F^A(Q,a^{-1}(Q)).
\]
That is, the group element  $a^{-1}(Q)$ carries the point $p$ to the submanifold  $\Sigma$. 
The coordinates of a point $(p,v)\in \tilde{\Sigma}$ are $(Q^{\ast A},\tilde f^a)$.
Therefore, the transition to adapted coordinates is given by the mapping
\[
 \tilde \varphi_i^{-1} :(Q^A,f^b)\to (Q^{\ast}{}^A(Q),\tilde f^b(Q),a^{\alpha}(Q)\,),
\]
where
\[
\tilde f^b(Q) = D^b_c(a(Q))\,f^c,
\]
($\bar D^b_c(a^{-1})\equiv D^b_c(a))$.

Replacement of the  $(Q^A,f^a)$ of a point $(p,v)\in \mathcal P\times V$  with adapted coordinates $(Q^{\ast}{}^B, \tilde f^b, a^{\alpha})$, 
\begin{equation}
Q^A=F^A(Q^{\ast}{}^B,a^{\alpha}),\;\;\;f^a=\bar D^a_b(a)\tilde f^b,
\label{transf_coord}
\end{equation}
 leads to the following transformations of the local coordinate vector fields: 
\begin{eqnarray}
\displaystyle 
&&\!\!\!\!\!\!\!\!\frac{\partial}{\partial f^a}=D^b_a(a)\frac{\partial}{\partial {\tilde f}^b},
\nonumber\\
&&\!\!\!\!\!\!\!\!\frac{\partial}{\partial Q^B}=\frac{\partial Q^{\ast}{}^A}{ \partial Q^B}\frac{\partial}{\partial Q^{\ast}{}^A}+\frac{\partial a^{\alpha}}{\partial Q^B}\frac{\partial}{\partial a^{\alpha}}+\frac{\partial {\tilde f}^b}{\partial Q^B}\frac{\partial}{\partial {\tilde f}^b}
\nonumber\\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!=\check F^C_B\Biggl(N^A_C(Q^{\ast})\frac{\partial}{\partial Q^{\ast}{}^A}+{\chi}^{\mu}_C({\Phi}^{-1})^{\beta}_{\mu}\bar{v}^{\alpha}_{\beta}(a)\frac{\partial}{\partial a^{\alpha}}-{\chi}^{\mu}_C({\Phi}^{-1})^{\nu}_{\mu}(\bar J_{\nu})^b_p\tilde f^p\frac{\partial}{\partial {\tilde f}^b}\Biggr)\!.
\label{vectfield}
\end{eqnarray}
Here $\check F^C_B\equiv F^C_B(F(Q^{\ast},a),a^{-1})$ is an inverse matrix to the matrix $F^A_B(Q^{\ast},a)$,
${\chi}^{\mu}_C\equiv \frac{\partial {\chi}^{\mu}(Q)}{\partial Q^C}|_{Q=Q^{\ast}}$, $({\Phi}^{-1})^{\beta}_{\mu}\equiv({\Phi}^{-1})^{\beta}_{\mu}(Q^{\ast})$ -- the matrix which is inverse to the Faddeev--Popov matrix:
\[
 ({\Phi})^{\beta}_{\mu}(Q)=K^A_{\mu}(Q)\frac{\partial {\chi}^{\beta}(Q)}{\partial Q^A},
\]
the matrix $\bar{v}^{\alpha}_{\beta}(a)$ is inverse of the matrix $\bar{u}^{\alpha}_{\beta}(a)$.\footnote{$\det \bar{u}^{\alpha}_{\beta}(a)$ is   the density of the right-invariant measure  given on the group $\mathcal G$.}


The operator $N^A_C$, defined  as
 $N^A_C(Q)=\delta^A_C-K^A_{\alpha}(Q)({\Phi}^{-1})^{\alpha}_{\mu}(Q){\chi}^{\mu}_C(Q)$, 
 is the projection operator ($N^A_BN^B_C=N^A_C$) onto the subspace which is orthogonal to the Killing vector fields $ K^A_{\alpha}(Q)\frac{\partial}{\partial Q^A}$ of the manifold ${\cal P}$. $N^A_C(Q^{\ast})$ is the restriction of $N^A_C(Q)$ to the submanifold $ \Sigma $:
\[
 N^A_C(Q^{\ast})\equiv N^A_C(F(Q^{\ast},e))\;\;\;N^A_C(Q^{\ast})=F^B_C(Q^{\ast},a)N^M_B(F(Q^{\ast},a))\check F_M^A(Q^{\ast},a),
\]
where $e$ is the unity element of the group. 



In the new coordinate basis $\displaystyle(\partial/\partial Q{}^{\ast A},\partial/\partial \tilde f^m,\partial/\partial a^{\alpha})$
the metric (\ref{metr_orig}) of the original manifold $\mathcal P \times \mathcal V$ is written as follows
\begin{equation}
\displaystyle
{\tilde G}_{\cal A\cal B}(Q{}^{\ast},\tilde f,a)=
\left(
\begin{array}{ccc}
 G_{CD}(P_{\bot})^C_A (P_{\bot})^D_B & 0 & G_{CD}(P_{\bot})^C_AK^D_{\nu}\bar u^{\nu}_{\alpha}\\
 0 & G_{ab} & G_{ap}K^p_{\nu}\bar u^{\nu}_{\alpha}\\
G_{BC}K^C_{\mu}\bar u^{\mu}_{\beta} & G_{bp}K^p_{\nu}\bar u^{\nu}_{\beta} & d_{\mu\nu}\bar u^{\mu}_{\alpha}\bar u^{\nu}_{\beta}\\
\end{array}
\right)
\label{metric2c}
\end{equation}
where $G_{CD}(Q{}^{\ast})\equiv G_{CD}(F(Q{}^{\ast},e))$:
\[
 G_{CD}(Q{}^{\ast})=F^M_C(Q{}^{\ast},a)F^N_D(Q{}^{\ast},a)G_{MN}(F(Q{}^{\ast},a)), 
\]
the projection operator $P_\bot$ and the components $K^A_{\mu}$ of the Killing vector fields  depend on $Q{}^{\ast}$, $\bar u^{\mu}_{\beta}=\bar u^{\mu}_{\beta}(a)$,  $K^p_{\nu}=K^p_{\nu}(\tilde f)$,  $d_{\mu\nu}(Q{}^{\ast},\tilde f)\bar u^{\mu}_{\alpha}(a)\bar u^{\nu}_{\beta}(a)$ is 
the metric on  $\mathcal G$--orbit through the point $(p,v)$. The components $d_{\mu\nu}$ of this metric are  given by
\begin{eqnarray*}
 d_{\mu\nu}(Q{}^{\ast},\tilde f)&=&K^A_{\mu}(Q{}^{\ast})G_{AB}(Q{}^{\ast})K^B_{\nu}(Q{}^{\ast})+K^a_{\mu}(\tilde f)G_{ab}K^b_{\nu}(\tilde f)
\nonumber\\
&\equiv&\gamma_{\mu \nu}(Q^{\ast})+\gamma'_{\mu \nu}(\tilde f).
\end{eqnarray*}



The pseudoinverse matrix ${\tilde G}^{\cal A\cal B}(Q{}^{\ast},\tilde f,a)$ to  the matrix (\ref{metric2c}) is given by the following expression:
\begin{equation}
\displaystyle
\left(
\begin{array}{ccc}
{G}^{EF}N_E^AN_F^B & -G^{EF}N^A_E{\Lambda}^{\nu}_FK^a_{\nu} &  G^{EF}N^A_E{\Lambda}^{\beta}_F\bar v^{\alpha}_{\beta}\\
-G^{EF}N^B_F{\Lambda}^{\nu}_EK^b_{\nu} & G^{ba}+G^{EF}{\Lambda}^{\nu}_E{\Lambda}^{\mu}_FK^b_{\nu}K^a_{\mu} & -G^{EF}{\Lambda}^{\nu}_E{\Lambda}^{\mu}_FK^b_{\nu}{\bar v}^{\alpha}_{\mu}\\
G^{EF}N^B_F{\Lambda}^{\mu}_E\bar v_{\mu}^{\beta} & -G^{EF}{\Lambda}^{\nu}_E{\Lambda}^{\mu}_FK^a_{\mu}{\bar v}^{\beta}_{\nu} &G^{EF}{\Lambda}^{\nu}_E{\Lambda}^{\mu}_F{\bar v}^{\alpha}_{\nu}{\bar v}^{\beta}_{\mu}  \\
\end{array}
\right).
\label{metric2b}
\end{equation}
Here  ${\Lambda}^{\nu}_E\equiv({\Phi}^{-1})^{\nu}_{\mu}(Q{}^{\ast}){\chi}^{\mu}_E(Q{}^{\ast})$.

The pseudoinversion of ${\tilde G}_{\cal A\cal B}$ means that
\[
\displaystyle
{\tilde G}^{\tilde{\cal A}\tilde{\cal D}}{\tilde G}_{\tilde{\cal D}\tilde{\cal B}}=
\left(
\begin{array}{ccc}
  (P_{\bot})^A_B & 0 & 0\\
 0 & {\delta}^a_b & 0\\
0 & 0 & {\delta}^{\alpha}_{\beta}\\
\end{array}
\right).
\]
  
The determinant of the matrix  (\ref{metric2c}) is defined by the following expression:
$\det {\tilde  G}_{{\cal A\cal B}}=(\det d_{\mu\nu})\,(\det {\bar u}^{\mu}_{\nu})^2\,H$ with
\begin{eqnarray}
 H= \displaystyle
\det \left(
\begin{array}{cc}
(P_{\bot})^{A'}_A\tilde G^{\rm H}_{A'B'}(P_{\bot})^{B'}_B & (P_{\bot})^{A'}_A\tilde G^{\rm H}_{A' a}\\
(P_{\bot})^{B'}_B\tilde G^{\rm H}_{b B'} & \tilde G^{\rm H}_{ab}\\
\end{array}
\right),
\label{det}
\end{eqnarray}
where
${\tilde G}^{\rm H}_{AB}=G_{AB}-G_{AC}K^C_{\mu}d^{\mu\nu}K^D_{\nu}G_{DB},$
$\tilde G^{\rm H}_{Aa}=-G_{AB}K^B_{\mu}d^{\mu\nu}K^b_{\nu}G_{ba}$, $\tilde G^{\rm H}_{ba}=G_{ba}-G_{bc}K^c_{\mu}d^{\mu\nu}K_{\nu}^pG_{pa}$.

Note that $\det {\tilde  G}_{{\cal A\cal B}}$ does not vanish only on the surface $\tilde \Sigma $. On this surface $\det(P_{\bot})^{A}_B$ is equal to unity. And in what follows, we denote by $d\equiv\det d_{\mu\nu}$.

The reduction of the path integral consists of several transformations of the original path integral. The first transformation of the path integral is connected with the introduction of new integration variables. In our case, these are variables locally given by adapted coordinates. The second transformation leads to factorization of the measure of the path integral, which makes it possible to separate the group variables from the variables describing the motion on the orbit space. This transformation is based on the use of the nonlinear filtering equation from the theory of the stochastic processes. And also we use the Girsanov transformation, which makes it possible to avoid the appearance of stochastic integrals in the exponent of the resulting Jacobian.
  
  As a result of all these transformations, we have obtained in \cite{Storchak_2021} an integral relation between the path integral given on $\tilde{\Sigma}$ and the path integral on $\mathcal P$, which are used to represent the corresponding Green's functions. For   reduction to the level of zero momentum this integral relation is
\begin{eqnarray*}
 &&d_b^{-1/4}d_a^{-1/4}G_{\tilde{\Sigma}}(Q^{\ast}_b,\tilde f_b,t_b;
 Q^{\ast}_a,\tilde f_a,t_a)=
\displaystyle\int _{\cal G}G_{\tilde{\cal P}}(p_b\theta,v_b\theta,t_b;
p_a,v_a,t_a) d\mu(\theta) , 
\end{eqnarray*}
where $(Q^{\ast}{},\tilde f)={\pi'}_{\tilde{\Sigma}}(p,v)$, $d_b=d(Q^{\ast}_b,\tilde f_b)$ and $d_a=d(Q^{\ast}_a,\tilde f_a)$.

The Green's function $G_{\tilde{\mathcal P}}(Q_b,f_b,t_b;Q_a,f_a,t_a)$ representing the kernal of the evolution semigroup ({\ref{orig_path_int}) acts in the Hilbert space of  functions with the scalar product $(\psi_1,\psi_2)=\int \psi_1(Q,f)\psi_2(Q,f)dv_{\tilde{\mathcal P}}$, where $dv_{\tilde{\mathcal P}}(Q,f)$ is a volume measure on $\tilde{\mathcal P}$.

The global semigroup  determined by the Green's function $G_{\tilde{\Sigma}}$ acts in the Hilbert  space of the scalar functions on $\tilde{\Sigma}$ with the following scalar product $(\psi_1,\psi_2)=\int \psi_1(Q^{\ast},\tilde f) \psi_2(Q^{\ast},\tilde f)dv_{\tilde{\Sigma}}.$ This 
 Green's function $G_{\tilde{\Sigma}}$ is presented by the following path integral:  
\begin{eqnarray*}
 &&G_{\tilde{\Sigma}}(Q^{\ast}_b,\tilde f_b,t_b;
 Q^{\ast}_a,\tilde f_a,t_a)\\
 &&=\int\limits_{{\tilde{\xi}_{\tilde \Sigma} (t_a)=\pi'_{\tilde\Sigma} (p_a,v_a)}\atop  
{\tilde{\xi}_{\tilde \Sigma} (t_b)=\pi'_{\tilde\Sigma} (p_b,v_b)}} d{\mu}^{\tilde{\xi}_{\tilde \Sigma}}
\exp \Bigl\{\frac 1{\mu ^2\kappa
m}\int_{t_a}^{t_b}\Bigl[\tilde{V}(\tilde{\xi}_{\tilde \Sigma}(u))+J(\sigma (\tilde{\xi}_{\tilde \Sigma}(u)))\Bigr]du\Bigr\}.
\nonumber\\
\end{eqnarray*}
The  measure in this path integral is generated by the stochastic process $\tilde{\xi}_{\tilde \Sigma}$
determined by the stochastic differential equations
\begin{eqnarray*}
 &&dQ^{\ast}{}^A(t)=\mu^2\kappa\Bigl(-\frac12h^{\tilde B\tilde M}\,{}^{ \mathrm  H}{\tilde \Gamma}^{ A}_{\tilde B\tilde M}+j^{ A}_{\Roman 1}\Bigr)dt


 +\mu\sqrt{\kappa}N^A_C\mathscr X^C_{\bar M}dw^{\bar M}_t,
 \label{sde_Q_ast_j}\,\,\rm{and}
\end{eqnarray*}
 \begin{eqnarray*}
  &&d\tilde f^a(t)=\mu^2\kappa\Bigl(-\frac12h^{\tilde B\tilde M}\,{}^{ \mathrm  H}{\tilde \Gamma}^{ a}_{\tilde B\tilde M}+j^{ a}_{\Roman 1}\Bigr)dt
 
 
  +\mu\sqrt{\kappa}\bigl(N^a_C\mathscr X^C_{\bar M}dw^{\bar M}_t+\mathscr X^a_{\bar b}dw^{\bar b}_t\bigr).
  \label{sde_f_j} 
 \end{eqnarray*}
where $ h^{\tilde B\tilde M}\equiv G^{EF}N^{\tilde B}_EN^{\tilde M}_F$ and $j^{ A}_{\Roman 1}$ and $j^{ a}_{\Roman 1}$ are the components of the mean curvature normal of the orbit space. They are given by the following expressions:
\begin{align*}
 j^{ A}_{\Roman 1}&=\frac12 h^{BM}N^A_{B,M}+\frac12h^{\tilde B\tilde M}\,{}^{ \mathrm  H}{\tilde \Gamma}^{ A}_{\tilde B\tilde M}-\frac12h^{\tilde B\tilde M}N^A_C\,{}^{ \mathrm  H}{\tilde \Gamma}^{ C}_{\tilde B\tilde M},
\\
j^{ a}_{\Roman 1}&=-\frac12N^a_Ch^{BM}N^C_{B,M}-\frac12N^a_Ch^{\tilde B\tilde M}\,{}^{ \mathrm  H}{\tilde \Gamma}^{ C}_{\tilde B\tilde M}.
\end{align*}


The reduction Jacobian $J$ is give by
\begin{equation}
J=-\frac18\mu^2\kappa\bigl(\tilde\triangle_{\tilde{\Sigma}}\sigma +\frac14<\partial\sigma,\partial \sigma>_{\tilde{\Sigma}}\bigr),
\label{jacobian_reduct}
\end{equation}
where $$\tilde\triangle_{\tilde{\Sigma}}\sigma= \triangle_{\tilde{\Sigma}}^{\scriptscriptstyle \rm H}\sigma+2j^{ A}_{\Roman 1}\sigma_A+2j^{ a}_{\Roman 1}\sigma_a,$$ $\triangle_{\tilde{\Sigma}}^{\scriptscriptstyle \rm H}$ is the Laplacian on $\tilde{\Sigma}$, and 
$<\partial\sigma,\partial \sigma>_{\tilde{\Sigma}}$ is a quadratic form in first partial derivatives of $\sigma\equiv \ln d$ obtained by using the horizontal metric on $\tilde{ \Sigma}$.




