Reports of the Academy of Sciences of the USSR

1. Volume 132, No. 6

MATHEMATICS

L. V. SABININ

ON AN EXPLICIT EXPRESSION OF THE CONNECTION FORMS OF A QUASISYMMETRIC SPACE THROUGH THE VALUES OF THE CURVATURE AND TORSION TENSORS AT A CERTAIN POINT

(Presented by Academician S. L. Sobolev on 20 I 1960)

A quasisymmetric space of affine connection is a space of affine connection with covariantly constant curvature and torsion tensors. P. K. Rashevskii showed that every quasisymmetric space is a homogeneous space \((^1)\), and every homogeneous space \(G/H\) with transformation group \(G\) and stationary group \(H\), and such that the Cartan metric of the group \(G\) on \(H\) is nondegenerate, is quasisymmetric \((^2)\). The connection in an affine connection space \(A_n\) in a moving frame is specified by smooth linear differential forms \(\omega^i(d)\) and \(\omega^i_j(d)\), depending on the coordinates of the space \(u^1, u^2,\ldots,u^n\), on the secondary parameters of the moving frame \(u^{n+1},\ldots,u^L\), and on the differentials \(du^i, du^\alpha\) \((i=1,2,\ldots,n;\ \alpha=n+1,\ldots,L)\); the forms \(\omega^i\) are linearly independent forms into which only the differentials \(du^i\) enter.

Introduce the notation:
\[ [\delta\omega(d)] = \delta\omega(d)-d\omega(\delta), \]
\[ [\omega_1(\delta)\omega_2(d)] = \omega_1(\delta)\omega_2(d)-\omega_1(d)\omega_2(\delta). \tag{1} \]

In the affine connection space \(A_n\) the structure equations hold:
\[ [\delta\omega^i(d)] + [\omega^i_k(\delta)\omega^k(d)] = S^i_{pq}\omega^p(\delta)\omega^q(d); \tag{2} \]
\[ [\delta\omega^i_j(d)] + [\omega^i_k(\delta)\omega^k_j(d)] = -R^i_{\cdot j,pq}\omega^p(\delta)\omega^q(d), \tag{3} \]
where \(R^i_{\cdot j,pq}\) is the curvature tensor; \(S^i_{pq}\) is the torsion tensor of the space \(A_n\) in the moving frame; \(\delta\) and \(d\) are symbols of differentials of infinitesimal linearly independent displacements, moreover such that \(\delta df-d\delta f=0\) for any at least twice continuously differentiable \(f=f(u^1,\ldots,u^n,u^{n+1},\ldots,u^L)\). As shown in \((^1)\), the structure equations of the group of motions \(G\) of the quasisymmetric space \(A_n=G/H\) have the form
\[ [\delta\psi^i(d)] + d^i_{k\gamma}[\psi^\gamma(\delta)\psi^k(d)] = S^i_{pq}\psi^p(\delta)\psi^q(d); \tag{4} \]
\[ [\delta\psi^\gamma(d)] + c^\gamma_{\beta\alpha}\psi^\alpha(\delta)\psi^\beta(d) = -b^\gamma_{pq}\psi^p(\delta)\psi^q(d), \tag{5} \]
\[ p,q,i,k,i=1,2,\ldots,n;\qquad \gamma,\alpha,\beta=n+1,\ldots,r. \]

In addition,
\[ \omega^i_j(d)=d^i_{j\alpha}\psi^\alpha(d),\qquad \omega^i(d)=\psi^i(d). \tag{6} \]

The quantities \(S^i_{pq}, d^i_{k\gamma}, -b^\gamma_{pq}, c^\gamma_{\beta\alpha}\) are the structural constants of the group \(G\). Comparing (4), (5), (6) with (1) and (2), we obtain
\[ R^i_{\cdot j,pq}=d^i_{j\gamma}b^\gamma_{pq}. \tag{7} \]

If the structure equations of a certain Lie group have the form

\[ [d\psi^I(d)] = B^I_{JK}\psi^J(\delta)\psi^K(d); \qquad I,J,K=1,2,\ldots,r, \tag{8} \]

then, as is well known from the general theory of Lie groups \({}^{(3)}\), the coefficients \(\psi^I_K(u)\) of the basic forms \(\psi^I(d)=\psi^I_K(u)\,du^K\) in canonical coordinates \(u^K\) \((K=1,2,\ldots,r)\) have, in matrix notation, the form

\[ \bar{\psi}(u)=\sum_{m=0}^{\infty}\frac{2}{(m+1)!}\,\bar{B}^{\,m},\qquad \bar{\psi}(u)=\|\psi^I_K(u)\|,\qquad \bar{B}=\|B^I_{KJ}u^J\|. \tag{9} \]

