Abstract
Full Text
UDC 513.82
MATHEMATICS
N. S. SINYUKOV
ON THE THEORY OF GEODESIC MAPPINGS OF RIEMANNIAN SPACES
(Presented by Academician A. N. Kolmogorov on 13 XI 1965)
The paper considers and in principle solves the problem of determining all Riemannian spaces admitting a nontrivial geodesic mapping onto a given (V_n). In addition, an estimate is given for the degree of arbitrariness in the solution of this problem, and necessary and sufficient conditions of algebraic character are obtained, in invariant form, for (V_n)’s admitting a nontrivial geodesic mapping. The investigation is carried out in the class of analytic functions, locally, without restriction on the signature of the spaces.
- A Riemannian space (V_n) with metric tensor (g_{ij}) ((i,j=1,2,\ldots,n)) admits a nontrivial geodesic mapping if and only if there exists a solution (\check g_{ij}(=\check g_{ji})) of the system of equations
[
\check g_{ij,k}=\lambda_i g_{kj}+\lambda_j g_{ki},
\tag{A_1}
]
where (\lambda_i\ne 0) is a gradient vector, and the comma denotes the sign of covariant differentiation in (V_n). In this case the metric tensor (\bar g_{ij}) of the corresponding space (\bar V_n) is determined from the relations
[
\bar g_{ij}=e^{2\psi}\check g^{\alpha\beta}g_{\alpha i}g_{\beta j},\qquad
\psi_{,i}=-\lambda_\alpha \check g^{\alpha\beta}g_{\beta i},
\tag{1}
]
where (\check g^{ij}) are the elements of the inverse matrix for (|\check g_{ij}|) (1). Nonvanishing of the determinant of the solution (\check g_{ij}) of equations ((A_1)) can always be achieved by adding to it a term of the form (c g_{ij}) ((c=\mathrm{const})).
From the integrability conditions of system ((A_1)) we find
[
\check g_{\alpha j}R^\alpha{}{ikl}
+\check g}R^\alpha{{ilj}
+\check g=0,}R^\alpha{}_{ijk
\tag{2}
]
[
\check g_{\alpha i}R^\alpha{}{j}
-\check g=0,}R^\alpha{}_{i
\tag{3}
]
[
n\lambda_{i,j}=\mu g_{ij}
+\check g_{\alpha i}R^\alpha{}{j}
-\check g^\beta,}R^\alpha{}_{ji}{
\tag{A_2}
]
as a consequence of which they reduce to the form
[
\check g_{\alpha_3}T^{\alpha_3}{}_{ij\,kl}=0,
\tag{B_1}
]
[
T^{\alpha\beta}{}{ij\,kl}
=
n\left(\delta_i^\alpha R^\beta{}
-\delta_k^\alpha R^\beta{}{l(ij)}\right)
-g}\left(\delta_i^\alpha R^\beta{l
-R^\alpha{}^\beta\right)}{
+g_{il}\left(\delta_k^\alpha R^\beta{}j
-R^\alpha{}^\beta\right).}{
]
In turn, from the integrability conditions of equations (A_2) it follows that
[
(n-1)\mu_{,k}
=
2(n+1)\lambda_\alpha R^\alpha{}k
-\check g}\left(R^{\alpha\beta}{{,k}-2R^\alpha{}^\beta\right),}{
\tag{A_3}
]
and they are represented in the form
[
\begin{aligned}
(n+3)\lambda_\alpha R^\alpha{}{ilk}
&=
\frac{1}{n-1}g
\left[(n+3)\lambda_\alpha R^\alpha{}k
-\check g}\left(R^{\alpha_3}{{,k}
-2R^\alpha{}^\beta\right)\right] \}{
&\quad
-\frac{1}{n-1}g_{ik}
\left[(n+3)\lambda_\alpha R^\alpha{}l
-\check g}\left(R^{\alpha_3}{{,l}
-2R^\alpha{}^\beta\right)\right] \}{
&\quad
+\check g_{\alpha i}R^\alpha{}{[l,k]}
+\check g.}R_{lki}{}^{\beta,\alpha
\end{aligned}
\tag{B_2}
]
Finally, the integrability conditions of the equations (A_3) (taking into account (A_1) and (A_2)) give the relations
[
\frac{n+1}{n}\,\check g_{\alpha\beta}
\left(R^\alpha{}{k\gamma\cdot}{}^\beta R^\gamma{}_l
-
R^\alpha{}}{}^\beta R^\gamma{k\right)
+(n+3)\lambda\alpha R^\alpha{}{[k,l]}
-
\check g
\left(
\frac{1}{2}R^{\alpha\beta}{}{,[kl]}
-
R^\alpha{}^\beta}{
+
R^\alpha{}_{\cdot l,\cdot k}{}^\beta
\right).
\tag{B_3}
]
Thus, we obtain a system of differential equations (A_{123}) of the first order with respect to (g_{ij}), (\lambda_i), (\mu), solved with respect to their derivatives, with coefficients from (V_n) and integrability conditions (B_{123}), and hence also the theorem:
Theorem 1. In order that (V_n) admit a nontrivial geodesic mapping, it is necessary and sufficient that the system of equations (A) have a nontrivial solution (\check g_{ij}\ne c g_{ij}), (\lambda_i\ne 0).
Since the general solution of the system (A) is determined from it (in the form of a Taylor series) uniquely up to the initial conditions (\overset{0}{g}_{ij}), (\overset{0}{\lambda}_i), (\overset{0}{\mu}), which can be chosen arbitrarily only in the case of its complete integrability, it follows from (B) that
Theorem 2. The general space (\bar V_n) admitting a nontrivial geodesic mapping onto a given (V_n) is determined by this space uniquely up to (r\le (n+1)(n+2)/2) arbitrary parameters. This maximum is attained if and only if (V_n) is a space of constant curvature.
