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MATHEMATICS
N. S. SINYUKOV
ON AN INVARIANT TRANSFORMATION OF RIEMANNIAN SPACES WITH COMMON GEODESICS
(Presented by Academician A. N. Kolmogorov on 1 XII 1960)
In the paper, first any two Riemannian spaces with common geodesics are considered. From the metric tensors of these spaces and the function determining their geodesic mapping, the metric tensors of another pair of Riemannian spaces, also with common geodesics, are constructed. Then the spaces obtained by applying this transformation to spaces of constant curvature and, generally speaking, not themselves spaces of constant curvature, are considered. Finally, for somewhat more general spaces, the existence is proved of a nontrivial geodesic mapping onto Riemannian spaces with a metric tensor of a specified form.
- Let the Riemannian space \(V_n\) with metric tensor \(g_{ij}\) \((i,j=1,2,\ldots,n)\) admit a nontrivial geodesic mapping onto the Riemannian space \(\overline V_n\) with metric tensor \(\overline g_{ij}\), determined by the gradient \(\psi_i \ne 0\), i.e.
\[ \overline g_{ij,k}=2\psi_k \overline g_{ij}+\psi_i\overline g_{kj}+\psi_j\overline g_{ki}, \tag{1} \]
where the comma denotes covariant differentiation in \(V_n\). Setting
\[ \check g_{ij}=e^{2\psi}\overline g^{\alpha\beta}g_{\alpha i}g_{\beta j}, \tag{2} \]
where \(\overline g^{ij}\) are the elements of the inverse matrix to \(\|\overline g_{ij}\|\), from (1) we obtain
\[ \check g_{ij,k}=\lambda_i g_{kj}+\lambda_j g_{ki},\qquad \lambda^i=-\psi_\alpha \overline g^{\alpha i} e^{2\psi}. \tag{3} \]
It is easy to see that the reverse passage from (3) to (1) is always possible if \(|\check g_{ij}|\ne 0\). Equations (3) show that the tensor \(\check g_{ij}\) is determined from them by \(\lambda_i\) and \(g_{kj}\) up to a covariantly constant tensor in \(V_n\). Consequently, the problem of geodesic mapping of \(V_n\) is also solved up to covariantly constant tensors. This fact, in a special case, was found earlier in (ยน). Introducing the tensor
\[ \hat g_{ij}=e^{2\varphi}\check g_{ij}, \tag{4} \]
from (3) we obtain in \(\check V_n\), with metric tensor \(\check g_{ij}\), equations of the form (1).
Thus, the following has been proved.
Theorem 1. If the Riemannian space \(V_n\) with metric tensor \(g_{ij}\) admits a nontrivial geodesic mapping onto \(\overline V_n\) with metric tensor \(\overline g_{ij}\), determined by the gradient \(\psi_i\), then the Riemannian space \(\check V_n\) admits a nontrivial geodesic mapping onto \(\hat V_n\), determined
by the same vector \(\psi_i\), and their metric tensors are determined by formulas (2) and (4), respectively.
The transformation thus found of a pair of Riemannian spaces with common geodesics will, for brevity, be called a \(\Gamma\)-transformation. Since under the conditions of the theorem \(\bar V_n\) admits a geodesic mapping onto \(V_n\), determined by the vector \(-\psi_i\), we similarly obtain one more pair of spaces with common geodesics.
Verification shows that application of a \(\Gamma\)-transformation to the spaces \(\check V_n\) and \(\hat V_n\) returns us to \(V_n\) and \(\bar V_n\), i.e., the \(\Gamma\)-transformation is involutive in character.
- Let us consider the \(\Gamma\)-transformation as applied to spaces \(V_n\) of constant curvature \(K\) and \(\bar V_n\) of constant curvature \(\bar K\) \((n>2)\). Then, in addition to equations (1), we have the condition
\[ K g_{ij}-\psi_{ij}=\bar K \bar g_{ij}, \qquad \psi_{ij}=\psi_{i,j}-\psi_i\psi_j . \tag{5} \]
Using the relation between the curvature tensors of the conformal spaces \(V_n,\hat V_n\) and relation (5), we obtain
\[ \hat R_{ij}=A g_{ij}+B\bar g_{ij}, \qquad A=(n-1)(K+\Delta_1\psi)-\bar K\bar g, \qquad B=-\bar K(n-2), \tag{6} \]
where \(\hat R_{ij}\) is the Ricci tensor of \(\hat V_n\); \(\Delta_1\psi\) is the first differential parameter of Beltrami; \(\bar g=g^{\alpha\beta}\bar g_{\alpha\beta}\). But if \(\hat V_n\) is also a space of constant curvature \(\hat K\), then from (6) it follows that
\[ \left[-(n-1)\hat K e^{2\psi}-A\right]g_{ij}=B\bar g_{ij}, \]
and, for a nontrivial geodesic mapping, this is possible only when \(B=0\), or, what is the same, when \(\bar K=0\). It is not difficult to see that this condition is also sufficient for \(\hat V_n\) to be a space of constant curvature. Thus we have proved:
Theorem 2. In order that \(\hat V_n\) be a space of constant curvature, it is necessary and sufficient that \(\bar V_n\) be a flat space.