The Green's function $G_{\tilde{\Sigma}}$ satisfies the forward Kolmogorov equation with the operator
\[
\hat{H}_{\kappa}=
\frac{\hbar \kappa}{2m}\tilde\triangle _{\tilde{\Sigma}}-\frac{\hbar \kappa}{8m}\Bigl[\tilde\triangle_{\tilde{\Sigma}}\sigma +\frac14<\partial\sigma,\partial \sigma>_{\tilde{\Sigma}}\Bigr]+\frac{1}{\hbar \kappa}\tilde V.
\]
Note that at $\kappa =i$ the forward Kolmogorov equation becomes
the Schr\"odinger equation with the Hamilton operator 
$\hat H=-\frac{\hbar}{\kappa}{\hat H}_{\kappa}\bigl|_{\kappa =i}$.






\section{The scalar curvature  of the manifold  $\tilde {\mathcal P}$}
The scalar curvature of $\tilde {\mathcal P}$ will be calculated by using the special nonholonomic  basis, also known  as the horizontal lift basis. 
This basis consists of the horizontal vector fields $H_A$  and $H_p$ together with the left-invariant vector fields $L_{\alpha}$.
 In terms of the principal fiber bundle coordinates, the horizontal vector fields are 
\begin{eqnarray*}
 &&H_A(Q^{\ast},\tilde f,a)= \Bigl[N^T_A\Bigl(\frac{\partial}{\partial Q^{\ast T}}
-\tilde {\mathscr A}^{\alpha }_T L_{\alpha}\Bigl)+N^p_A\Bigl(\frac{\partial}{\partial {\tilde f}^p}-\tilde {\mathscr A}^{\alpha }_pL_{\alpha}\Bigl)\Bigr],\\
&&H_p(Q^{\ast},\tilde f,a)=\Bigl( \frac{\partial}{\partial {\tilde f}^p}-\tilde {\mathscr A}^{\alpha }_pL_{\alpha}\Bigl).\\
\end{eqnarray*}
These vector fields are defined with the help of the  components of the projection operator $N^{\tilde A}_{\tilde C}$ :
$N^{\tilde A}_{\tilde C}=(N^A_C,N^A_p,N^q_A,N^q_p).$ 

The commutation relations of the vector fields have the following form:
\begin{eqnarray*}
 &&[H_A,H_B]={\mathbb C}^T_{AB}\,H_T+{\mathbb C}^p_{AB}\,H_p+{\mathbb C}^{\alpha}_{AB}L_{\alpha}, \\
 &&[H_A,H_p]={\mathbb C}^q_{Ap}\,H_q+{\mathbb C}^{\alpha}_{Ap}L_{\alpha},\\
 &&[H_p,H_q]={\mathbb C}^{\alpha}_{pq}L_{\alpha},\\
&&[H_A,L_{\alpha}]=0,\;\;\;[H_p,L_{\alpha}]=0,\;\;\;[L_{\alpha},L_{\beta}]=c^{\gamma}_{\alpha \beta}L_{\gamma},\\
 \end{eqnarray*}
where the structure constants of the nonholonimic basis  are 
\begin{eqnarray*}
&&{\mathbb C}^T_{AB}=({\Lambda}^{\gamma}_A N^R_B-{\Lambda}^{\gamma}_BN^R_A) K^{T}_{{\gamma}, R},\\
&&{\mathbb C}^p_{AB}=-N^D_AN^R_B({\Lambda}^{\alpha}_{R,D}-{\Lambda}^{\alpha}_{D,R}) K^p_{\alpha}  \;\;  -c^{\sigma}_{\alpha \beta}{\Lambda}^{\beta}_A{\Lambda}^{\alpha}_BK^p_{\sigma}, \\
&&{\mathbb C}^{\alpha}_{AB}=-N^S_AN^P_B\,\tilde{\mathscr F}^{\alpha}_{SP}-(N^E_AN^p_B-N^E_BN^p_A)\tilde{\mathscr F}^{\alpha}_{Ep}+N^m_AN^p_B\tilde{\mathscr F}^{\alpha}_{pm},\\
&&{\mathbb C}^q_{Ap}=({\bar J}_{\alpha})^q_p{\Lambda}^{\alpha}_A=K^q_{\alpha,p}{\Lambda}^{\alpha}_A,\;\;\;
{\mathbb C}^{\alpha}_{Ap}=-N^E_A\tilde{\mathscr F}^{\alpha}_{Ep}-N^q_A\tilde{\mathscr F}^{\alpha}_{qp},\;\;\;{\mathbb C}^{\alpha}_{pq}=-\tilde{\mathscr F}^{\alpha}_{pq}\,.\\
\end{eqnarray*}
The curvature tensor $\tilde{\mathscr F}^{\alpha}_{SP}$ of the connection ${\tilde{\mathscr A}^{\alpha}_P}$ is given by
\[
\tilde{\mathscr F}^{\alpha}_{SP}=\displaystyle\frac{\partial}{\partial Q^{\ast}{}^S}\,\tilde{\mathscr A}^{\alpha}_P- 
\frac{\partial}{\partial {Q^{\ast}}^P}\,\tilde{\mathscr A}^{\alpha}_S
+c^{\alpha}_{\nu\sigma}\, \tilde{\mathscr A}^{\nu}_S\,
\tilde{\mathscr A}^{\sigma}_P,\;\;(\tilde{\mathscr F}^{\alpha}_{SP}({Q^{\ast}},a)={\bar{\rho}}^{\alpha}_{\mu}(a)\,{\mathscr F}^{\mu}_{SP}(Q^{\ast})).  
\]
  The tensors $\tilde{\mathscr F}^{\alpha}_{Ep}$ and $\tilde{\mathscr F}^{\alpha}_{pm}$ are defined in a similar way. Note  that ${\mathbb C}^q_{Ap}=-{\mathbb C}^q_{pA}$. It can also be shown that the first term on the right in ${\mathbb C}^p_{AB}$ is zero.


In our nonholonomic basis $(H_A,H_p,L_{\alpha})$,  the original metric (\ref{metric2c}) can be  written as 
\begin{equation}
\displaystyle
{\check G}_{\cal A\cal B}(Q^{\ast},\tilde f,a)=
\left(
\begin{array}{ccc}
{\tilde G}^{\rm H}_{AB} & {\tilde G}^{\rm H}_{Am} & 0\\
{\tilde G}^{\rm H}_{nB} & {\tilde G}^{\rm H}_{nm} & 0\\
0 & 0 &\tilde{d}_{\alpha \beta }  \\
\end{array}
\right)=\left( \begin{array}{cc}
{\tilde G}^{\rm H}_{\tilde A \tilde B}  & 0 \\
0 & \tilde{d}_{\alpha \beta }  \\
\end{array}
\right),
\label{metric2cc}
\end{equation}
with
\[
 {\check G}(H_A,H_B)\equiv{\tilde G}^{\rm H}_{AB},\;{\check G}(H_A,H_b)\equiv{\tilde G}^{\rm H}_{Ab},\;{\check G}(L_{\alpha},L_{\beta})\equiv\tilde{d}_{\alpha \beta }=\rho^{\alpha'}_{\alpha}\rho^{\beta'}_{\beta}d_{\alpha' \beta' }
\]

The metric tensor ${\tilde G}^{\rm H}_{\tilde{ A} \tilde{ B}}(Q^{\ast}{}^A,\tilde f ^m)$ of the ``horizontal metric'' has the following components:
\begin{eqnarray*}
&&{\tilde G}^{\rm H}_{AB}={{\tilde \Pi}}^{\tilde A}_{A}\,{{\tilde \Pi}}^{\tilde B}_B \,G_{\tilde A\tilde B}=G_{AB}-G_{AD}K^{D}_{\alpha}d^{\alpha \beta}K^R_{\beta}\,G_{RB},\\
&&{\tilde G}^{\rm H}_{Am}=-G_{AB}K^{B}_{\alpha}\,d^{\alpha \beta}K^p_{\beta}G_{pm},\;\;\;{\tilde G}^{\rm H}_{Am}={\tilde G}^{\rm H}_{mA},\\
&&{\tilde G}^{\rm H}_{mn}={{\tilde \Pi}}^r_mG_{rn}={{\tilde \Pi}}^{\tilde C}_{m}\,{{\tilde \Pi}}^{\tilde D}_n \,G_{\tilde C\tilde D}=G_{mn}-G_{mr}K^r_{\alpha}d^{\alpha \beta}K_{\beta}^pG_{pn}.\\
\end {eqnarray*}
 Note that metric ${\tilde G}^{\rm H}$  is given on the local surface $\tilde \Sigma$ and gives rise the metric on the orbit space $\mathcal P\times _\mathcal G V$ provided that the submanifold  $\tilde \Sigma$ is given parametrically.


The pseudoinverse matrix ${\check G}^{\cal A\cal B}$ to the matrix (\ref{metric2cc}) is represented as
\begin{equation*}
\displaystyle
{\check G}^{\cal A\cal B}=
\left(
\begin{array}{ccc}
{G}^{EF}N^A_EN^B_F & {G}^{EF}N^A_EN^q_F & 0\\
{G}^{EF}N^p_F N^B_E & {G}^{pq}+G^{AB}N^p_AN^q_B & 0\\
0 & 0 &\tilde{d}^{\alpha \beta }  \\
\end{array}
\right)\equiv
\left(
\begin{array}{ccc}
 h^{AB} & h^{Aq} & 0\\
 h^{pB} & h^{pq} & 0 \\
 0 & 0 & \tilde{d}^{\alpha \beta }\\
 \end{array}
 \right).
\end{equation*}
This matrix is defined from the following orthogonality condition:
\[
\displaystyle
{\check G}^{\cal A\cal B}{\check G}_{\cal B\cal D}=
\left(
\begin{array}{ccc}
N^A_D& 0 & 0\\
 N^p_D & {\delta}^p_m & 0\\
0 & 0 & {\delta}^{\alpha}_{\beta} \\
\end{array}
\right)=
\left(
\begin{array}{cc}
N^{\tilde A}_{\tilde D} & 0 \\
0 & {\delta}^{\alpha}_{\beta} \\
\end{array}
\right),
\]
where 
\[
\displaystyle
N^{\tilde A}_{\tilde D}=
\left(
\begin{array}{cc}
N^A_D& N^A_m\\
 N^p_D & N^p_m\\
\end{array}
\right)
\]
($N^A_m=0, N^p_m={\delta}^p_m$).


\subsection{The Christoffel symbols}
In the article, the calculation of the Christoffel symbols is carried out according to the following formula:
\begin{eqnarray}
&&2{\check{\rm\Gamma}}_{\mathscr A\mathscr B}^{\mathscr D}\check G(\partial_{\mathscr D},\partial_{\mathscr C})=\partial_{\mathscr A}\check G(\partial_{\mathscr B},\partial_{\mathscr C})+\partial_{\mathscr B}\check G(\partial_{\mathscr A},\partial_{\mathscr C})-\partial_{\mathscr C}\check G(\partial_{\mathscr A},\partial_{\mathscr B})\nonumber\\
&&\;\;\;\;\;-\check G(\partial_{\mathscr A},[\partial_{\mathscr B},\partial_{\mathscr C}])-\check G(\partial_{\mathscr B},[\partial_{\mathscr A},\partial_{\mathscr C}])+\check G(\partial_{\mathscr C},[\partial_{\mathscr A},\partial_{\mathscr B}]),
\label{christoff_main}
\end{eqnarray}
where by indices we mean generalized indices, such as, for example, $\mathscr A=(A,a,\alpha)$, and by $\partial_{\mathscr A}$ we denote one of the vector fields of our coordinate basis.
Summation over repeated indices is assumed.

In the case where \underline{ $\mathscr A=A, \mathscr B=B, \mathscr C=p$ } this formula is written as
\begin{eqnarray*}
 &&2[{\check\Gamma}^D_{AB}{\tilde G}^{\rm H}_{Dp}+{\check\Gamma}^a_{AB}{\tilde G}^{\rm H}_{ap}]=
H_A{\tilde G}^{\rm H}_{Bp}+H_B{\tilde G}^{\rm H}_{Ap}-H_p{\tilde G}^{\rm H}_{AB}\nonumber\\
&&-\check G(H_A,[H_B,H_p])-\check G(H_B,[H_A,H_p])+\check G(H_p,[H_A,H_B]).
\end{eqnarray*}
This can be represented in the following form:
\begin{eqnarray*}
 &&2[{\check\Gamma}^D_{AB}{\tilde G}^{\rm H}_{Dp}+{\check\Gamma}^a_{AB}{\tilde G}^{\rm H}_{ap}]\\
&&\;\;\;=N^E_A{\tilde G}^{\rm H}_{Bp,E}+N^q_A{\tilde G}^{\rm H}_{Bp,q}+N^E_B{\tilde G}^{\rm H}_{Ap,E}+N^q_B{\tilde G}^{\rm H}_{Ap,q}-{\tilde G}^{\rm H}_{AB,p}
\nonumber\\
&&\;\;\;-\mathbb C^q_{Bp}{\tilde G}^{\rm H}_{Aq}-\mathbb C^q_{Ap}{\tilde G}^{\rm H}_{Bq}
+\mathbb C^T_{AB}{\tilde G}^{\rm H}_{pT}+\mathbb C^q_{AB}{\tilde G}^{\rm H}_{pq}
\end{eqnarray*}
or, explicitly, as follows:
\begin{eqnarray*}
 &&2[{\check\Gamma}^D_{AB}{\tilde G}^{\rm H}_{Dp}+{\check\Gamma}^a_{AB}{\tilde G}^{\rm H}_{ap}]=\\
&&N^E_A{\tilde G}^{\rm H}_{Bp,E}+N^q_A{\tilde G}^{\rm H}_{Bp,q}+N^E_B{\tilde G}^{\rm H}_{Ap,E}+N^q_B{\tilde G}^{\rm H}_{Ap,q}-{\tilde G}^{\rm H}_{AB,p}
\nonumber\\
&&-K^q_{\alpha,p}\Lambda^{\alpha}_B{\tilde G}^{\rm H}_{Aq}- K^q_{\alpha,p}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{Bq}
+(\Lambda^{\gamma}_AN^R_B-\Lambda^{\gamma}_BN^R_A)K^T_{\gamma, R}{\tilde G}^{\rm H}_{pT}-c^{\sigma}_{\alpha \beta}\Lambda^{\beta}_A\Lambda^{\alpha}_BK^q_{\sigma}{\tilde G}^{\rm H}_{pq}.
\end{eqnarray*}









In the case where \underline{ $\mathscr A=A, \mathscr B=b, \mathscr C=C$, $(A,b,C)$ } the formula (\ref{christoff_main}) gives us the equation
\begin{eqnarray*}
 &&2[{\check\Gamma}^D_{Ab}{\tilde G}^{\rm H}_{DC}+{\check\Gamma}^a_{Ab}{\tilde G}^{\rm H}_{aC}]=
H_A{\tilde G}^{\rm H}_{bC}+H_B{\tilde G}^{\rm H}_{AC}-H_C{\tilde G}^{\rm H}_{Ab}\nonumber\\
&&-\check G(H_A,[H_b,H_C])-\check G(H_b,[H_A,H_C])+\check G(H_C,[H_A,H_b]).
\end{eqnarray*}

The right-hand side of this equation has the following terms:
\begin{eqnarray*}
 &&
H_A{\tilde G}^{\rm H}_{bC}+H_b{\tilde G}^{\rm H}_{AC}-H_C{\tilde G}^{\rm H}_{Ab}\nonumber\\
&&-\mathbb C^q_{bC}{\tilde G}^{\rm H}_{Aq}-\mathbb C^T_{AC}{\tilde G}^{\rm H}_{bT}-\mathbb C^q_{AC}{\tilde G}^{\rm H}_{qb}+\mathbb C^q_{Ab}{\tilde G}^{\rm H}_{Cq}.
\end{eqnarray*}
They can be rewritten as
\begin{eqnarray*}
 &&N^E_A{\tilde G}^{\rm H}_{bC,E}+N^q_A{\tilde G}^{\rm H}_{bC,q}+{\tilde G}^{\rm H}_{AC,b}-N^E_C{\tilde G}^{\rm H}_{Ab,E}-N^q_C{\tilde G}^{\rm H}_{Ab,q}
\nonumber\\
&&
-(\Lambda^{\gamma}_AN^R_C-\Lambda^{\gamma}_CN^R_A)K^T_{\gamma, R}{\tilde G}^{\rm H}_{bT}+
c^{\sigma}_{\alpha \beta}\Lambda^{\beta}_A\Lambda^{\alpha}_CK^q_{\sigma}{\tilde G}^{\rm H}_{qb}  
+K^q_{\alpha,b}\Lambda^{\alpha}_C{\tilde G}^{\rm H}_{Aq}+ K^q_{\alpha,b}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{Cq}.
\end{eqnarray*}
The terms of this expression can be represented in a different form.
The first term is rewritten as
\[
 N^E_A({\tilde G}^{\rm H}_{bC,E}+{\tilde G}^{\rm H}_{EC,b}-{\tilde G}^{\rm H}_{bE,C})-N^E_A({\tilde G}^{\rm H}_{EC,b}-{\tilde G}^{\rm H}_{bE,C}).
\]
This leads to 
\[
 2N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEC}-{\tilde G}^{\rm H}_{AC,b}+K^E_{\alpha}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{EC,b}+{\tilde G}^{\rm H}_{bA,C}-K^E_{\alpha}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{bE,C}.
\]