The relations (B), the first (B^1), the second (B^2), and so on, their continuations, are linear algebraic equations with respect to (\check g_{ij}), (\lambda_i), and (\mu), with coefficients from (V_n). Since the solution of the system (A) corresponding to initial values satisfying the indicated relations satisfies them identically, we obtain the theorem:
Theorem 3. In order that (V_n) admit a nontrivial geodesic mapping, it is necessary and sufficient that the system of equations (B), (B^1), (B^2,\ldots) have a nontrivial solution with respect to (\check g_{ij}) and (\lambda_i).
Analysis of the result of covariant differentiation of the conditions (B_1) shows that when (\check g_{\alpha\beta}T^{\alpha\beta}_{ij\,kl,m}=0) by virtue of (B_1), (\lambda_i=0). This occurs for (V_n) satisfying the relations
[
T^{(\alpha3)}{ij\ \ kl,m}
=
Q^{pqrs}.}T^{(\alpha3)}_{pq\ \ rs
\tag{4}
]
Consequently, we have the theorem:
Theorem 4. Riemannian spaces satisfying conditions (4) do not admit a nontrivial geodesic mapping.
- Consider the relations (2), writing them in the form
[
\check g_{\alpha3}\,\tilde T^{\alpha3}_{ij\,kl}=0,
]
[
\tilde T^{\alpha3}{ij\,kl}
=
\delta_j^{(\alpha}R^{3)}{}
+
\delta_k^{(\alpha}R^{\beta)}{}{ilj}
+
\delta_l^{(\alpha}R^{3)}{}.
\tag{5}
]
In the matrix (|\tilde T^{\alpha\beta}_{ij\,kl}|) we shall take combinations of upper indices as the column numbers, and the lower indices as the row numbers. If (\alpha_1,\alpha_2,\ldots,\alpha_n) are distinct numbers from (1) to (n), then formula (5) shows that the determinant of the minor of order ((n-2)) of the matrix corresponding to the column numbers ((\alpha_3\alpha_3)), ((\alpha_3\alpha_4)), (\ldots), ((\alpha_3\alpha_n)) and to the row numbers ((\alpha_1\alpha_3,\alpha_1\alpha_2)), ((\alpha_1\alpha_4,\alpha_1\alpha_2)), (\ldots), ((\alpha_1\alpha_n,\alpha_1\alpha_2)) is
[
\Delta=2\left(R^{\alpha_3}{}_{\cdot\,\alpha_1\alpha_1\alpha_2}\right)^{\,n-2}.
\tag{6}
]
For the determinant of the minor of order ((n-2)) of the same matrix corresponding to the column numbers ((\alpha_1\alpha_4)), ((\alpha_4\alpha_4)), (\ldots), ((\alpha_4\alpha_n)) and to the row numbers ((\alpha_1\alpha_1,\alpha_2\alpha_3)), ((\alpha_1\alpha_4,\alpha_2\alpha_3)), (\ldots), ((\alpha_1\alpha_n,\alpha_2\alpha_3)), we obtain the formula
[
D_1=2\left(R^{\alpha_4}{}_{\cdot\,\alpha_1\alpha_2\alpha_3}\right)^{\,n-2}.
\tag{7}
]
Hence it follows
Lemma 1. If the rank of ( \left|\tilde T^{\alpha\beta}_{ijkl}\right| \rho < n-2), then (V_n) is a space of constant curvature.
A joint consideration of relations (B^1) and (B^2) leads to the lemma:
Lemma 2. For (V_n) distinct from spaces of constant curvature, (\lambda_i) and (\mu) are determined uniquely in terms of the objects (V_n), the components of the tensor (g_{ij}), and one of the components of the vector (\lambda_i).
Lemmas 1 and 2 give the theorem:
Theorem 5. The totality of all Riemannian spaces (\bar V_n) admitting a nontrivial geodesic mapping onto (V_n), distinct from a space of constant curvature, depends on
(r \leq n(n+1)/2 - (n-4)) essential parameters.
In a similar way, from (B^1), (B^2), and (3) ((^2)) it follows
Theorem 6. The totality of all Riemannian spaces (\bar V_n) admitting a nontrivial geodesic mapping onto (V_n), distinct from an Einstein space, depends on
(r \leq n(n+1)/2 - (n-2)) essential parameters.
3. Finally, the displacement vector (\xi^i) of a nontrivial infinitesimal geodesic transformation of (V_n) satisfies the equations ((^3))
[
h_{ij,k}=2\psi_k g_{ij}+\psi_i g_{kj}+\psi_j g_{ki},
]
where (h_{ij}=\xi_{i,j}+\xi_{j,i}), (\psi_i\ne 0) is a gradient vector. Therefore the tensor (\hat g_{ij}=h_{ij}-2\psi g_{ij}) satisfies the equations (A_1), and (V_n) admits a nontrivial geodesic mapping. On this basis, Theorems 5 and 6 give an estimate of the number of essential parameters on which the displacement vector of a general nontrivial infinitesimal geodesic transformation of (V_n) depends. Namely, in the first case (\tilde r \leq n^2+4), and in the second (\tilde r \leq n^2+2) (cf. ((^4))).
Odessa State University
named after I. I. Mechnikov
Received
10 X 1965
REFERENCES
- N. S. Sinyukov, DAN, 137, No. 6 (1961).
- I. P. Egorov, DAN, 66, No. 5 (1949).
- L. P. Eisenhart, Riemannian Geometry, IL, 1948.
- I. P. Egorov, DAN, 61, No. 4 (1948).