We pass to the canonical frame (one at each point) by means of the condition

\[ u^\alpha=0,\qquad \alpha=n+1,\ n+2,\ldots,r. \tag{10} \]

Since in our case \(u^\alpha=0\), in order to find \(\psi^i(d)\), \(\psi^\alpha(d)\) it is enough to find \(\psi^i_j\), \(\psi^\alpha_i\). From (9) it is seen that for this it is necessary to consider \((\bar{B}^{\,m})^i_j\), \((\bar{B}^{\,m})^\alpha_j\). We shall use the obvious equalities

\[ (\bar{B}^{m+1})^i_j=(\bar{B})^i_k(\bar{B}^{m})^k_j+ (\bar{B})^i_\alpha(\bar{B}^{m})^\alpha_j,\qquad (\bar{B}^{m+1})^\alpha_j=(\bar{B})^\alpha_k(\bar{B}^{m})^k_j+ (\bar{B})^\alpha_\beta(\bar{B}^{m})^\beta_j. \tag{11} \]

For the structures (4) and (5), owing to condition (10), we have

\[ \underset{0}{B}{}^k_j=S^k_{jl}u^l,\qquad \underset{0}{B}{}^\alpha_j=-b^\alpha_{jl}u^l,\qquad \underset{0}{B}{}^i_\alpha=a^i_{\alpha l}u^l,\qquad \underset{0}{B}{}^\alpha_\beta=0, \tag{12} \]

where \(\underset{0}{B}{}^I_J\) is the value of the matrix \(B^I_J\) when \(u^\alpha=0\). Therefore (11) has the form

\[ \left(\underset{0}{\bar{B}}{}^{\,m+1}\right)^i_j = \underset{0}{B}{}^i_k \left(\underset{0}{\bar{B}}{}^{\,m}\right)^k_j + \underset{0}{B}{}^i_\beta \left(\underset{0}{\bar{B}}{}^{\,m}\right)^\beta_j, \qquad \left(\underset{0}{\bar{B}}{}^{\,m+1}\right)^\alpha_j = \underset{0}{B}{}^\alpha_k \left(\underset{0}{\bar{B}}{}^{\,m}\right)^k_j . \tag{13} \]

Eliminating \(\left(\underset{0}{\bar{B}}{}^{\,m}\right)^\beta_j\) from (13), we obtain

\[ \left(\underset{0}{\bar{B}}{}^{\,m+2}\right)^i_j = S^i_k \left(\underset{0}{\bar{B}}{}^{\,m+1}\right)^k_j + N^i_k \left(\underset{0}{\bar{B}}{}^{\,m}\right)^k_j, \qquad N^i_k=R^i{}_{.l,ks}u^l u^s,\qquad S^i_k=\underset{0}{B}{}^i_k . \tag{14} \]

The expressions (14) make sense starting with \(m=0\). Using also (6), (12), (14), we obtain

\[ \omega^i_{jk}=a^i_{j\alpha}\psi^\alpha_k = -\,R^i{}_{.j,lq}u^q \sum_{m=0}^{\infty}\frac{1}{(m+2)!} \left(\underset{0}{\bar{B}}{}^{\,m}\right)^l_k . \tag{15} \]

Thus,

\[ \omega^i=\sum_{m=0}^{\infty}\frac{1}{(m+1)!}\,b^i_m,\qquad \omega^i_j=-R^i{}_{.j,lq}u^q\varphi^l, \]

\[ \varphi^l=\sum_{m=0}^{\infty}\frac{1}{(m+2)!}\,b^l_m,\qquad b^l_m=\left(\underset{0}{\bar{B}}{}^{\,m}\right)^l_k\,du^k, \tag{16} \]

where, with the help of (14) and (12), we have

\[ b^i_{m+2}=S^i_k b^k_{m+1}+N^i_k b^k_m,\qquad b^k_0=du^k,\qquad b^k_1=S^k_j\,du^j,\qquad i,j,k=1,2,\ldots,n. \tag{17} \]