The sixth term is rewritten as
\[
 -[(\Lambda^{\gamma}_A(\delta ^R_C-K^R_{\varphi} \Lambda^{\varphi}_C)-(\Lambda^{\gamma}_C(\delta ^R_A-K^R_{\mu} \Lambda^{\mu}_A)]
K^T_{\gamma, R}{\tilde G}^{\rm H}_{bT}, 
\]
where, according to the $\rm C$--identity from Appendix A, one should use
\[
 K^T_{\gamma, R}{\tilde G}^{\rm H}_{bT}=-K^T_{\gamma}{\tilde G}^{\rm H}_{bT,R}-K^q_{\gamma}{\tilde G}^{\rm H}_{bq,R}.
\]
First we consider the transformation of  the $``\Lambda \Lambda$---terms''. They are given by
\[
 \Lambda^{\gamma}_A\Lambda^{\varphi}_CK^R_{\varphi}K^T_{\gamma}{\tilde G}^{\rm H}_{bT,R}+\Lambda^{\gamma}_A\Lambda^{\varphi}_CK^R_{\varphi}K^q_{\gamma}{\tilde G}^{\rm H}_{bq,R}\\
-\Lambda^{\gamma}_C\Lambda^{\mu}_AK^R_{\mu}K^T_{\gamma}{\tilde G}^{\rm H}_{bT,R}-\Lambda^{\gamma}_C\Lambda^{\mu}_AK^R_{\mu}K^q_{\gamma}{\tilde G}^{\rm H}_{bq,R}.
\]
Using the $\rm C$--identity from Appendix A, we have
\[
 K^T_{\gamma}{\tilde G}^{\rm H}_{bT,R}+K^q_{\gamma}{\tilde G}^{\rm H}_{bq,R}=-{\tilde G}^{\rm H}_{bT}K^T_{\gamma, R}.
\]
So, we get 
\[
 -\Lambda^{\gamma}_C\Lambda^{\mu}_A{\tilde G}^{\rm H}_{bT}(K^R_{\mu}K^T_{\gamma,R}-K^R_{\gamma}K^T_{\mu,R})=-\Lambda^{\gamma}_C\Lambda^{\mu}_A{\tilde G}^{\rm H}_{bT}K^T_{\sigma} c^{\sigma}_{\mu \gamma}.
\]
Combining this with the seventh term of the original expression, we get
\[
 -\Lambda^{\mu}_A\Lambda^{\gamma}_C c^{\sigma}_{\mu \gamma}(K^T_{\sigma}{\tilde G}^{\rm H}_{bT}+K^q_{\sigma}{\tilde G}^{\rm H}_{qb}).
\]
But the expression in the bracket  is zero by the identity B from the Appendix A.
This means that there are no  $``\Lambda \Lambda$---terms'' in the final expression.

The remaining terms of the right-hand side of the considered equation are
\begin{enumerate}
 \item  
 $2N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEC}-{\tilde G}^{\rm H}_{AC,b}+K^E_{\alpha}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{EC,b}+{\tilde G}^{\rm H}_{bA,C}-K^E_{\alpha}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{bE,C}$
\item $-K^q_{\alpha}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{bC,q}$
\item ${\tilde G}^{\rm H}_{AC,b}$
\item $-{\tilde G}^{\rm H}_{Ab,C}+K^E_{\alpha}\Lambda^{\alpha}_C{\tilde G}^{\rm H}_{Ab,E}$
\item $K^q_{\mu}\Lambda^{\mu}_C{\tilde G}^{\rm H}_{Ab,q}$
\item $\Lambda^{\gamma}_AK^T_{\gamma}{\tilde G}^{\rm H}_{bT,C}+\Lambda^{\gamma}_AK^q_{\gamma}{\tilde G}^{\rm H}_{bq,C}-\Lambda^{\gamma}_CK^T_{\gamma}{\tilde G}^{\rm H}_{bT,A}-\Lambda^{\gamma}_CK^q_{\gamma}{\tilde G}^{\rm H}_{bq,A}$
\item $0$
\item $\Lambda^{\alpha}_CK^q_{\alpha,b}{\tilde G}^{\rm H}_{Aq}=-\Lambda^{\alpha}_CK^q_{\alpha}{\tilde G}^{\rm H}_{Aq,b}-\Lambda^{\alpha}_CK^R_{\alpha}{\tilde G}^{\rm H}_{AR,b}$ (by the identity $D$)
\item $\Lambda^{\alpha}_AK^q_{\alpha,b}{\tilde G}^{\rm H}_{Cq}=-\Lambda^{\alpha}_AK^q_{\alpha}{\tilde G}^{\rm H}_{Cq,b}-\Lambda^{\alpha}_AK^R_{\alpha}{\tilde G}^{\rm H}_{CR,b}$ (by the identity $D$)
\end{enumerate}
 Note that there are  mutual cancellations between some terms: $(1.2)$ is cancelled with $(3.1)$,  $(1.3)$ with $(9.2)$, $(1.4)$ with $(4.1)$, $(1.5)$ with $(6.1)$.  

The sum of  three terms $(6.3)+(4.2)+(8.2)$ is rewritten according to the formula
\[
\Lambda^{\gamma}_CK^T_{\gamma}(-{\tilde G}^{\rm H}_{bT,A}+{\tilde G}^{\rm H}_{Ab,T}-{\tilde G}^{\rm H}_{AT,b})=-2\Lambda^{\gamma}_CK^T_{\gamma}\,{}^{\mathrm  H}{\tilde \Gamma}_{AbT}.
\]

For the sum $(6.2)+(9.1)+(2.1)$ we have
\[
 \Lambda^{\gamma}_AK^q_{\gamma}({\tilde G}^{\rm H}_{bq,C}-{\tilde G}^{\rm H}_{Cq,b}-{\tilde G}^{\rm H}_{bC,q})
=-2\Lambda^{\gamma}_AK^q_{\gamma}\,{}^{\mathrm  H}{\tilde \Gamma}_{bqC}.
\]
And for $(6.4)+(5.1)+(8.1)$ --
\[
 \Lambda^{\gamma}_CK^q_{\gamma}(-{\tilde G}^{\rm H}_{bq,A}+{\tilde G}^{\rm H}_{Ab,q}-{\tilde G}^{\rm H}_{Aq,b})
=-2\Lambda^{\gamma}_CK^q_{\gamma}\,{}^{\mathrm  H}{\tilde \Gamma}_{Abq}.
\]
So we get
\[
 2(N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEC}+N^q_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bqC}-\Lambda^{\gamma}_CK^T_{\gamma}\,{}^{\mathrm  H}{\tilde \Gamma}_{AbT}-\Lambda^{\gamma}_CK^q_{\gamma}\,{}^{\mathrm  H}{\tilde \Gamma}_{Abq}).
\]
It can be shown that
\[
 K^T_{\gamma}\,{}^{\mathrm  H}{\tilde \Gamma}_{AbT}+K^q_{\gamma}\,{}^{\mathrm  H}{\tilde \Gamma}_{Abq}=0.
\]
This is done in the following way. We first rewrite the expression as
\[
 K^T_{\gamma}({\tilde G}^{\rm H}_{AT,b}+{\tilde G}^{\rm H}_{Tb,A}-{\tilde G}^{\rm H}_{Ab,T})+K^q_{\gamma}({\tilde G}^{\rm H}_{Aq,b}+{\tilde G}^{\rm H}_{qb,A}-{\tilde G}^{\rm H}_{Ab,q}).
\]
Making use of the identities D and C from  Appendix A, we have the representations 
\[
 K^T_{\gamma}{\tilde G}^{\rm H}_{AT,b}=-K^q_{\gamma}{\tilde G}^{\rm H}_{Aq,b}-K^q_{\gamma,b}{\tilde G}^{\rm H}_{Aq}
\]
and
\[
 K^T_{\gamma}{\tilde G}^{\rm H}_{Tb,A}=-K^q_{\gamma}{\tilde G}^{\rm H}_{bq,A}-K^R_{\gamma,A}{\tilde G}^{\rm H}_{bR}.
\]
Taking them into account, we obtain an expression equal to zero by the Killing identity $\Roman{4}$ from Appendix A:
\[
 K^T_{\gamma}{\tilde G}^{\rm H}_{Ab,T}+K^q_{\gamma,b}{\tilde G}^{\rm H}_{Aq}+K^R_{\gamma,A}{\tilde G}^{\rm H}_{bR}+K^q_{\gamma}{\tilde G}^{\rm H}_{Ab,q}.
\]
Therefore, as a result,  we have the following relation:
\[
 {\check\Gamma}^D_{Ab}{\tilde G}^{\rm H}_{DC}+{\check\Gamma}^a_{Ab}{\tilde G}^{\rm H}_{aC}=N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEC}+N^q_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bqC}, 
\]
which can be rewritten as 
\[
 {\check\Gamma}^{\tilde D}_{Ab}{\tilde G}^{\rm H}_{{\tilde D}C}=N^{\tilde E}_A\,{}^{\mathrm  H}{\tilde \Gamma}_{b{\tilde E}C}.
\]

In the case where \underline{ $\mathscr A=A, \mathscr B=b, \mathscr C=p$ } the formula (\ref{christoff_main}) has the form
\begin{eqnarray*}
&&2[{\check\Gamma}^D_{Ab}{\tilde G}^{\rm H}_{Dp}+{\check\Gamma}^a_{Ab}{\tilde G}^{\rm H}_{ap}]=
H_A{\tilde G}^{\rm H}_{bp}+H_b{\tilde G}^{\rm H}_{Ap}-H_p{\tilde G}^{\rm H}_{Ab}\nonumber\\
&&-\check G(H_A,[H_b,H_p])-\check G(H_b,[H_A,H_p])+\check G(H_p,[H_A,H_b]).
\end{eqnarray*}

The right-hand side of this formula, 
\begin{eqnarray*}N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEp}
 H_A{\tilde G}^{\rm H}_{bp}+H_b{\tilde G}^{\rm H}_{Ap}-H_p{\tilde G}^{\rm H}_{Ab}
-\mathbb C^q_{Ap}{\tilde G}^{\rm H}_{bq}+\mathbb C^q_{Ab}{\tilde G}^{\rm H}_{pq},
\end{eqnarray*}
can be expressed explicitly as 
\begin{eqnarray*}
 N^E_A{\tilde G}^{\rm H}_{bp,E}+N^q_A{\tilde G}^{\rm H}_{bp,q}+{\tilde G}^{\rm H}_{Ap,b}-{\tilde G}^{\rm H}_{Ab,p}
-K^q_{\alpha,p}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{bq}+ K^q_{\alpha,b}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{pq}.
\end{eqnarray*}
Representing the first term of this expression in the following form
\[
  N^E_A({\tilde G}^{\rm H}_{bp,E}+{\tilde G}^{\rm H}_{pE,b}-{\tilde G}^{\rm H}_{bE,p})-N^E_A{\tilde G}^{\rm H}_{pE,b}+N^E_A{\tilde G}^{\rm H}_{bE,p},
\]
we write it as
\[
 2N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEp}-N^E_A{\tilde G}^{\rm H}_{pE,b}+N^E_A{\tilde G}^{\rm H}_{bE,p}.
\]
Therefore, on the right side of the formula we have the following terms:
\begin{enumerate}
 \item $2N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEp}-{\tilde G}^{\rm H}_{pA,b}+K^E_{\alpha}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{pE,b}+{\tilde G}^{\rm H}_{bA,p}-K^E_{\alpha}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{bE,p}$
\item $-K^q_{\alpha,p}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{bp,q}$
\item ${\tilde G}^{\rm H}_{Ap,b}$
\item $-{\tilde G}^{\rm H}_{Ab,p}$
\item $-K^q_{\alpha,p}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{bq}=\Lambda^{\alpha}_A({\tilde G}^{\rm H}_{bq,p}K^q_{\alpha} +{\tilde G}^{\rm H}_{bE,p}K^E_{\alpha})$ by identity B
\item $K^q_{\alpha,b}\Lambda^{\alpha}_A{\tilde G}^{\rm H}_{pq}=-\Lambda^{\alpha}_A({\tilde G}^{\rm H}_{pq,b}K^q_{\alpha} +{\tilde G}^{\rm H}_{pE,p}K^E_{\alpha})$ by identity B
\end{enumerate}
There are some mutual cancellations between terms: (1.2) with (3.1), (1.3) with (6.2), (1.4) with (4.1), (1.5) with (5.2).
As a result, we get
\[
 2N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEp}+K^q_{\alpha}\Lambda^{\alpha}_A(-{\tilde G}^{\rm H}_{bp,q} +{\tilde G}^{\rm H}_{bq,p}-{\tilde G}^{\rm H}_{pq,b}),
\]
or
\[
 2N^E_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bEp}+2N^q_A\,{}^{\mathrm  H}{\tilde \Gamma}_{bqp}\equiv 2N^{\tilde E}_A\,{}^{\mathrm  H}{\tilde \Gamma}_{b{\tilde E}p}.
\]
The final relation of this subsection is 
\[
 {\check\Gamma}^{\tilde D}_{Ab}{\tilde G}^{\rm H}_{{\tilde D}p}=N^{\tilde E}_A\,{}^{\mathrm  H}{\tilde \Gamma}_{b{\tilde E}p}.
\]
This relation can be combined with the relation from $(A, b, C)$ -- case:
\[
 {\check\Gamma}^{\tilde D}_{Ab}{\tilde G}^{\rm H}_{{\tilde D}\tilde C}=N^{\tilde E}_A\,{}^{\mathrm  H}{\tilde \Gamma}_{b{\tilde E}\tilde C}.
\]
To determine ${}^{\rm H}{\tilde \Gamma}_{b{\tilde E}}^{\tilde R}$, we use the following relation:
\[
 {\tilde \Gamma}_{b{\tilde E}\tilde C}={\tilde G}^{\rm H}_{\tilde R \tilde C}{}^{\rm H}{\tilde \Gamma}_{b{\tilde E}}^{\tilde R}.
\]
Note that this Christoffel symbol on the right side is determined modulo terms $T$ satisfying ${\tilde G}^{\rm H}_{\tilde A \tilde M}T^{\tilde M}_{b\tilde C}=0$.
With this definition we have
\[
 {\check\Gamma}^{\tilde D}_{Ab}{\tilde G}^{\rm H}_{{\tilde D}\tilde C}=N^{\tilde E}_A{\tilde G}^{\rm H}_{\tilde R \tilde C}{}^{\rm H}{\tilde \Gamma}_{b{\tilde E}}^{\tilde R}.
\]
Multiplying both sides of this relation by $G^{SF}N^T_SN^{\tilde C}_F$ and taking into account that
\[
 N^{\tilde C}_F{\tilde G}^{\rm H}_{{\tilde D}\tilde C}={\tilde G}^{\rm H}_{{\tilde D}F},\,\,\,\,G^{SF}{\tilde G}^{\rm H}_{{\tilde D}F}=\tilde {\Pi}^S_{\tilde D},\,\,\,\,\tilde {\Pi}^S_{\tilde D}N^T_S=N^T_{\tilde D},
\]
\[
 N^T_{\tilde D}{\check\Gamma}^{\tilde D}_{Ab}\equiv N^T_{ D}{\check\Gamma}^{D}_{Ab}+N^T_{r}{\check\Gamma}^{r}_{Ab}=N^T_{ D}{\check\Gamma}^{D}_{Ab},\;\; \rm{since}\;\; N^T_{r}=0,
\]
we get
\[
 N^T_D{\check\Gamma}^{ D}_{Ab}=N^T_RN^{\tilde E}_A\,{}^{\rm H}{\tilde \Gamma}_{b{\tilde E}}^{R}.
\]
Hence it follows that
\[
 {\check\Gamma}^{ D}_{Ab}=N^{\tilde E}_A\,{}^{\rm H}{\tilde \Gamma}_{b{\tilde E}}^{D}
\]
(modulo the terms $X^D_{AbC}$ for which $N^T_DX^D_{AbC}=0$).