Denote by \(L_n^1\) the class of all Riemannian spaces \(\hat V_n\) obtained from spaces of constant curvature by a \(\Gamma\)-transformation.
- Joint consideration of equations (1) and (6) gives the following condition on spaces of class \(L_n^1\):
\[ \hat R_{ij\Lambda K}=\sigma_k\hat g_{ij}+\varphi_i\hat g_{kj}+\varphi_j\hat g_{ki}, \tag{7} \]
where we have set
\[ \sigma_k=e^{-2\psi}(A_{,k}-2A\psi_k), \qquad \varphi_i=e^{2\psi}(\psi^\alpha\hat R_{\alpha i}-A\psi_i), \qquad \psi^\alpha=\psi_\beta g^{\beta\alpha}, \tag{8} \]
and directly from (7) it follows that
\[ \sigma_i=N_1\hat R_{,i}, \qquad \varphi_i=N_2\hat R_{,i}, \qquad N_1=\frac{n}{n(n+1)-2}, \qquad N_2=\frac{n-2}{2[n(n+1)-2]} . \tag{9} \]
Calculations show that in \(\hat V_n\) the tensor \(\check g_{ij}\), determined by the equalities (2) and (6), as a consequence of (7) and (8) satisfies equations of the form (1). At the same time, from (5) and (6) we have
\[ \psi_{ij}=\left(K+\frac{A}{n-2}\right)e^{-2\psi}\hat g_{ij}-\frac{1}{n-2}\hat R_{ij}. \tag{10} \]
These equations may be regarded as conditions in \(\hat V_n\) on \(\psi\) and \(A\). Now it is not difficult to verify that \(V_n\), determined by formula (4), will be a space-
with constant curvature, if \(\hat V_n\) is conformally flat and \(\psi_i\) satisfies condition (10).
Thus, we have:
Theorem 3. In order that \(\hat V_n\) belong to the class \(L_n^1\), it is necessary and sufficient that it be conformally flat, satisfy condition (7), and that there exist a nontrivial solution of the system of equations (8), (10) with respect to \(\psi_i\) and \(A\).
- Finally, let us consider Riemannian spaces \(\hat V_n\) satisfying only relation (7). We shall denote this class of spaces by \(L_n^2\). Putting
\[ \rho_{ij}=\hat R_{ij}-\hat\sigma \hat g_{ij}+\alpha \hat g_{ij},\quad (\alpha=\mathrm{const}), \tag{11} \]
we see that the tensor \(\rho_{ij}\) satisfies equations of the form (3), and, moreover, because of the arbitrariness of \(\alpha\), one may always assume \(|\rho_{ij}|\ne0\). But then, as was indicated earlier, a transition is possible to relations of the form (1) in \(\hat V_n\), which will be satisfied by the tensor
\[ \bar\rho_{ij}=e^{2\psi}\rho^{\alpha\beta}\hat g_{\alpha i}\hat g_{\beta j}, \tag{12} \]
where \(\psi_i=-\varphi^\gamma \rho^{\alpha\beta}\hat g_{\alpha\gamma}\hat g_{\beta i}\), and \(\rho^{ij}\) are the elements of the inverse matrix for \(\|\rho_{ij}\|\).
Taking (9) into account, we obtain the theorem:
Theorem 4. Every Riemannian space \(\hat V_n\) of the class \(L_n^2\) \((n>2)\) with nonconstant scalar curvature admits a nontrivial geodesic mapping onto a Riemannian space with metric tensor \(\bar\rho_{ij}\), determined by equalities (11), (12).
Equations (7) and (9) give a tensorial characteristic of Riemannian spaces of the class \(L_n^2\).
Odessa State University
named after I. I. Mechnikov
Received
16 XI 1960
REFERENCES
- N. S. Sinyukov, DAN, 111, No. 4 (1956).