To solve equation (17), let us pass to the vector space of \(2n\) variables \(\xi^{a}\) \((a=1,2,\ldots,2n)\). Consider vectors \(\underset{m}{\xi^{a}}\) such that \(\underset{m}{\xi^{i}}=\underset{m}{b^{i}}\), \(\underset{m}{\xi^{n+i}}=\underset{m+1}{b^{i}}\) \((i=1,2,\ldots,n)\), and a matrix \(D^{a}_{b}\) such that \(D^{i}_{j}=0\), \(D^{n+i}_{n+j}=S^{i}_{j}\), \(D^{n+i}_{j}=N^{i}_{j}\), \(D^{i}_{n+j}=\delta^{i}_{j}\), \(i=1,2,\ldots,n\). Then equation (17) can be rewritten in the form

\[ \underset{m+1}{\xi^{a}}=D^{a}_{b}\underset{m}{\xi^{b}}, \qquad \underset{0}{\xi^{i}}=du^{i}, \qquad \underset{0}{\xi^{n+i}}=S^{i}_{j}du^{j}. \tag{18} \]

From (18) it follows that

\[ \underset{s}{\xi^{a}}=(\overline{D}^{\,s+1})^{a}_{b}\xi^{b}, \qquad \xi^{i}=0, \qquad \xi^{n+i}=du^{i}, \qquad i=1,2,\ldots,n. \tag{19} \]

Introduce a form \(\Omega^{a}\) such that

\[ \Omega^{i}=\varphi^{i}, \qquad \Omega^{n+i}=\omega^{i}. \tag{20} \]

Then, obviously, the expansion

\[ \Omega^{a}=\xi^{a}+\sum_{m=0}^{\infty}\frac{1}{(m+2)!}\,\underset{m}{\xi^{a}} \tag{21} \]

holds.

Let us note that if a function \(f(\overline{A})\) is representable by a power series \(\sum_{m=1}^{\infty} a_m \overline{A}^{m}\), then division by \(\overline{A}\), \(\frac{1}{\overline{A}}f(\overline{A})\), is possible even if \(\overline{A}\) is a degenerate matrix. By \(\frac{1}{\overline{A}}f(\overline{A})\) we shall mean the series \(\sum_{m=1}^{\infty} a_m \overline{A}^{m-1}\).

Taking (19) into account, we obtain further

\[ \overline{\Omega} = \binom{\overline{\varphi}}{\overline{\omega}} = \left(\frac{e^{\overline{D}}-\overline{E}}{\overline{D}}\right) \binom{\overline{0}}{\overline{du}}, \qquad \overline{\Omega}_{1} = \frac{1}{2} \left( \frac{e^{\overline{D}}-\overline{E}}{\overline{D}} + \frac{e^{(\overline{B}\overline{D}\overline{B})}-\overline{E}}{(\overline{B}\overline{D}\overline{B})} \right) \binom{\overline{0}}{\overline{du}} = \binom{\overline{0}}{\overline{\omega}} \]

\[ \overline{\Omega}_{2} = \frac{1}{2} \left( \frac{e^{\overline{D}}-\overline{E}}{\overline{D}} - \frac{e^{(\overline{B}\overline{D}\overline{B})}-\overline{E}}{(\overline{B}\overline{D}\overline{B})} \right) \binom{\overline{0}}{\overline{du}} = \binom{\overline{\varphi}}{\overline{0}}, \qquad \overline{B}= \begin{pmatrix} -\overline{I} & \overline{0}\\ \overline{0} & \overline{I} \end{pmatrix}, \qquad \overline{E}= \begin{pmatrix} \overline{I} & \overline{0}\\ \overline{0} & \overline{I} \end{pmatrix} \tag{22,} \]

\[ \overline{D}= \begin{pmatrix} \overline{0} & \overline{I}\\ \overline{N} & \overline{S} \end{pmatrix}, \qquad \overline{B}\,\overline{D}\,\overline{B} = - \begin{pmatrix} \overline{0} & \overline{I}\\ \overline{N} & -\overline{S} \end{pmatrix} = -\overline{\mathfrak{D}}, \qquad \overline{S}=\|S^{i}_{jq}u^{q}\|, \qquad \overline{I}=\|\delta^{i}_{j}\|, \]

\[ \overline{N}=\|R^{i}_{\cdot j,ls}u^{l}u^{s}\|, \qquad \overline{\varphi}= \begin{pmatrix} \varphi^{1}\\ \vdots\\ \varphi^{n} \end{pmatrix}, \qquad \overline{\omega}= \begin{pmatrix} \omega^{1}\\ \vdots\\ \omega^{n} \end{pmatrix}, \qquad \overline{du}= \begin{pmatrix} du^{1}\\ \vdots\\ du^{n} \end{pmatrix}, \qquad \omega^{i}_{j}=-R^{i}_{\cdot j,lq}u^{q}\varphi^{l}. \]