One can similarly obtain representations for other Christoffel symbols of the manifold $\tilde{\mathcal P}$, independent of the indices associated with the group manifold:
\[
 {\check\Gamma}^{D}_{AB}=N^{\tilde E}_A\,{}^{\rm H}{\Gamma}^D_{B\tilde{E}}\;\;\;\;{\check\Gamma}^{D}_{ab}={}^{\rm H}{\Gamma}^D_{ab}
 \]
\[
 {\check\Gamma}^{a}_{AB}=N^{\tilde E}_A\,{}^{\rm H}{\Gamma}^a_{B\tilde{E}}\;\;\;\;{\check\Gamma}^{q}_{ab}={}^{\rm H}{\Gamma}^q_{ab}
 \]
\[
 \;\;{\check\Gamma}^{D}_{Ab}=N^{\tilde E}_A\,{}^{\rm H}{\Gamma}^D_{b\tilde{E}}\;\;\;\;\;\;\;{\check\Gamma}^{D}_{aB}={}^{\rm H}{\Gamma}^D_{aB}
\]
\[
 \;\;{\check\Gamma}^{p}_{Ab}=N^{\tilde E}_A\,{}^{\rm H}{\Gamma}^p_{b\tilde{E}}\;\;\;\;\;\;\;{\check\Gamma}^{D}_{aB}={}^{\rm H}{\Gamma}^D_{aB}
\]



Concerning to Christoffel symbols with indices related to the group manifold, first of all they include those originated from the commutator relations  of  the basis vector fields: 

\begin{eqnarray*}
 &&{\check\Gamma}^{\epsilon}_{ab}=-\frac12\tilde {\mathscr F}^{\epsilon}_{ab},\;\;\;\;{\check\Gamma}^{\epsilon}_{A\mu}=\frac12\tilde d^{\epsilon\nu}H_A\tilde d_{\mu\nu},\\
 &&{\check\Gamma}^{\epsilon}_{AB}=\frac12\mathbb{C}^{\epsilon}_{AB}=\frac12[-N^S_AN^R_B\tilde {\mathscr F}^{\epsilon}_{SR}-(N^E_AN^p_B-N^E_BN^p_A)\tilde {\mathscr F}^{\epsilon}_{Ep}-N^q_AN^p_B\tilde {\mathscr F}^{\epsilon}_{qp}],\\
 &&{\check\Gamma}^{\epsilon}_{a\alpha}=\frac12\tilde d^{\epsilon\mu}H_a\tilde d_{\alpha\mu},\;\;\;{\check\Gamma}^{\epsilon}_{\mu b}=\frac12\tilde d^{\epsilon\nu}H_b\tilde d_{\mu\nu},\;\;\;{\check\Gamma}^{\epsilon}_{\mu B}=\frac12\tilde d^{\epsilon\nu}H_B\tilde d_{\mu\nu},\\
&& {\check\Gamma}^{\nu}_{Ab}=\frac12(-N^E_A\tilde {\mathscr F}^{\nu}_{Eb}-N^q_A\tilde {\mathscr F}^{\nu}_{qb}),\;\;\;
{\check\Gamma}^{\nu}_{aB}=-\frac12(-N^E_B\tilde {\mathscr F}^{\nu}_{Ea}-N^q_B\tilde {\mathscr F}^{\nu}_{qa}),\\
&&( \mathbb C^{\alpha}_{Bp}=-\mathbb C^{\alpha}_{pB}).
\end{eqnarray*}

Other such Christoffel symbols are obtained from equations that are special cases of equation (\ref{christoff_main}):
\begin{enumerate}
\item 
${\check\Gamma}^{D}_{a \nu}{\tilde G}^{\rm H}_{DC}+{\check\Gamma}^{p}_{a \nu}{\tilde G}^{\rm H}_{pC}=-\frac12\mathbb{C}^{\mu}_{aC}\tilde d_{\nu\mu}$
\item
${\check\Gamma}^{D}_{a \mu}{\tilde G}^{\rm H}_{Da}+{\check\Gamma}^{p}_{A\mu}{\tilde G}^{\rm H}_{pA}=-\frac12\mathbb{C}^{\epsilon}_{Aa}\tilde d_{\mu\epsilon}$
\item 
${\check\Gamma}^{D}_{a \nu}{\tilde G}^{\rm H}_{Db}+{\check\Gamma}^{p}_{a \nu}{\tilde G}^{\rm H}_{pb}=-\frac12\mathbb{C}^{\mu}_{ab}\tilde d_{\nu\mu}$
\item
${\check\Gamma}^{D}_{\mu b}{\tilde G}^{\rm H}_{Da}+{\check\Gamma}^{p}_{\mu b}{\tilde G}^{\rm H}_{pa}=-\frac12\mathbb{C}^{\alpha}_{ba}\tilde d_{\mu\alpha}$
\item
${\check\Gamma}^{D}_{\mu b}{\tilde G}^{\rm H}_{DC}+{\check\Gamma}^{p}_{\mu b}{\tilde G}^{\rm H}_{pC}=-\frac12\mathbb{C}^{\alpha}_{bC}\tilde d_{\mu\alpha}$
\item
${\check\Gamma}^{D}_{\mu B}{\tilde G}^{\rm H}_{DC}+{\check\Gamma}^{p}_{\mu B}{\tilde G}^{\rm H}_{pC}=-\frac12\mathbb{C}^{\alpha}_{BC}\tilde d_{\mu\alpha}$
\item
${\check\Gamma}^{D}_{\mu B}{\tilde G}^{\rm H}_{Db}+{\check\Gamma}^{p}_{\mu B}{\tilde G}^{\rm H}_{pb}=-\frac12\mathbb{C}^{\alpha}_{Bb}\tilde d_{\mu\alpha}$
\item
${\check\Gamma}^{D}_{\mu \nu}{\tilde G}^{\rm H}_{Db}+{\check\Gamma}^{p}_{\mu \nu}{\tilde G}^{\rm H}_{pb}=-\frac12H_b\tilde d_{\mu\nu}$
\item
${\check\Gamma}^{D}_{A \mu}{\tilde G}^{\rm H}_{DC}+{\check\Gamma}^{p}_{A\mu}{\tilde G}^{\rm H}_{pC}=-\frac12\mathbb{C}^{\epsilon}_{AC}\tilde d_{\mu\epsilon}$
\item
${\check\Gamma}^{D}_{\mu \nu}{\tilde G}^{\rm H}_{DC}+{\check\Gamma}^{p}_{\mu \nu}{\tilde G}^{\rm H}_{pC}=-\frac12H_C\tilde d_{\mu\nu}$
\end{enumerate}
Combining the equations of the first and third rows of the table, we rewrite these equations in matrix form:
\[
 \left(\begin{matrix}{\tilde G}^H_{CD}&{\tilde G}^H_{Cp}\cr{\tilde G}^H_{bD}&{\tilde G}^H_{bp}\cr\end{matrix}\right)
\left(\begin{matrix}{\check\Gamma}^{D}_{a \nu}\cr {\check\Gamma}^{p}_{a \nu}\cr\end{matrix}\right)=-\frac12\left(\begin{matrix}
\mathbb{C}^{\mu}_{aC}\tilde d_{\nu\mu}\cr\mathbb{C}^{\mu}_{ab}\tilde d_{\nu\mu}\cr\end{matrix}\right),
\]
where
\[
 \mathbb{C}^{\mu}_{aC}=-\mathbb{C}^{\mu}_{Ca}=N^E_C\tilde{\mathscr F}^{\mu}_{Ea}+N^p_C\tilde{\mathscr F}^{\mu}_{pa}, \;\;\;\mathbb{C}^{\mu}_{ab}=-\frac12\tilde{\mathscr F}^{\mu}_{ab}.
\]
Multiplying the matrix equation (from the left) by the matrix
\[
 \left(\begin{matrix}{\tilde G}^{EF}N^A_EN^C_F&{\tilde G}^{EF}N^A_EN^b_F\cr {\tilde G}^{EF}N^q_FN^C_E & {\tilde G}^{bq}+{\tilde G}^{FE}N^b_FN^q_E\cr\end{matrix}\right),
\]
it can be shown that the left-hand side of the transformed equation will be
\[
 \left(\begin{matrix} N^A_D&0\cr N^q_D&\delta^q_p\cr \end{matrix}\right)\left(\begin{matrix}{\check\Gamma}^{D}_{a \nu}\cr {\check\Gamma}^{p}_{a \nu}\cr\end{matrix}\right).
\]
On the right-hand side of the matrix equation, we have in the first row
\[
\frac12 \tilde d_{\nu\mu}[{\tilde G}^{EF}N^A_EN^C_F(N^{E'}_C\tilde{\mathscr F}^{\mu}_{E'a}+N^p_C\tilde{\mathscr F}^{\mu}_{pa})+{\tilde G}^{EF}N^A_EN^b_F\tilde{\mathscr F}^{\mu}_{ba}]=
\]
\[
 =\frac12 \tilde d_{\nu\mu}[{\tilde G}^{EF}N^A_EN^T_F\tilde{\mathscr F}^{\mu}_{Ta}+{\tilde G}^{EF}N^A_EN^b_F\tilde{\mathscr F}^{\mu}_{ba}]\equiv\frac12 \tilde d_{\nu\mu}{\tilde G}^{EF}N^A_EN^{\tilde T}_F\tilde{\mathscr F}^{\mu}_{\tilde{T}a},\,\tilde{T}=(T,b).
\]
Here we have used the properties of the projectors $N$: $N^{\tilde A}_{\tilde B} N^{\tilde B}_{\tilde C}=N^{\tilde A}_{\tilde C}$
from which $N^p_CN^C_F=0$.

Thus, the first equation is
\begin{equation}
 N^A_D{\check\Gamma}^{D}_{a \nu}=-\frac12 \tilde d_{\nu\mu}{\tilde G}^{EF}N^A_EN^{\tilde T}_F\tilde{\mathscr F}^{\mu}_{\tilde{T}a}.
\label{eq_for_part_christoff}
\end{equation}
From this it follows that 
\begin{equation}
 {\check\Gamma}^{D}_{a \nu}=-\frac12 {\tilde G}^{DF}N^{\tilde T}_F\tilde{\mathscr F}^{\mu}_{\tilde{T}a}\tilde d_{\nu\mu}
\label{sol_part_christoff_eq}
\end{equation}
(modulo the terms for which $N^A_DX^D_{a\nu}=0$).

Then the second equation is
\[
 {\check\Gamma}^{q}_{a \nu}=-\frac12 {\tilde G}^{qb}\tilde{\mathscr F}^{\mu}_{ba}\tilde d_{\nu\mu}.
\]

In the same way, the following Christoffel symbols can be obtained:
\begin{eqnarray*}
&&{\check\Gamma}^{D}_{\mu b}=-\frac12 {\tilde G}^{DF}N^{\tilde T}_F\tilde{\mathscr F}^{\sigma}_{\tilde{T}b}\tilde d_{\mu\sigma} \;\;\;\;\;\;\;
{\check\Gamma}^{q}_{\mu b}=\frac12 {\tilde G}^{qa}\tilde{\mathscr F}^{\sigma}_{ba}\tilde d_{\mu\sigma}\\
&&{\check\Gamma}^{D}_{A\mu }=\frac12 {\tilde G}^{DF}N^{\tilde R}_FN^{\tilde S}_A\tilde{\mathscr F}^{\sigma}_{\tilde{S}\tilde{R}}\tilde d_{\mu\sigma} \;\;\;
{\check\Gamma}^{q}_{A\mu }=\frac12 {\tilde G}^{qa}N^{\tilde E}_A\tilde{\mathscr F}^{\sigma}_{\tilde{E}a}\tilde d_{\mu\sigma}\\
&&{\check\Gamma}^{q}_{\mu B}=\frac12 {\tilde G}^{qb}N^{\tilde E}_B\tilde{\mathscr F}^{\sigma}_{\tilde{E}b }\tilde d_{\mu\sigma} \;\;\;\;\;\;\;\;\;\;
{\check\Gamma}^{D}_{\mu B}=\frac12 {\tilde G}^{DF}N^{\tilde R}_FN^{\tilde S}_B\tilde{\mathscr F}^{\sigma}_{\tilde{S}\tilde{R} }\tilde d_{\mu\sigma}\\
&&{\check\Gamma}^{D}_{\mu \nu}=-\frac12 {\tilde G}^{DF}N^{\tilde C}_F(H_{\tilde{C}}\,\tilde d_{\mu\nu}) \;\;\;\;\;
{\check\Gamma}^{q}_{\mu \nu}=-\frac12 {\tilde G}^{qb}(H_b\,\tilde d_{\mu\nu}).
\end{eqnarray*}
And also from (\ref{christoff_main}) it follows that
\begin{eqnarray*}
 {\check\Gamma}^{\sigma}_{\mu\nu}&=&\frac12{\tilde d}^{\sigma\gamma}(L_{\mu}{\tilde d}_{\nu\gamma}+L_{\nu}{\tilde d}_{\mu\gamma}-L_{\gamma}{\tilde d}_{\mu\nu}-c^{\varphi}_{\nu\gamma}{\tilde d}_{\mu\varphi}-c^{\varphi}_{\mu\gamma}{\tilde d}_{\nu\varphi}+c^{\varphi}_{\mu\nu}{\tilde d}_{\varphi\gamma})\\
 &=&\frac12{\tilde d}^{\sigma\gamma}(c^{\varphi}_{\mu\nu}{\tilde d}_{\varphi\gamma}-c^{\varphi}_{\gamma\nu}{\tilde d}_{\mu\varphi}-c^{\varphi}_{\gamma\mu}{\tilde d}_{\nu\varphi}).
\end{eqnarray*}


For the Christoffel symbols of the manifold $\tilde{\mathcal P}$, which have "group" indices, there is another representation. It turns out that in this representation the calculation of the curvature of the manifold can be done with less difficulty. Let us explain how these representations are obtained.

In our above computation of the Christoffel symbol ${\check\Gamma}^{D}_{a \nu}$, we have obtained equation (\ref{eq_for_part_christoff})
 which has the ``solution'' (\ref{sol_part_christoff_eq}). One can transform the right-hand side of this equation so that the equation becomes
$$N^A_R{\check\Gamma}^{R}_{a \nu}=-\frac12 \tilde d_{\nu\mu}{\tilde G}^{DF}N^A_RN^R_DN^{\tilde T}_F\tilde{\mathscr F}^{\mu}_{\tilde{T}a}.$$
Then, as before, omitting $N^A_R$, we get the ``solution''
\begin{eqnarray*}
{\check\Gamma}^{R}_{a \nu}&=&-\frac12 \tilde d_{\nu\mu}{\tilde G}^{DF}N^R_DN^{\tilde T}_F\tilde{\mathscr F}^{\mu}_{\tilde{T}a}\\
&=&-\frac12 \tilde d_{\nu\mu}({\tilde G}^{DF}N^R_DN^{T}_F\tilde{\mathscr F}^{\mu}_{{T}a}+{\tilde G}^{DF}N^R_DN^{b}_F\tilde{\mathscr F}^{\mu}_{{b}a})\\
&=&-\frac12 \tilde d_{\nu\mu}(h^{RT}\tilde{\mathscr F}^{\mu}_{{T}a}+h^{Rb}\tilde{\mathscr F}^{\mu}_{{b}a}).
\end{eqnarray*}
The difference between the new ``solution'' and the previous one is equal to
$-\frac12 K^R_{\sigma}(G^{EF}\Lambda ^{\sigma}_EN^{\tilde T}_F\tilde{\mathscr F}^{\mu}_{\tilde{T}a}\tilde d_{\nu\mu})$. So,
$N^A_RK^R_{\sigma}(...)=0$. And two ``solutions'' belong to the same class.

The Christoffel symbols in the new representation are 
\begin{eqnarray*}
&&{\check\Gamma}^{D}_{AB}=N^{\tilde E}_A\,{}^{\rm H}{\Gamma}^D_{B\tilde{E}}\;\;\;\;{\check\Gamma}^{D}_{ab}={}^{\rm H}{\Gamma}^D_{ab}
\nonumber\\
&&{\check\Gamma}^{a}_{AB}=N^{\tilde E}_A\,{}^{\rm H}{\Gamma}^a_{B\tilde{E}}\;\;\;\;{\check\Gamma}^{q}_{ab}={}^{\rm H}{\Gamma}^q_{ab}
\nonumber\\
&&{\check\Gamma}^{D}_{Ab}=N^{\tilde E}_A\,{}^{\rm H}{\Gamma}^D_{b\tilde{E}}\;\;\;\;\;\;\;{\check\Gamma}^{D}_{aB}={}^{\rm H}{\Gamma}^D_{aB}
\nonumber\\
&&{\check\Gamma}^{p}_{Ab}=N^{\tilde E}_A\,{}^{\rm H}{\Gamma}^p_{b\tilde{E}}\;\;\;\;\;\;\;{\check\Gamma}^{p}_{aB}={}^{\rm H}{\Gamma}^p_{aB}
\nonumber\\
 &&{\check\Gamma}^{D}_{a\mu }=-\frac12( h^{DT}\tilde{\mathscr F}^{\sigma}_{{T}a}+h^{Db}\tilde{\mathscr F}^{\sigma}_{ba})\tilde d_{\mu\sigma},\;\;\;\;{\check\Gamma}^{D}_{\mu a}={\check\Gamma}^{D}_{a\mu}
 \nonumber\\
 &&{\check\Gamma}^{q}_{a\mu }=-\frac12( h^{qE}\tilde{\mathscr F}^{\sigma}_{Ea}+h^{qb}\tilde{\mathscr F}^{\sigma}_{ba})\tilde d_{\mu\sigma},\;\;\;\;\;\;{\check\Gamma}^{q}_{\mu a}={\check\Gamma}^{q}_{a\mu}
 \nonumber\\
 &&{\check\Gamma}^{D}_{A\mu }=\frac12[N^S_A( h^{DR}\tilde{\mathscr F}^{\sigma}_{SR}+h^{Dp}\tilde{\mathscr F}^{\sigma}_{Sp})+N^q_A(h^{DR}\tilde{\mathscr F}^{\sigma}_{qR}+h^{Dp}\tilde{\mathscr F}^{\sigma}_{qp})]
 \tilde d_{\mu\sigma}
 \nonumber\\
 &
 {\check\Gamma}^{D}_{\mu A}={\check\Gamma}^{D}_{A\mu}
 \nonumber\\
&&{\check\Gamma}^{q}_{A\mu }=\frac12[N^S_A( h^{qR}\tilde{\mathscr F}^{\sigma}_{SR}+h^{qa}\tilde{\mathscr F}^{\sigma}_{Sa})+N^p_A(h^{qR}\tilde{\mathscr F}^{\sigma}_{pR}+h^{qa}\tilde{\mathscr F}^{\sigma}_{pa})]
 \tilde d_{\mu\sigma}
 \nonumber\\
 &
 {\check\Gamma}^{q}_{\mu A}={\check\Gamma}^{q}_{A\mu}
 \nonumber\\
 &&{\check\Gamma}^{\epsilon}_{AB}=\frac12\mathbb{C}^{\epsilon}_{AB}=\frac12N^E_A(N^D_B\tilde{\mathscr F}^{\epsilon}_{DE}+N^p_B\tilde{\mathscr F}^{\epsilon}_{pE})+\frac12N^p_A(N^E_B\tilde{\mathscr F}^{\epsilon}_{Ep}+N^q_B\tilde{\mathscr F}^{\epsilon}_{qp})
 \nonumber\\


 && {\check\Gamma}^{\epsilon}_{Ab}=\frac12\mathbb{C}^{\epsilon}_{Ab}=\frac12(-N^E_A\tilde {\mathscr F}^{\epsilon}_{Eb}-N^q_A\tilde {\mathscr F}^{\epsilon}_{qb}),\;\;\;\;{\check\Gamma}^{\epsilon}_{bA}=-{\check\Gamma}^{\epsilon}_{Ab}
 \nonumber\\
 &&{\check\Gamma}^{\epsilon}_{aB}=-\frac12\mathbb{C}^{\epsilon}_{Ba}=\frac12(N^E_B\tilde {\mathscr F}^{\epsilon}_{Ea}+N^q_B\tilde {\mathscr F}^{\epsilon}_{qa}),\;\;\;\;{\check\Gamma}^{\epsilon}_{BA}=-{\check\Gamma}^{\epsilon}_{aB}
 \nonumber\\
 &&{\check\Gamma}^{\epsilon}_{ab}=-\frac12\tilde {\mathscr F}^{\epsilon}_{ab},\;\;\;\;{\check\Gamma}^{\epsilon}_{A\mu}=\frac12\tilde d^{\epsilon\nu}H_A\tilde d_{\mu\nu},\;\;\;{\check\Gamma}^{\epsilon}_{a\mu}=\frac12\tilde d^{\epsilon\nu}H_a\tilde d_{\mu\nu},\;\;{\check\Gamma}^{\epsilon}_{\mu a}={\check\Gamma}^{\epsilon}_{a\mu}
 \nonumber\\
 &&{\check\Gamma}^{D}_{\mu\nu}=-\frac12(h^{DC}H_C\,\tilde d_{\mu\nu}+h^{Db}H_b\,\tilde d_{\mu\nu})
 \nonumber\\
 &&{\check\Gamma}^{q}_{\mu\nu}=-\frac12(h^{qC}H_C\,\tilde d_{\mu\nu}+h^{qb}H_b\,\tilde d_{\mu\nu})
 \nonumber\\
 &&{\check\Gamma}^{\alpha}_{\beta\gamma}=\frac12\tilde d^{\alpha\mu}(c^{\epsilon}_{\beta\gamma}\tilde d_{\epsilon\mu}-c^{\epsilon}_{\mu\gamma}\tilde d_{\epsilon\beta}-c^{\epsilon}_{\mu\beta}\tilde d_{\epsilon\gamma})
 \end{eqnarray*}

\subsection{The Ricci curvature tensor}
In the article, the Riemannian tensor $\check{\mathcal R}_{\mathcal A\mathcal P\mathscr A \mathscr C}$  of the manifold $\tilde{\mathcal P}$ is defined by the Riemannian curvature operator $  \Omega(X,Y)=[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$ as follows
$$\check R(X,Y,Z,Z')=\check{G}_{\tilde{\mathcal P}}(\Omega(X,Y)Z,Z').$$

The components $\check R_{\mathscr A\mathscr C}=\check R_{\mathscr A\mathscr M\mathscr C}^{\;\;\;\;\;\;\;\;\;\;\mathscr M}$ of the Ricci curvature tensor of the metric (\ref{metric2cc}) will be evaluated in accordance with the following formula:
$$\check R_{\mathscr A\mathscr C}=\hat\partial_{\mathscr A}\check{\rm \Gamma}^{\mathscr P}_{\mathscr P\mathscr C}-\hat\partial_{\mathscr P}\check{\rm \Gamma}^{\mathscr P}_{\mathscr A\mathscr C}+\check{\rm \Gamma}^{\mathscr D}_{\mathscr P\mathscr C}\check{\rm \Gamma}^{\mathscr P}_{\mathscr A\mathscr D}-\check{\rm \Gamma}^{\mathscr E}_{\mathscr A\mathscr C}\check{\rm \Gamma}^{\mathscr P}_{\mathscr P\mathscr E}-\mathbb C^{\mathscr E}_{\mathscr A\mathscr P}\check{\rm \Gamma}^{\mathscr P}_{\mathscr E\mathscr C}.$$
In this formula, each  index given by a capital script letter denotes   three different indices, for example, $\mathscr A=(A,a,\alpha)$. This means that 
$\check R_{\mathscr A\mathscr C}$ is presented by the following components: $(\check R_{ A C}, \check R_{ A b}, \check R_{ b,A}, \check R_{ ab}, \check R_{ \alpha,\beta})$.



First,  consider the calculation  of the components  $\check R_{ A C}$ defined by 
$$\check R_{ A C}=\hat\partial_{A}\check{\rm \Gamma}^{\mathscr P}_{\mathscr PC}-\hat\partial_{\mathscr P}\check{\rm \Gamma}^{\mathscr P}_{AC}+\check{\rm \Gamma}^{\mathscr D}_{\mathscr PC}\check{\rm \Gamma}^{\mathscr P}_{A\mathscr D}-\check{\rm \Gamma}^{\mathscr E}_{AC}\check{\rm \Gamma}^{\mathscr P}_{\mathscr P\mathscr E}-\mathbb C^{\mathscr E}_{A\mathscr P}\check{\rm \Gamma}^{\mathscr P}_{\mathscr EC}.$$

They  are written as follows:
\begin{eqnarray}
 \check R_{ A C}&=&H_{A}\check{\rm \Gamma}^{ E}_{ EC}-H_{P}\check{\rm \Gamma}^{ P}_{ AC}
+H_{A}\check{\rm \Gamma}^{ p}_{ pC}-H_{q}\check{\rm \Gamma}^{q}_{ AC}
+\check{\rm \Gamma}^{ E}_{ DC}\check{\rm \Gamma}^{ D}_{A E}+\check{\rm \Gamma}^{ E}_{qC}\check{\rm \Gamma}^{q}_{ D C}
\nonumber\\
&&
+\check{\rm \Gamma}^{ q}_{ BC}\check{\rm \Gamma}^{ B}_{A q}+\check{\rm \Gamma}^{ q}_{ pC}\check{\rm \Gamma}^{ p}_{A q}
-\check{\rm \Gamma}^{ E}_{ AC}\check{\rm \Gamma}^{ P}_{ PE}-\check{\rm \Gamma}^{ E}_{ AC}\check{\rm \Gamma}^{ q}_{ qE}
-\check{\rm \Gamma}^{ q}_{ AC}\check{\rm \Gamma}^{ R}_{ Rq}-\check{\rm \Gamma}^{ q}_{ AC}\check{\rm \Gamma}^{ p}_{ pq}
\nonumber\\
&&-\mathbb C^{E}_{AR}\check{\rm \Gamma}^{R}_{ EC}-\mathbb C^{q}_{AR}\check{\rm \Gamma}^{R}_{qC}-\mathbb C^{q}_{Ap}\check{\rm \Gamma}^{p}_{qC}
\nonumber\\
&&+ H_{A}\check{\rm \Gamma}^{\alpha}_{ \alpha C}-L_{\alpha}\check{\rm \Gamma}^{\alpha}_{AC}+\check{\rm \Gamma}^{ E}_{ \mu C}\check{\rm \Gamma}^{\mu}_{ AE}+\check{\rm \Gamma}^{ q}_{\alpha C}\check{\rm \Gamma}^{\alpha}_{A q}
+\check{\rm \Gamma}^{\mu}_{ BC}\check{\rm \Gamma}^{ B}_{A \mu}+\check{\rm \Gamma}^{ \mu}_{ pC}\check{\rm \Gamma}^{ p}_{A \mu}
\nonumber\\
&&+\check{\rm \Gamma}^{ \mu}_{\nu C}\check{\rm \Gamma}^{\nu}_{A \mu}-\check{\rm \Gamma}^{ E}_{ AC}\check{\rm \Gamma}^{ \mu}_{\mu E}
-\check{\rm \Gamma}^{ q}_{ AC}\check{\rm \Gamma}^{ \mu}_{\mu q}
-\check{\rm \Gamma}^{ \alpha}_{ AC}\check{\rm \Gamma}^{ R}_{ R\alpha}
-\check{\rm \Gamma}^{ \alpha}_{ AC}\check{\rm \Gamma}^{ \mu}_{\mu \alpha}
\nonumber\\
&&-\mathbb C^{\mu}_{AR}\check{\rm \Gamma}^{R}_{\mu C}-\mathbb C^{\mu}_{Aq}\check{\rm \Gamma}^{q}_{\mu C}.
\label{R_AC}
\end{eqnarray}
Using the obtained Christoffel symbols $\check{\rm \Gamma}$ and the structure constants
$\mathbb C^{E}_{AR}, \mathbb C^{q}_{AR}$, $\mathbb C^{q}_{Ap}$, it  can be shown that the expression standing at the  first three lines of the right-hand side of (\ref{R_AC}) coinsides  with  
\[
 N^{\tilde S}_AN^{\tilde E}_{ M}\,{}^{\rm H}\!R_{\tilde S \tilde E C}{}^{ M},
\]
where 
\[
 {}^{\rm H}\!R_{\tilde S \tilde E C}{}^{M}=\partial_{\tilde S}{}^{\rm H}\!{\rm \Gamma}^{M}_{C\tilde E}-\partial_{\tilde E}{}^{\rm H}\!{\rm \Gamma}^{M}_{C\tilde S}+{}^{\rm H}\!{\rm \Gamma}^{\tilde K}_{C\tilde E}{}^{\rm H}\!{\rm \Gamma}^{M}_{\tilde K\tilde S}-{}^{\rm H}\!{\rm \Gamma}^{\tilde P}_{C\tilde S}{}^{\rm H}\!{\rm \Gamma}^{M}_{\tilde P\tilde E}.
\]
Also note that the expression for ${}^{\rm H}\!R_{\tilde S \tilde E C}{}^{M}$ is similar to the expression used for the Riemannian curvature tensor of the manifold. But in our case the metric of the manifold is degenerate. 

Therefore,  $\check R_{ A C}$ is represented as
 $\check R_{AC}=N^{\tilde S}_AN^{\tilde E}_{ M}\,{}^{\rm H}\!R_{\tilde S\tilde E C}{}^{ M}+\check R'_{AC}$. 

It can be verified that the same is true for other components  of the Ricci tensor $\check R_{\mathscr A\mathscr C}$.  For   $\check R_{ A b}$  defined as 
$$\check R_{ A b}=\hat\partial_{A}\check{\rm \Gamma}^{\mathscr P}_{\mathscr Pb}-\hat\partial_{\mathscr P}\check{\rm \Gamma}^{\mathscr P}_{Ab}+\check{\rm \Gamma}^{\mathscr D}_{\mathscr Pb}\check{\rm \Gamma}^{\mathscr P}_{A\mathscr D}-\check{\rm \Gamma}^{\mathscr E}_{Ab}\check{\rm \Gamma}^{\mathscr P}_{\mathscr P\mathscr E}-\mathbb{C}^{\mathscr E}_{A\mathscr P}\check{\rm \Gamma}^{\mathscr P}_{\mathscr Eb}$$
and given by the following expression:
\begin{eqnarray}
 \check R_{ A b}&=&H_{A}\check{\rm \Gamma}^{ B}_{ Bb}-H_{B}\check{\rm \Gamma}^{ B}_{ Ab}
+H_{A}\check{\rm \Gamma}^{ p}_{ pb}-H_{p}\check{\rm \Gamma}^{p}_{ Ab}
+\check{\rm \Gamma}^{ B}_{ Cb}\check{\rm \Gamma}^{ C}_{A B}+\check{\rm \Gamma}^{ B}_{pb}\check{\rm \Gamma}^{p}_{AB}
\nonumber\\
&&
+\check{\rm \Gamma}^{ q}_{ Cb}\check{\rm \Gamma}^{ C}_{A q}+\check{\rm \Gamma}^{ q}_{ pb}\check{\rm \Gamma}^{ p}_{A q}
-\check{\rm \Gamma}^{ E}_{ Ab}\check{\rm \Gamma}^{ B}_{ BE}-\check{\rm \Gamma}^{ E}_{ AC}\check{\rm \Gamma}^{ p}_{ pE}
-\check{\rm \Gamma}^{ q}_{ Ab}\check{\rm \Gamma}^{ B}_{ Bq}-\check{\rm \Gamma}^{ q}_{ Ab}\check{\rm \Gamma}^{ p}_{ pq}
\nonumber\\
&&-\mathbb C^{E}_{AB}\check{\rm \Gamma}^{B}_{ Eb}-\mathbb C^{q}_{AB}\check{\rm \Gamma}^{B}_{qB}-\mathbb C^{q}_{Ap}\check{\rm \Gamma}^{p}_{qb}
\nonumber\\
&&+ H_{A}\check{\rm \Gamma}^{\alpha}_{ \alpha b}-L_{\alpha}\check{\rm \Gamma}^{\alpha}_{Ab}+\check{\rm \Gamma}^{ B}_{ \alpha b}\check{\rm \Gamma}^{\alpha}_{ AB}+\check{\rm \Gamma}^{ q}_{\alpha b}\check{\rm \Gamma}^{\alpha}_{A q}
+\check{\rm \Gamma}^{\alpha}_{Cb}\check{\rm \Gamma}^{ C}_{A \alpha}+\check{\rm \Gamma}^{ \alpha}_{ pb}\check{\rm \Gamma}^{ p}_{A \alpha}
\nonumber\\
&&+\check{\rm \Gamma}^{ \alpha}_{\mu b}\check{\rm \Gamma}^{\mu}_{A \alpha}-\check{\rm \Gamma}^{ E}_{ Ab}\check{\rm \Gamma}^{ \alpha}_{\alpha E}
-\check{\rm \Gamma}^{ q}_{ Ab}\check{\rm \Gamma}^{ \alpha}_{\alpha q}
-\check{\rm \Gamma}^{ \alpha}_{ Ab}\check{\rm \Gamma}^{ B}_{ B\alpha}-\check{\rm \Gamma}^{ \alpha}_{ Ab}\check{\rm \Gamma}^{ p}_{ p\alpha}
-\check{\rm \Gamma}^{ \alpha}_{ Ab}\check{\rm \Gamma}^{ \mu}_{\mu \alpha}
\nonumber\\
&&-\mathbb C^{\alpha}_{AB}\check{\rm \Gamma}^{B}_{\alpha b}-\mathbb C^{\alpha}_{Ap}\check{\rm \Gamma}^{p}_{\alpha b},
\label{R_Ab}
\end{eqnarray}
  the  first three lines on the right-hand side of (\ref{R_Ab}) are  the same as in\\  
$N^{\tilde S}_AN^{\tilde E}_{ M}\,{}^{\rm H}\!R_{\tilde S \tilde E b}{}^{ M}$.
 Therefore, we can represent $\check R_{ A b}$ as
\[
 \check R_{Ab}=N^{\tilde S}_AN^{\tilde E}_{ M}\,{}^{\rm H}\!R_{\tilde S\tilde E b}{}^{ M}+\check R'_{Ab}. 
\]