If the quasi-symmetric space \(A_n\) under consideration admits the existence of a covariantly constant nondegenerate positive-definite tensor field \(g_{ij}(u)\), then, as can be shown, the basic forms \(\omega^{i}\) can be chosen so that

\[ ds^{2}=\sum_{i=1}^{n}(\omega^{i})^{2} = (\overline{0}\ \overline{\omega})\cdot \binom{\overline{0}}{\overline{\omega}} = \overline{\Omega}_{1}^{*}\cdot\overline{\Omega}_{1}. \tag{23} \]

We shall make use of this choice of basis. Then

\[ ds^{2} = \frac{1}{4}\, \overline{\xi}^{\,*} \left( \frac{e^{\overline{D}^{*}}-\overline{E}}{\overline{D}^{*}} - \frac{e^{-\overline{\mathfrak{D}}^{*}}-\overline{E}}{\overline{\mathfrak{D}}^{*}} \right) \left( \frac{e^{\overline{D}}-\overline{E}}{\overline{D}} - \frac{e^{-\overline{\mathfrak{D}}}-\overline{E}}{\overline{\mathfrak{D}}} \right) \overline{\xi}, \qquad \overline{\xi}= \binom{\overline{0}}{\overline{du}}. \tag{24} \]

Expressions analogous to (24), but somewhat more complicated, can also be given in the case of an indefinite metric.

If, in particular, the space is symmetric, then

\[ \bar{\bar S}=0,\qquad \overline D= \begin{pmatrix} \bar{\bar 0} & \bar{\bar I}\\ \bar{\bar N} & \bar{\bar 0} \end{pmatrix} =\overline{\mathscr D},\qquad \bar{\bar N}^{*}=\bar{\bar N},\qquad \overline D^{\,2}= \begin{pmatrix} \bar{\bar N} & \bar{\bar 0}\\ \bar{\bar 0} & \bar{\bar N} \end{pmatrix}, \]

and thus:

\[ \begin{aligned} \bar\omega &=\sum_{m=0}^{\infty}\frac{\bar{\bar N}^{m}\,d\bar u}{(2m+1)!} =\frac{\operatorname{sh}\sqrt{\bar{\bar N}}}{\sqrt{\bar{\bar N}}}\,d\bar u,\\[6pt] \bar\varphi &=\sum_{m=0}^{\infty}\frac{\bar{\bar N}^{m}\,d\bar u}{(2m+2)!} =\frac{\operatorname{ch}\sqrt{\bar{\bar N}}-\bar{\bar I}}{\bar{\bar N}}\,d\bar u . \end{aligned} \tag{25} \]

Using (24), we obtain

\[ ds^{2}=\bar\omega^{*}\bar\omega =d\bar u^{*}\left(\frac{\operatorname{sh}^{2}\sqrt{\bar{\bar N}}}{\bar{\bar N}}\right)d\bar u =d\bar u^{*}\left(\frac{\operatorname{ch}\left(2\sqrt{\bar{\bar N}}\right)-\bar{\bar I}}{2\bar{\bar N}}\right)d\bar u . \tag{26} \]

Formulas equivalent to (26) were first obtained by P. A. Shirokov\({}^{4}\), but by an entirely different method.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
18 I 1960

REFERENCES

\({}^{1}\) P. K. Rashevsky, Proceedings of the Seminar on Vector and Tensor Analysis, vol. VIII, 82 (1950).
\({}^{2}\) P. K. Rashevsky, Proceedings of the Seminar on Vector and Tensor Analysis, vol. IX, 49 (1952).
\({}^{3}\) N. G. Chebotarev, Theory of Lie Groups, 1940.
\({}^{4}\) P. A. Shirokov, Matematicheskii Sbornik, 41 (83), no. 3, 361 (1957).