The components $\check R_{ bA}$ are defined as
\begin{eqnarray}
 \check R_{ bA}&=&H_{b}\check{\rm \Gamma}^{ B}_{ BA}-H_{B}\check{\rm \Gamma}^{ B}_{ bA}
+H_{b}\check{\rm \Gamma}^{ p}_{ pA}-H_{p}\check{\rm \Gamma}^{p}_{b A}
+\check{\rm \Gamma}^{ B}_{ CA}\check{\rm \Gamma}^{ C}_{b B}+\check{\rm \Gamma}^{ B}_{pA}\check{\rm \Gamma}^{p}_{bB}
\nonumber\\
&&
+\check{\rm \Gamma}^{ q}_{ CA}\check{\rm \Gamma}^{ C}_{b q}+\check{\rm \Gamma}^{ q}_{ pA}\check{\rm \Gamma}^{ p}_{b q}
-\check{\rm \Gamma}^{ E}_{ bA}\check{\rm \Gamma}^{ B}_{ BE}-\check{\rm \Gamma}^{ E}_{ bA}\check{\rm \Gamma}^{ p}_{ pE}
-\check{\rm \Gamma}^{ q}_{ bA}\check{\rm \Gamma}^{ B}_{ Bq}-\check{\rm \Gamma}^{ q}_{ bA}\check{\rm \Gamma}^{ p}_{ pq}
\nonumber\\
&&-\mathbb C^{q}_{bB}\check{\rm \Gamma}^{B}_{qA}
\nonumber\\
&&+ H_{b}\check{\rm \Gamma}^{\alpha}_{ \alpha A}-L_{\alpha}\check{\rm \Gamma}^{\alpha}_{bA}+\check{\rm \Gamma}^{ B}_{ \alpha A}\check{\rm \Gamma}^{\alpha}_{ bB}+\check{\rm \Gamma}^{ q}_{\alpha A}\check{\rm \Gamma}^{\alpha}_{b q}
+\check{\rm \Gamma}^{\alpha}_{CA}\check{\rm \Gamma}^{ C}_{b \alpha}+\check{\rm \Gamma}^{ \alpha}_{ pA}\check{\rm \Gamma}^{ p}_{b \alpha}
\nonumber\\
&&+\check{\rm \Gamma}^{ \alpha}_{\mu A}\check{\rm \Gamma}^{\mu}_{b \alpha}-\check{\rm \Gamma}^{ E}_{ bA}\check{\rm \Gamma}^{ \alpha}_{\alpha E}
-\check{\rm \Gamma}^{ q}_{ bA}\check{\rm \Gamma}^{ \alpha}_{\alpha q}
-\check{\rm \Gamma}^{ \alpha}_{ bA}\check{\rm \Gamma}^{ B}_{ B\alpha}-\check{\rm \Gamma}^{ \alpha}_{ bA}\check{\rm \Gamma}^{ p}_{ p\alpha}
-\check{\rm \Gamma}^{ \alpha}_{ bA}\check{\rm \Gamma}^{ \mu}_{\mu \alpha}
\nonumber\\
&&-\mathbb C^{\alpha}_{bB}\check{\rm \Gamma}^{B}_{\alpha A}-\mathbb C^{\alpha}_{bp}\check{\rm \Gamma}^{p}_{\alpha A},
\label{R_bA}
\end{eqnarray}
and we have
\[
 \check R_{bA}=N^{\tilde E}_{ M}\,{}^{\rm H}\!R_{b\tilde E A}{}^{ M}+\check R'_{bA}. 
\]

The components $\check R_{ ab}$ are defined as
\begin{eqnarray}
 \check R_{ ab}&=&H_{a}\check{\rm \Gamma}^{ B}_{ Bb}-H_{B}\check{\rm \Gamma}^{ B}_{ ab}
+H_{a}\check{\rm \Gamma}^{ p}_{ pb}-H_{q}\check{\rm \Gamma}^{q}_{ab }
+\check{\rm \Gamma}^{ B}_{ Db}\check{\rm \Gamma}^{ D}_{aB }+\check{\rm \Gamma}^{ B}_{qb}\check{\rm \Gamma}^{q}_{aB}
\nonumber\\
&&
+\check{\rm \Gamma}^{ p}_{ Bb}\check{\rm \Gamma}^{B}_{ap}+\check{\rm \Gamma}^{ p}_{ qb}\check{\rm \Gamma}^{ q}_{ap}
-\check{\rm \Gamma}^{ R}_{ ab}\check{\rm \Gamma}^{ B}_{ BR}-\check{\rm \Gamma}^{ R}_{ ab}\check{\rm \Gamma}^{ q}_{ qR}
-\check{\rm \Gamma}^{ p}_{ ab}\check{\rm \Gamma}^{ R}_{ Rp}-\check{\rm \Gamma}^{ p}_{ ab}\check{\rm \Gamma}^{q}_{ qp}
\nonumber\\
&&-\mathbb C^{p}_{aD}\check{\rm \Gamma}^{D}_{pb}
\nonumber\\
&&+ H_{a}\check{\rm \Gamma}^{\alpha}_{ \alpha b}-L_{\alpha}\check{\rm \Gamma}^{\alpha}_{ab}+\check{\rm \Gamma}^{ B}_{ \alpha b}\check{\rm \Gamma}^{\alpha}_{ aB}+\check{\rm \Gamma}^{ p}_{\alpha b}\check{\rm \Gamma}^{\alpha}_{ap}
+\check{\rm \Gamma}^{\alpha}_{Bb}\check{\rm \Gamma}^{ B}_{a \alpha}+\check{\rm \Gamma}^{ \alpha}_{ pb}\check{\rm \Gamma}^{ p}_{a \alpha}
\nonumber\\
&&+\check{\rm \Gamma}^{ \alpha}_{ \mu b}\check{\rm \Gamma}^{\mu}_{a \alpha}-\check{\rm \Gamma}^{ R}_{ ab}\check{\rm \Gamma}^{ \alpha}_{\alpha R}-\check{\rm \Gamma}^{ p}_{ ab}\check{\rm \Gamma}^{ \alpha}_{\alpha p}-\check{\rm \Gamma}^{ \alpha}_{ ab}\check{\rm \Gamma}^{ R}_{R \alpha}-\check{\rm \Gamma}^{ \alpha}_{ ab}\check{\rm \Gamma}^{ p}_{p \alpha}
 -\check{\rm \Gamma}^{ \alpha}_{ ab}\check{\rm \Gamma}^{ \mu}_{\mu \alpha}
\nonumber\\
&&-\mathbb C^{\alpha}_{aB}\check{\rm \Gamma}^{B}_{\alpha b}-\mathbb C^{\alpha}_{ap}\check{\rm \Gamma}^{p}_{\alpha b},
\label{R_ab}
\end{eqnarray}
and in this case
\[
 \check R_{ab}=N^{\tilde E}_{ M}\,{}^{\rm H}\!R_{a\tilde E b}{}^{ M}+\check R'_{ab}. 
\]



The components  $\check R_{\alpha \beta}$ of the Ricci curvature tensor  are  defined by the following formula
$$\check R_{\alpha \beta}=L_{\alpha}\check{\rm \Gamma}^{\mathscr P}_{\mathscr P\beta}-\hat\partial_{\mathscr P}\check{\rm \Gamma}^{\mathscr P}_{\alpha\beta}+\check{\rm \Gamma}^{\mathscr D}_{\mathscr P\beta}\check{\rm \Gamma}^{\mathscr P}_{\alpha\mathscr D}-\check{\rm \Gamma}^{\mathscr E}_{\alpha\beta}\check{\rm \Gamma}^{\mathscr P}_{\mathscr P\mathscr E}-\mathbb C^{\mathscr E}_{\alpha\mathscr P}\check{\rm \Gamma}^{\mathscr P}_{\mathscr E\beta},$$
which means that
\begin{eqnarray}
 \check R_{ \alpha\beta}&=&L_{\alpha}\check{\rm \Gamma}^{\mu}_{\mu\beta}-L_{\mu}\check{\rm \Gamma}^{ \mu}_{\alpha\beta}+\check{\rm \Gamma}^{ \mu}_{\nu\beta}\check{\rm \Gamma}^{\nu}_{\alpha\mu}-\check{\rm \Gamma}^{ \mu}_{\alpha\beta}\check{\rm \Gamma}^{ \nu}_{\nu\mu}-c^{\mu}_{\alpha\nu}\check{\rm \Gamma}^{ \nu}_{\mu\beta}
 \nonumber\\
&&+L_{\alpha}\check{\rm \Gamma}^{B}_{B\beta}-H_{B}\check{\rm \Gamma}^{ B}_{ \alpha\beta}+L_{\alpha}\check{\rm \Gamma}^{q}_{q\beta}-H_{q}\check{\rm \Gamma}^{ q}_{ \alpha\beta}+
+\check{\rm \Gamma}^{ B}_{ Db}\check{\rm \Gamma}^{ D}_{\alpha\beta }+\check{\rm \Gamma}^{ B}_{qb}\check{\rm \Gamma}^{q}_{\alpha B}
\nonumber\\
&&+\check{\rm \Gamma}^{ B}_{\mu\beta}\check{\rm \Gamma}^{\mu}_{\alpha B}+\check{\rm \Gamma}^{ q}_{R\beta }\check{\rm \Gamma}^{R}_{\alpha q}+\check{\rm \Gamma}^{ q}_{p\beta }\check{\rm \Gamma}^{p}_{\alpha q}+\check{\rm \Gamma}^{ q}_{\mu\beta }\check{\rm \Gamma}^{\mu}_{\alpha q}+\check{\rm \Gamma}^{\mu}_{B\beta }\check{\rm \Gamma}^{B}_{\alpha \mu}+\check{\rm \Gamma}^{\mu}_{q\beta }\check{\rm \Gamma}^{q}_{\alpha \mu}
\nonumber\\
&&-\check{\rm \Gamma}^{ E}_{ \alpha\beta}(\check{\rm \Gamma}^{ R}_{ RE}+\check{\rm \Gamma}^{ q}_{ qE}+\check{\rm \Gamma}^{ \mu}_{ \mu E})
-\check{\rm \Gamma}^{ q}_{ \alpha\beta}(\check{\rm \Gamma}^{ R}_{ Rq}+\check{\rm \Gamma}^{ p}_{ pq}+\check{\rm \Gamma}^{ \mu}_{ \mu q})
\nonumber\\
&&-\check{\rm \Gamma}^{\mu}_{ \alpha\beta}(\check{\rm \Gamma}^{ R}_{ R\mu}+\check{\rm \Gamma}^{ p}_{ p\mu}).
\label{R_alphbet}
\end{eqnarray}
From this it follows that
$\check R_{\alpha \beta}$ can also be represented as
$$\check R_{\alpha \beta}=R_{\alpha \beta}+\check R'_{\alpha \beta},$$
where
\[
 R_{\alpha \beta}=L_{\alpha}\check{\rm \Gamma}^{\mu}_{\mu\beta}-L_{\mu}\check{\rm \Gamma}^{\mu}_{\alpha\beta}+\check{\rm \Gamma}^{\mu}_{\nu\beta}\check{\rm \Gamma}^{\nu}_{\alpha\mu}-\check{\rm \Gamma}^{\mu}_{\alpha\beta}\check{\rm \Gamma}^{\nu}_{\nu\mu}-c^{\mu}_{\alpha\nu}\check{\rm \Gamma}^{\nu}_{\mu\beta}
\]
are  the components of the Ricci curvature tensor  for a Lie group $\mathcal G$.
The scalar curvature of the orbit 
\[
 R_{\mathcal G}\equiv\tilde{d}^{\alpha\beta}R_{\alpha \beta}=\frac12{d}^{\mu\nu}c^{\sigma}_{\mu\alpha}c^{\alpha}_{\nu\sigma}+\frac14d_{\mu\sigma}d^{\alpha\beta}d^{\epsilon\nu}c^{\mu}_{\epsilon\alpha}c^{\sigma}_{\nu\beta}.
\]
\subsection{The calculation of the scalar curvature}
In the horizontal lift basis, the scalar curvature of the original manifold $\tilde{\mathcal P} $ is defined  by the following formula:
\[
\check R_{\tilde{\mathcal P}}=h^{AC}\check R_{AC}+h^{Ab}\check R_{Ab}+h^{bA}\check R_{bA}+h^{ab}\check R_{ab}+\tilde{d}^{\alpha\beta}\check R_{\alpha\beta}.
\]
Since the scalar curvature of the manifold ${\tilde{\Sigma}}$ with the metric tensor ${\tilde G}^{\rm H}_{\tilde{ A} \tilde{ B}} $ is 
\begin{equation}
{}^{\scriptscriptstyle \rm H}\!R_{\tilde{\Sigma}}=h^{SC}N^{\tilde E}_{ M}\,{}^{\rm H}\!R_{ S\tilde E C}{}^{ M}+h^{Sb}N^{\tilde E}_{ M}\,{}^{\rm H}\!R_{\tilde S\tilde E b}{}^{ M}+h^{bA}N^{\tilde E}_{ M}\,{}^{\rm H}\!R_{b\tilde E A}{}^{ M}+h^{ab}N^{\tilde E}_{ M}\,{}^{\rm H}\!R_{a\tilde E b}{}^{ M}
\label{scal_curv_sigma}
\end{equation}
and $\tilde{d}^{\alpha\beta}R_{\alpha \beta}=R_{\mathcal G}$, it remains to compute
\[
 h^{AC}\check R'_{AC}+h^{Ab}\check R'_{Ab}+h^{bA}\check R'_{bA}+h^{ab}\check R'_{ab}+
 \tilde{d}^{\alpha\beta}\check R'_{\alpha \beta}.
\]
An explicit representation of the terms of this sum is given in Appendix B.

To find the expression for the scalar curvature in the required form, we must regroup its terms   related to the tensors $\check R'_{ A C},\check R'_{ A b},\check R'_{b A}$ etc. Consider the terms from the first rows of these tensors.

Such a term from  $\check R'_{AC}$ is
\begin{eqnarray*}
&H_{A}\check{\rm \Gamma}^{\alpha}_{ \alpha C}&=\frac12H_A(\tilde d^{\alpha\nu}H_C\tilde d_{\alpha\nu})
=\frac12N^{\tilde E}_A\partial_{\tilde E}( d^{\alpha\nu}N^{\tilde F}_C\partial_{\tilde F} d_{\alpha\nu}).
\end{eqnarray*}
It can be written as  follows
\begin{eqnarray*}
&&\frac12(N^E_AN^B_C\partial_E(d^{\alpha\nu}\partial_Bd_{\alpha\nu})+N^E_AN^p_C\partial_E(d^{\alpha\nu}\partial_pd_{\alpha\nu})
\nonumber\\
 &&+N^q_AN^B_C\partial_q(d^{\alpha\nu}\partial_Bd_{\alpha\nu})+N^q_AN^p_C\partial_q(d^{\alpha\nu}\partial_pd_{\alpha\nu})
 \nonumber\\
 &&+N^E_AN^B_{C,E}(d^{\alpha\nu}\partial_Bd_{\alpha\nu})+     N^E_AN^p_{C,E}(d^{\alpha\nu}\partial_pd_{\alpha\nu})+N^q_AN^p_{C,q}(d^{\alpha\nu}\partial_pd_{\alpha\nu}).
\end{eqnarray*}

Multiplying the previous expression by $h^{AC}$, we get
\[
 \frac12[h^{EB}\partial_E(d^{\alpha\nu}\partial_Bd_{\alpha\nu})+h^{EC}N^B_{C,E}(d^{\alpha\nu}\partial_Bd_{\alpha\nu})+h^{EC}N^p_{C,E}(d^{\alpha\nu}\partial_pd_{\alpha\nu})].
\]
From  $\check R'_{Ab}$ we have
$$h^{Ab}H_A \check{\rm \Gamma}^{\alpha}_{ \alpha b}=\frac12h^{Ab}H_A(\tilde d^{\alpha\nu} H_b\tilde d_{\nu \alpha})=\frac12h^{Eb}\partial_E(d^{\alpha\nu} \partial_b\, d_{\nu \alpha}).$$
The next contribution comes from $\check R'_{bA}$:
\begin{align*}
h^{bA}H_b \check{\rm \Gamma}^{\alpha}_{ \alpha A}&=\frac12h^{bA}H_b(\tilde d^{\alpha\nu} H_A\tilde d_{\nu \alpha})\nonumber\\
&=\frac12[h^{bE}\partial_b(d^{\alpha\nu} \partial_E d_{\nu \alpha})+h^{bA}\partial_b(N^p_A)(d^{\alpha\nu} \partial_p d_{\nu \alpha})]\nonumber\\
&=\frac12h^{bE}\partial_b(d^{\alpha\nu} \partial_E d_{\nu \alpha}),{\rm since}\,h^{bA}N^p_{A,b}=-G^{LM}N^b_LN^A_{M,B}N^p_A=0.
\end{align*}
And from  $\check R'_{ab}$ we have
\[
h^{ab}H_a \check{\rm \Gamma}^{\alpha}_{ \alpha b}=\frac12h^{ab}H_a(\tilde d^{\alpha\nu} H_b\tilde d_{\nu \alpha})=\frac12h^{ab}\partial_a( d^{\alpha\nu} \partial_b d_{\nu \alpha}).
\]
From $\check R'_{\alpha\beta}$ we obtain the sum of two terms:
\begin{eqnarray*}
&&-\tilde d^{\alpha\beta}H_B\check{\rm \Gamma}^{B}_{\alpha\beta}=-\tilde d^{\alpha\beta}H_B(-\frac12h^{BC}H_C\tilde d_{\alpha\beta}-\frac12h^{Bb}H_b\tilde d_{\alpha\beta}),
 \\
 &&-\tilde d^{\alpha\beta}H_q\check{\rm \Gamma}^{q}_{\alpha\beta}=-\tilde d^{\alpha\beta}H_q(-\frac12h^{qC}H_C\tilde d_{\alpha\beta}-\frac12h^{qb}H_b\tilde d_{\alpha\beta}).
 \end{eqnarray*}

Consider the transformation of the first term (without the first factor).
\begin{align*}
 &\frac12H_B(h^{BC}H_C\tilde d_{\alpha\beta}+h^{Bb}H_b\tilde d_{\alpha\beta})=\frac12H_B(h^{BC})(H_C\tilde d_{\alpha\beta})+\frac12H_B(h^{Bb})(H_b\tilde d_{\alpha\beta})
 \nonumber\\
 &+\frac12h^{BC}(H_BH_C\tilde d_{\alpha\beta})+\frac12h^{Bb}(H_BH_b\tilde d_{\alpha\beta})
\nonumber\\
 &=\frac12H_B(h^{BC})(N^E_C\tilde {\mathscr D}_E\tilde d_{\alpha\beta}+N^p_C\tilde {\mathscr D}_p\tilde d_{\alpha\beta})+\frac12H_B(h^{Bb})(\tilde {\mathscr D}_b\tilde d_{\alpha\beta})
 \nonumber\\
 &+\frac12h^{BC}(N^R_B\tilde {\mathscr D}_R(H_C\tilde d_{\alpha\beta})+N^q_B\tilde {\mathscr D}_q(H_C\tilde d_{\alpha\beta}))\\
 &+\frac12h^{Bb}(N^R_B\tilde {\mathscr D}_R(\tilde {\mathscr D}_b\tilde d_{\alpha\beta})+N^q_B\tilde {\mathscr D}_q(\tilde {\mathscr D}_b\tilde d_{\alpha\beta})).\\
 \end{align*}
In this formulas by $\tilde {\mathscr D}_E\tilde d_{\alpha\beta}$ we denote the covariant derivative defined as
\[
 \tilde {\mathscr D}_E\tilde d_{\alpha\beta}=\partial_E\tilde d_{\alpha\beta}-c^{\sigma}_{\mu\alpha}\tilde{\mathscr A}^{\mu}_E\tilde d_{\sigma\beta}-c^{\sigma}_{\mu\beta}\tilde{\mathscr A}^{\mu}_E\tilde d_{\sigma\alpha},
\]
$\tilde{\mathscr A}^{\mu}_E=\bar{\rho}^{\mu}_{\alpha}{\mathscr A}^{\alpha}_E(Q^{\ast},\tilde f)$. Also, $\tilde {\mathscr D}_E\tilde d_{\alpha\beta}={\rho}^{\mu}_{\alpha}{\rho}^{\nu}_{\beta}{\mathscr D}_Ed_{\mu\nu}$.
  A similar definition is used for $\tilde {\mathscr D}_p$.
 
 Since $h^{BC}N^q_B=0$ and $h^{Bb}N^q_B=0$, and also $h^{BC}N^R_B=h^{RC}$, this expression can be rewritten as
\begin{align*}
 &\frac12H_B(h^{BC})(N^E_C\tilde {\mathscr D}_E\tilde d_{\alpha\beta}+N^p_C\tilde {\mathscr D}_p\tilde d_{\alpha\beta})+\frac12H_B(h^{Bb})(\tilde {\mathscr D}_b\tilde d_{\alpha\beta})\\
 &+\frac12h^{RC}N^E_{C,R}\tilde {\mathscr D}_E\tilde d_{\alpha\beta}+\frac12h^{RC}N^p_{C,R}\tilde {\mathscr D}_p\tilde d_{\alpha\beta}\\
 &+\frac12h^{RE}\tilde {\mathscr D}_R\tilde {\mathscr D}_E\tilde d_{\alpha\beta}+\frac12h^{Rb}\tilde {\mathscr D}_R\tilde {\mathscr D}_b\tilde d_{\alpha\beta}.
\end{align*}

The second term (also without the first factor) can be rewritten as
\begin{align*}
 \frac12(H_q(h^{qE})\tilde {\mathscr D}_E\tilde d_{\alpha\beta}+H_q(h^{qb})\tilde {\mathscr D}_b\tilde d_{\alpha\beta}+h^{qE}\tilde {\mathscr D}_q\tilde {\mathscr D}_E\tilde d_{\alpha\beta}+h^{qb}\tilde {\mathscr D}_q\tilde {\mathscr D}_b\tilde d_{\alpha\beta}).
\end{align*}
To obtain such an expression, we used the following properties: $H_q(h^{qC})N^E_C=H_q(h^{qE})$, since $N^E_C$ depends only on $Q^{\ast}$, and $H_q(h^{qC})N^p_C=0$.

Note also that
$$\tilde d^{\alpha\beta}\tilde {\mathscr D}_E\tilde d_{\alpha\beta}=d^{\alpha\beta}\partial_Ed_{\alpha\beta}=\partial_E(\ln \det(d_{\alpha\beta}))\equiv\sigma_E$$
for a semisimple Lie groups, and a similar equality holds for $\tilde d^{\alpha\beta}\tilde {\mathscr D}_p\tilde d_{\alpha\beta}$.

Hence after taking into account the factor $\tilde d^{\alpha\beta}$,   we will use a new representation for the following expressions:
$$\tilde d^{\alpha\beta}(N^E_C\tilde {\mathscr D}_E\tilde d_{\alpha\beta}+N^p_C\tilde {\mathscr D}_p\tilde d_{\alpha\beta})=N^E_C\sigma_E+N^p_C\sigma_p=\sigma_C.$$
The last equality due to to the identity derived in Appendix C. 


Further transformations of the terms of the obtained expressions will be carried out using the following identity:
\[
 \tilde d^{\alpha\beta}(\tilde {\mathscr D}_E\tilde {\mathscr D}_C\tilde d_{\alpha\beta})
 =\partial_E(d^{\alpha\beta}\partial_Cd_{\alpha\beta})+d^{\alpha'\epsilon'}d^{\nu'\beta'}({\mathscr D}_Ed_{\epsilon'\nu'})({\mathscr D}_Cd_{\alpha'\beta'}).
\]
Then $-\tilde d^{\alpha\beta}H_B\check{\rm \Gamma}^{B}_{\alpha\beta}$ becomes 
\begin{align*}
 &\frac12h^{RE}(\partial _R\partial_E\sigma+d^{\alpha'\epsilon'}d^{\nu'\beta'}({\mathscr D}_Rd_{\epsilon'\nu'})({\mathscr D}_Ed_{\alpha'\beta'}))\\
 &+\frac12h^{Rb}(\partial _R\partial_b\sigma+d^{\alpha'\epsilon'}d^{\nu'\beta'}({\mathscr D}_Rd_{\epsilon'\nu'})({\mathscr D}_bd_{\alpha'\beta'}))\\
 &+\frac12h^{RC}(N^E_{C,R}\sigma_E+N^p_{C,R}\sigma_p)
 +\frac12(H_B(h^{BC})\sigma_C+H_B(h^{Bb})\sigma_b).
 \end{align*}

The second term,  $-\tilde d^{\alpha\beta}H_q\check{\rm \Gamma}^{q}_{\alpha\beta}$, is written as follows: 
\begin{align*}
 &\frac12h^{qE}(\partial _q\partial_E\sigma+d^{\alpha'\epsilon'}d^{\nu'\beta'}({\mathscr D}_qd_{\epsilon'\nu'})({\mathscr D}_Ed_{\alpha'\beta'}))\\
 &+\frac12h^{qb}(\partial _q\partial_b\sigma+d^{\alpha'\epsilon'}d^{\nu'\beta'}({\mathscr D}_qd_{\epsilon'\nu'})({\mathscr D}_bd_{\alpha'\beta'}))\\
 & +\frac12(\partial_q(h^{qE})\sigma_E+\partial_q(h^{qb})\sigma_b).
 \end{align*}


In addition, we need formulas representing partial derivatives of $h^{\tilde D\tilde E}$.  They are a consequence of the general formula
\[
 \partial_{\tilde A}h^{\tilde D\tilde E}=-h^{\tilde B\tilde D}\,{}^{\rm H}{\rm \Gamma}^{\tilde E}_{\tilde B\tilde A}-h^{\tilde C\tilde E}\,{}^{\rm H}{\rm \Gamma}^{\tilde D}_{\tilde C\tilde A}.
\]
For particular indexes, this formula gives the following:
\[
 \partial_{E}h^{EC}=-h^{EB}\,{}^{\rm H}{\rm \Gamma}^{C}_{BE}-h^{Eb}\,{}^{\rm H}{\rm \Gamma}^{C}_{bE}-h^{BC}\,{}^{\rm H}{\rm \Gamma}^{E}_{BE}-h^{bC}\,{}^{\rm H}{\rm \Gamma}^{E}_{bE}
\]
\[
 \partial_{E}h^{Eb}=-h^{EA}\,{}^{\rm H}{\rm \Gamma}^{b}_{AE}-h^{Ea}\,{}^{\rm H}{\rm \Gamma}^{b}_{aE}-h^{Ab}\,{}^{\rm H}{\rm \Gamma}^{E}_{AE}-h^{ab}\,{}^{\rm H}{\rm \Gamma}^{E}_{aE}
\]
\[
 \partial_{q}h^{qC}=-h^{qA}\,{}^{\rm H}{\rm \Gamma}^{C}_{Aq}-h^{qa}\,{}^{\rm H}{\rm \Gamma}^{C}_{aq}-h^{AC}\,{}^{\rm H}{\rm \Gamma}^{q}_{Aq}-h^{aC}\,{}^{\rm H}{\rm \Gamma}^{q}_{aq}
\]
\[
 \partial_{q}h^{qp}=-h^{qA}\,{}^{\rm H}{\rm \Gamma}^{p}_{Aq}-h^{qa}\,{}^{\rm H}{\rm \Gamma}^{p}_{aq}-h^{Ap}\,{}^{\rm H}{\rm \Gamma}^{q}_{Aq}-h^{ap}\,{}^{\rm H}{\rm \Gamma}^{q}_{aq}.
\]
 The terms of $-\tilde d^{\alpha\beta}H_B\check{\rm \Gamma}^{B}_{\alpha\beta}$ that include  partial derivatives of $h^{\tilde A\tilde B}$ are
 \[
  \frac12[N^E_B\partial_E(h^{BM})\sigma_M+N^p_B\partial_p(h^{BM})\sigma_M+N^E_B\partial_E(h^{Bb})\sigma_b+N^p_B\partial_p(h^{Bb})\sigma_b].
 \]
 The terms of this expression can be expressed as follows:







 \begin{align}
&N^E_B\partial_E(h^{BM})=-h^{ED}\,{}^{\rm H}\Gamma^M_{DE}-h^{Ep}\,{}^{\rm H}\Gamma^M_{pE}-h^{TM}N^E_B\,{}^{\rm H}\Gamma^B_{TE}
 -h^{pM}N^E_B\,{}^{\rm H}\Gamma^B_{pE}

 \nonumber\\
 & N^p_B\partial_p(h^{BM})=-h^{TM}N^p_B\,{}^{\rm H}\Gamma^B_{Tp}-h^{qM}N^p_B\,{}^{\rm H}\Gamma^B_{qp}

 \nonumber\\
&N^E_B\partial_E(h^{Bb})=-h^{ED}\,{}^{\rm H}\Gamma^b_{DE}-h^{Ep}\,{}^{\rm H}\Gamma^b_{pE}-h^{Tb}N^E_B\,{}^{\rm H}\Gamma^B_{TE}
-h^{pb}N^E_B\,{}^{\rm H}\Gamma^B_{pE}
\nonumber\\
&N^p_B\partial_p(h^{Bb})=-h^{Tb}N^p_B\,{}^{\rm H}\Gamma^B_{Tp}-h^{qb}N^p_B\,{}^{\rm H}\Gamma^B_{qp}
  \label{dif_h^{BM}}
  \end{align}
 
 
 The terms of
$-\tilde d^{\alpha\beta}H_q\check{\rm \Gamma}^{q}_{\alpha\beta}$ 
that include  partial derivatives of $h^{\tilde A\tilde B}$ are
\[
  \frac12 (\partial_q h^{qE})\sigma_E+\frac12(\partial_q h^{qb})\sigma_b,
 \]
where
 \begin{align}


 &\partial_q(h^{qE})=-h^{qD}\,{}^{\rm H}\Gamma^E_{Dq}-h^{qp}\,{}^{\rm H}\Gamma^E_{pq}-h^{TE}\,{}^{\rm H}\Gamma^q_{Tq}-h^{pE}\,{}^{\rm H}\Gamma^q_{pq}\nonumber\\

 &\partial_q(h^{qb})=-h^{qD}\,{}^{\rm H}\Gamma^b_{Dq}-h^{qp}\,{}^{\rm H}\Gamma^b_{pq}-h^{Tb}\,{}^{\rm H}\Gamma^q_{Tq}-h^{pb}\,{}^{\rm H}\Gamma^q_{pq}
\label{dif_h^{qb}} 
 \end{align}

We see that the terms we have considered belong to certain types. There are second partial derivatives of $\sigma$, terms with Christoffel symbols and $\mathscr D\mathscr D$-terms.  Gathering terms with second partial derivatives of $\sigma$ in all Ricci tensors, we find that the sum of such terms is the initial part of the Laplace-Beltrami operator on the manifold $\tilde {\Sigma}$  applied to $\sigma$. 

To obtain the second part of this Laplacian, consisting of terms with Christoffel coefficients, it is necessary to take into account the contribution of not only the first two rows of Ricci tensors. In particularly, from $h^{AC}\check R'_{ A C}$ we have  two such terms:
\begin{align*}
 -h^{AC}\check \Gamma^E_{AC}\check \Gamma^{\mu}_{\mu E}&=-\frac12h^{AC}N^{\tilde R}_A\,{}^{\rm H}\Gamma ^E_{C\tilde R}(N^D_E\sigma_D+N^p_E\sigma_p)=-\frac12h^{AC}N^{\tilde R}_A\,{}^{\rm H}\Gamma ^E_{C\tilde R}\sigma_E\\
 &=-\frac12h^{AC}N^{ R}_A\,{}^{\rm H}\Gamma ^E_{CR}\sigma_E\;\;\rm{and}
\end{align*}
\begin{align*}
 -h^{AC}\check \Gamma^q_{AC}\check \Gamma^{\mu}_{\mu q}&=-\frac12h^{AC}N^{\tilde E}_A\,{}^{\rm H}\Gamma ^q_{C\tilde E}\sigma_q
 =-\frac12h^{AC}N^{ E}_A\,{}^{\rm H}\Gamma ^q_{CE}\sigma_q.
\end{align*}
From $h^{Ab}\check R'_{ A b}$ they are
\begin{align*}
 -h^{Ab}\check \Gamma^E_{Ab}\check \Gamma^{\alpha}_{\alpha E}=-\frac12h^{Db}\,{}^{\rm H}\Gamma ^E_{bD}\sigma_E,\;\;\;\;
 -h^{Ab}\check \Gamma^q_{Ab}\check \Gamma^{\alpha}_{\alpha q}=-\frac12h^{Eb}\,{}^{\rm H}\Gamma ^E_{bE}\sigma_q.
\end{align*}
The same contributions will be from $h^{bA}\check R'_{b A }$, that is,  from $-h^{bA}\check \Gamma^E_{bA}\check \Gamma^{\alpha}_{\alpha E}$ and $-h^{bA}\check \Gamma^q_{bA}\check \Gamma^{\alpha}_{\alpha q}$. 

The contribution from $h^{ab}\check R'_{ a b}$ is 
$$-h^{ab}\check \Gamma^R_{ab}\check \Gamma^{\alpha}_{\alpha R}=-\frac12h^{ab}\,{}^{\rm H}\Gamma ^R_{ab}\sigma_R,\;\;\;\;-h^{ab}\check \Gamma^p_{ab}\check \Gamma^{\alpha}_{\alpha p}=-\frac12h^{ab}\,{}^{\rm H}\Gamma ^p_{ab}\sigma_p.$$


With regard to the Christoffel terms of $d^{\alpha\beta}\check R'_{\alpha\beta}$ obtained after using the substitution (\ref{dif_h^{BM}}) and (\ref{dif_h^{qb}}), note that all ``inconvenient'' terms, that is, like the following: $h^{TM}N^E_B\,{}^{\rm H}\Gamma^B_{TE}\sigma_M$, are canceled by  terms belonging to the contribution in $\check R_{\tilde {\mathcal P}}$ from this Ricci tensor and the terms originating from $-\tilde d^{\alpha\beta}\check \Gamma^E_{\alpha\beta}\check \Gamma^{R}_{RE}$, $ -\tilde d^{\alpha\beta}\check \Gamma^E_{\alpha\beta}\check \Gamma^{q}_{qE}$, $-\tilde d^{\alpha\beta}\check \Gamma^q_{\alpha\beta}\check \Gamma^{R}_{Rq}$ and $-\tilde d^{\alpha\beta}\check \Gamma^q_{\alpha\beta}\check \Gamma^{p}_{pq}$.

Hence gathering all the terms with the second partial derivatives of sigma, as well as all the above terms with the Christoffel coefficients at the first partial derivatives of sigma, we get the Laplace-Beltrami  operator $\triangle_{\tilde{\Sigma}}^{\scriptscriptstyle\rm H}$ for the manifold with the ``horizontal metric'' ${\tilde G}^{\scriptscriptstyle\rm H}(Q^{\ast}{}^A,\tilde f ^p)$ (\ref{metric2cc}), acting on $\sigma =\ln \det d_{\alpha\beta}$.

Together with $\triangle_{\tilde{\Sigma}}^{\scriptscriptstyle\rm H}\sigma$, we get a term with derivatives of projectors:
\[
 h^{RC}(N^E_{C,R}\sigma_E+N^p_{C,R}\sigma_p).
\]
But this term is vanish by virtue of the identity obtained in Appendix C.

Next consider the $\mathscr D\mathscr D$-terms. They are in  all Ricci tensors. But $\mathscr D\mathscr D$-terms from $\tilde d^{\alpha\beta}\check R'_{\alpha\beta} $ are mutually cancelled. So we are left with contributions from other Ricci tensors. Their sum is determined by the following expression: 
$$||j||^2=\frac14h^{A'B'}d^{\alpha'\epsilon'}d^{\nu'\beta'}({\mathscr D}_A'd_{\epsilon'\nu'})({\mathscr D}_B'd_{\alpha'\beta'}),$$ which, as can be shown, is the ``square'' of the second fundamental form of the orbit.

Note that in this expression we have used a condensed notation of indices, in which the capital  letters with a prime as a superscript mean two indices, for example,  $A'=(A,a)$.
 

There are two more types of terms that we must account for as contributions to the scalar curvature. One of them represents a quadratic form in  the first partial derivatives of $\sigma$:
\[
 \frac14{\tilde G}^{\scriptscriptstyle\rm H}(\partial \ln d,\partial \ln d)\equiv\frac14(h^{AB}\sigma_A\sigma_B+h^{Ab}\sigma_A\sigma_b+h^{bA}\sigma_b\sigma_A+h^{ab}\sigma_a\sigma_b)
\]
This contribution only comes from the Ricci tensor $\tilde d^{\alpha\beta}\check R'_{\alpha\beta} $. 

The last type of terms whose contribution to the 
ture must be taken into account are $\mathscr F\mathscr F $- terms. 

As a result of our computation, we get the following representation for the scalar curvature $\check R_{\tilde{\mathcal P}}$
\begin{eqnarray}
{\check R}_{\tilde{\mathcal P}}={}^{\scriptscriptstyle\rm H}{ R}_{\tilde{\Sigma}}+{ R}_{\mathcal G}
+\frac14\mathscr F^2
+||j||^2
+\triangle_{\scriptscriptstyle \tilde {\Sigma}}^{\scriptscriptstyle\rm H}\ln d+\frac14{\tilde G}^{\scriptscriptstyle\rm H}(\partial \ln d,\partial \ln d), 
\label{curvat_itog}
\end{eqnarray}
where ${}^{\scriptscriptstyle\rm H}{\rm R}_{\tilde{\Sigma}}$ is from (\ref{scal_curv_sigma}), and  we have used the evident  symbolical notation for
$$\mathscr F^2\equiv h^{A'B'} h^{C'D'}d_{\mu\nu}{\mathscr F}^{\mu}_{A'C'}{\mathscr F}^{\nu}_{B'D'}$$.




\section{The geometrical representation of the Jacobian}
Having obtained the expression for curvature (\ref{curvat_itog}), we can try to compare some of its terms with the expression for the reduction Jacobian (\ref{jacobian_reduct}).  We see that in both expressions there are a similar terms. But there are terms in the Jacobian that seem superfluous. These are terms with normal curvature of the orbit space. This is actually the case, because it can be shown that these terms are zero.

Neglecting the common factor, we have the following expression in the reduction Jacobian:
\[
 \tilde{ J}=\tilde{\triangle}_{\tilde \Sigma}\sigma+\frac14<\partial\sigma,\partial\sigma>.
\]
Its first part is
\[
 (h^{AB}\sigma_{AB}+2h^{Ab}\sigma_{Ab}+h^{ab}\sigma_{ab}-h^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^A_{\tilde B\tilde M}\sigma_{A}-h^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^a_{\tilde B\tilde M}\sigma_{a}+2j^{A}_{\Roman 1}\sigma_A+2j^{a}_{\Roman 1}\sigma_a),
 \]
where
\[
  j^{A}_{\Roman 1}=\frac12h^{BM}N^A_{B,M}+\frac12h^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^A_{\tilde B\tilde M}-\frac12N^A_Ch^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^C_{\tilde B\tilde M}
 \]
and
 \[
 j^{a}_{\Roman 1}=\frac12h^{CM}N^a_{C,M}-\frac12N^a_Ch^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^C_{\tilde B\tilde M}.
\]
Substituting $j^{A}_{\Roman 1}$ and $j^{a}_{\Roman 1}$ into the expression of the first part of the Jacobian, we get
\[
 h^{AB}\sigma_{AB}+2h^{Ab}\sigma_{Ab}+h^{ab}\sigma_{ab}-h^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^A_{\tilde B\tilde M}\sigma_{A}-h^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^a_{\tilde B\tilde M}\sigma_{a}+h^{BM}N^A_{B,M}+h^{CM}N^a_{C,M}
\]
\[
 ={\triangle}_{\tilde \Sigma}^{\scriptscriptstyle\rm H}\sigma+h^{BM}N^A_{B,M}\sigma_A+h^{CM}N^a_{C,M}\sigma_a.
\]
To obtain this we have used the equality
\[
 N^A_Ch^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^C_{\tilde B\tilde M}\sigma_A+N^a_Ch^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^C_{\tilde B\tilde M}\sigma_a=h^{\tilde B\tilde M}\,{}^{\rm H}\Gamma^C_{\tilde B\tilde M}\sigma_C,
\]
which follows from
\[
 N^A_C\sigma_A+N^a_C\sigma_a=\sigma_C.
\]
The last equality is due to the Killing relation and the property of the semisimple Lie group. This identity was derived in Appendix C.
 
 So, we obtain the Laplacian and two additional terms. Recalling that when calculating the curvature, we have already met such additional terms to the Laplacian. But there it was concluded that these terms are also zero.
 This follows from the corollary of the identity given in Appendix C.

 So we can rewrite the formula (\ref{curvat_itog}) as 
 \[
  {\check R}_{\tilde{\mathcal P}}={}^{\scriptscriptstyle\rm H}{ R}_{\tilde{\Sigma}}+{ R}_{\mathcal G}
+\frac14\mathscr F^2
+||j||^2+\tilde{ J}.
 \]
This means that The Hamilton operator of the Schr\"odinger equation on the manifold $\tilde{\Sigma}$ has the following form:
\[
 \hat H_{\tilde {\Sigma}}=-\frac{\hbar^2}{2m}{\triangle}_{\tilde \Sigma}^{\scriptscriptstyle\rm H}+\frac{\hbar ^2}{8m}\Bigl[{\check R}_{\tilde{\mathcal P}}-{}^{\scriptscriptstyle\rm H}{ R}_{\tilde{\Sigma}}-{ R}_{\mathcal G}
-\frac14\mathscr F^2
-||j||^2\Bigr]+\tilde{ V}.
\]
In conclusion, we note that ${\check R}_{\tilde{\mathcal P}}={ R}_{{\mathcal P}}$, since in our case we have the product manifold, $R_{\mathcal V}=0$.



\section*{Appendix A}
\section*{Projectors, their properties,\\ some identities and Killing relations}
\setcounter{equation}{0}
\defA.\arabic{equation}{A.\arabic{equation}}
\subsection*{The horizontal projector ${\tilde \Pi}^{\tilde A}_{\tilde B}$}

By definition
$${\tilde \Pi}^{\tilde A}_{\tilde B}={\delta}^{\tilde A}_{\tilde B}-K^{\tilde A}_{\alpha}d^{\alpha \beta}K^{\tilde D}_{\beta}G_{\tilde D \tilde B}.$$
Its components are
$${\tilde \Pi}^{\tilde A}_{\tilde B}\equiv({\tilde \Pi}^A_B,{\tilde \Pi}^A_n,{\tilde \Pi}^m_B,{\tilde \Pi}^m_n),$$

$${{\tilde \Pi}}^{A}_{B}={\delta}^A_B-K^{A}_{\alpha}d^{\alpha \beta}K^{D}_{\beta}G_{D B},\;\;\;\;\;
{{\tilde \Pi}}^{A}_{n}=-K^{A}_{\mu}d^{\mu \nu}K^{p}_{\nu}G_{pn},$$

$${{\tilde \Pi}}^{m}_{B}=-K^{m}_{\mu}d^{\mu \nu}K^{D}_{\nu}G_{D B},\;\;\;\;
{{\tilde \Pi}}^{m}_{n}={\delta}^m_n-K^{m}_{\mu}d^{\mu \nu}K^{r}_{\nu}G_{rn}.$$

The main properties:
\[
 {\tilde \Pi}^{\tilde A}_{\tilde B}{\tilde \Pi}^{\tilde B}_{\tilde C}={\tilde \Pi}^{\tilde A}_{\tilde C}, \;\;\;
{{\tilde \Pi}}^{\tilde L}_{\tilde B}N^{\tilde A}_{\tilde L}=N^{\tilde A}_{\tilde B},\;\;\;
{{\tilde \Pi}}_{\tilde L}^{\tilde A}N_{\tilde C}^{\tilde L}={{\tilde \Pi}}_{\tilde C}^{\tilde A},\;\;\;
{{\tilde \Pi}}^{\tilde A}_{\tilde E}K^{\tilde E}_{\alpha}=0.
\]

\subsection*{The projector $N^{\tilde A}_{\tilde B}$}

Its components are
$$N^{\tilde A}_{\tilde C}\equiv(N^A_B,N^A_n,N^m_B,N^m_n),$$ 

$$N^A_B={\delta}^A_B-K^A_{\mu}({\Phi}^{-1})^{\mu}_{\nu}\,{\chi}^{\nu}_B,\;\;
N^A_n=0,\;\;  N^m_B=-K^m_{\alpha}({\Phi}^{-1})^{\alpha}_{\mu}\,{\chi}^{\mu}_B,
\;\;  N^m_n={\delta}^m_n.$$
The main properties:
$$N^{\tilde A}_{\tilde B} N^{\tilde B}_{\tilde C}=N^{\tilde A}_{\tilde C},\;\;\;
(P_{\bot})^{\tilde L}_{\tilde B}\,N^{\tilde C}_{\tilde L}=(P_{\bot})^{\tilde C}_{\tilde B},\;\;\;      
N^{\tilde A}_{\tilde B}\,(P_{\bot})^{\tilde C}_{\tilde A}=N^{\tilde C}_{\tilde B}.
$$




\subsection*{The projector $(P_{\bot})^{\tilde A}_{\tilde B}$}


Its components are
\[
 (P_{\bot})^{\tilde A}_{\tilde B}\equiv(\,(P_{\bot})^A_B,\, (P_{\bot})^A_n,\, (P_{\bot})^m_B,\,(P_{\bot})^m_n\,),
\]
\[
 (P_\bot)^A_B=\delta^A_B-\chi ^{\alpha}_{B}\,(\chi \chi ^{\top})^{-1}{}^{\beta}_{\alpha}\,(\chi ^{\top})^A_{\beta},\;(\chi ^{\top})^A_{\mu}=G^{AB}{\gamma}_{\mu\nu}\chi ^{\nu}_B,\;{\gamma}_{\mu\nu}=K^A_{\mu}G_{AB}K^B_{\nu}, 
\]
 $(P_{\bot})^{A}_{n}=0,\;\;(P_{\bot})^{m}_{B}=0,\;\;(P_{\bot})^{m}_{n}=\delta ^m_n.$








\subsection*{Some identities derived from $K^{\tilde R}_{\gamma}{\tilde G}^H_{{\tilde R\tilde A} }=0$}

(1) $\tilde A\to A$
\[
 K^R_{\gamma}{\tilde G}^H_{RA}+K^p_{\gamma}{\tilde G}^H_{pA}=0\;\;\;\;\;{\rm or}\;\;\;K^{\tilde R}_{\gamma}{\tilde G}^H_{{\tilde R}A}=0
\]
\[
(A)\;\;\;\;K^R_{\gamma, D}{\tilde G}^H_{RA}+K^R_{\gamma}{\tilde G}^H_{RA,D}+K^p_{\gamma}{\tilde G}^H_{pA,D}=0.
\]
\[
\!\!\!\!(D)\;\;\;\;{\tilde G}^H_{AR,q}K^R_{\gamma}+{\tilde G}^H_{Ap,q}K^p_{\gamma}+{\tilde G}^H_{Ap}K^p_{\gamma,q}=0. 
\]

$\!\!\!\!\!\!\!(2)$ $\tilde A\to p$
\[
 {\tilde G}^H_{pq}K^q_{\mu}+{\tilde G}^H_{pA}K^A_{\mu}=0\;\;\;\;\;{\rm or}\;\;\;{\tilde G}^H_{p{\tilde R}}K^{\tilde R}_{\mu}=0
\]
\[
\!\!\!\!(B)\;\;\;\;{\tilde G}^H_{pR,n}K^R_{\mu}+{\tilde G}^H_{pr,n}K^r_{\mu}+{\tilde G}^H_{pr}K^r_{\mu,n}=0.
\]
\[
\!\!\!(C)\;\;\;\;{\tilde G}^H_{pR,D}K^R_{\mu}+{\tilde G}^H_{pr,D}K^r_{\mu}+{\tilde G}^H_{pR}K^R_{\mu,n}=0. 
\]
These relations are obtained as a result of the differentiations.



\subsection*{Killing relations for the horizontal metric $G^H_{\tilde A \tilde B}$}









\[
 {\tilde G}^H_{\tilde A \tilde B,\tilde D}K^{\tilde D}_{\alpha}+{\tilde G}^H_{\tilde R \tilde B}K^{\tilde R}_{\alpha,\tilde A}+{\tilde G}^H_{\tilde A \tilde R}K^{\tilde R}_{\alpha ,\tilde B}=0
\]

\begin{flushleft}
         $\;\;\;\;\;\;{\tilde A}\to A,\;\;\;{\tilde B} \to B$
\end{flushleft}

\begin{flushleft}
         $\rm {{\Roman 1}}\,.$ $\;\;\;\;{\tilde G}^H_{A B,D}K^{D}_{\alpha}+{\tilde G}^H_{A B,p}K^{p}_{\alpha}+{\tilde G}^H_{R B}K^{R}_{\alpha,A}+{\tilde G}^H_{AR }K^{R}_{\alpha,B}=0.$
\end{flushleft}



\begin{flushleft}
         $\;\;\;\;\;\;{\tilde A}\to p,\;\;\;{\tilde B} \to q$
\end{flushleft}

 \begin{flushleft}
         $\rm {{\Roman 2}}$\,. $\;\;\;\; {\tilde G}^H_{pq,D}K^{D}_{\alpha}+{\tilde G}^H_{pq,r}K^{r}_{\alpha}+{\tilde G}^H_{rq}K^{r}_{\alpha,p}+{\tilde G}^H_{pr }K^{R}_{\alpha,q}=0.$
\end{flushleft}





\begin{flushleft}
         $\;\;\;\;\;\;{\tilde A}\to p,\;\;\;{\tilde B} \to B$
\end{flushleft}

 \begin{flushleft}
         $\rm {{\Roman 3}}$\,. $\;\;\;\;{\tilde G}^H_{pB,D}K^{D}_{\alpha}+{\tilde G}^H_{pB,r}K^{r}_{\alpha}+{\tilde G}^H_{rB}K^{r}_{\alpha,p}+
{\tilde G}^H_{pR}K^{R}_{\alpha,B}=0.$
\end{flushleft}




\begin{flushleft}
         $\;\;\;\;\;\;{\tilde A}\to B,\;\;\;{\tilde B} \to p$
\end{flushleft}

\begin{flushleft}
         $\rm {{\Roman 4}}$\,. $\;\;\;\;{\tilde G}^H_{Bp,D}K^{D}_{\alpha}+{\tilde G}^H_{Bp,r}K^{r}_{\alpha}+{\tilde G}^H_{Rp}K^{R}_{\alpha,B}+{\tilde G}^H_{Br}K^{r}_{\alpha,p}=0.$
\end{flushleft}

$\rm{\Roman 4}=\rm{\Roman 3}$

