N. S. SINYUKOV

ALMOST GEODESIC MAPPINGS OF AFFINELY CONNECTED AND RIEMANNIAN SPACES

(Presented by Academician A. N. Kolmogorov on 13 II 1963)

At first the notion of almost geodesic lines of affinely connected spaces without torsion is introduced here, as curves closest to geodesic lines in the affine sense. Then mappings of affinely connected (and Riemannian) spaces are considered, under which almost geodesic or geodesic lines of one space pass into almost geodesic lines of another. The investigation is carried out locally in the class of analytic functions.

1. A curve \(C\), defined by the equations \(x^i = x^i(t)\) \((i = 1, 2, \ldots, n)\), will be called almost geodesic of the affinely connected space \(A_n\), if for it there exists a field of parallel planes \(E_2\{\lambda^i,\mu^i\}\), each of which at the corresponding point passes through the tangent vector \(\xi^i = dx^i/dt\). Thus an almost geodesic is characterized by the fact that along it the conditions are identically satisfied

\[ \xi^i = p\lambda^i + q\mu^i;\quad \lambda^i{}_{,\alpha}\xi^\alpha = a_1\lambda^i + b_1\mu^i;\quad \mu^i{}_{,\alpha}\xi^\alpha = a_2\lambda^i + b_2\mu^i . \tag{1} \]

It follows from this that the differential equation of almost geodesics has the form

\[ \xi^i_{2|} = a\xi^i + b\xi^i_{1|}, \tag{2} \]

where \(\xi^i_{1|} = \xi^i{}_{,\alpha}\xi^\alpha,\ \xi^i_{2|} = \xi^i_{1|,\alpha}\xi^\alpha\), the comma denotes the sign of covariant differentiation. Here \(a\) and \(b\) may be regarded as arbitrary functions of \(x^1, x^2, \ldots, x^n, t\). For given \(a\) and \(b\), system (2) has a unique solution for any initial values \(x^i|_{t=t_0},\ \xi^i|_{t=t_0},\ \xi^i_{1|}|_{t=t_0}\). When \(A_n\) is flat, its almost geodesics are curves lying in two-dimensional planes \((^1)\).

2. Considering a mapping \(\bar A_n\) onto \(A_n\), under which every almost geodesic of one passes into an almost geodesic of the other, it is not hard to find that it must necessarily be geodesic. Obviously, the converse is also true.

3. We therefore turn to a mapping \(\bar A_n\) onto \(A_n\), under which every geodesic of the first passes into an almost geodesic of the second, calling it almost geodesic. Obviously, in equation (2) of the almost geodesic of \(A_n\), corresponding to a geodesic of \(\bar A_n\), \(a\) and \(b\) depend on the direction of the latter. Assuming this dependence to be analytic (and then it will necessarily be rational), we find that an almost geodesic mapping is characterized by the conditions

\[ P^h_{(ij,k)} + 2P^h_{\alpha(i}P^\alpha_{jk)} = a_{(ij}\delta^h_{k)} + b_{(i}P^h_{jk)} . \tag{3} \]

Here \(P^h_{ij} = \Gamma^h_{ij} - \bar\Gamma^h_{ij}\), \(\Gamma^k_{ij}\) and \(\bar\Gamma^k_{ij}\) are the connection coefficients of \(A_n\) and \(\bar A_n\) in a common coordinate system under the mapping, \(a_{ij}\) is a symmetric tensor, \(b_i\) is a vector, and parentheses denote cyclic summation. When \(A_n\) is flat and referred to affine coordinates, (3) gives the fundamental equations of the theory of \(n - 2\) projective spaces \((^2)\). These equations are satisfied by \(\bar\Gamma^k_{ij} = f C^k_{ij}\), where \(f\) is an arbitrary function, and \(C^k_{ij}\) are constants connected by the conditions \(C^h_{\alpha(i}C^\alpha_{jk)} = 0\), i.e., they define a commutative algebra satisfying the Jacobi identity. The spaces \((^3)\) belong here.

4. An almost geodesic mapping \(\bar A_n\) onto \(A_n\) will be called linear if to each geodesic of \(\bar A_n\) there corresponds an almost geodesic of \(A_n\), for which the field of parallel planes \(E_2\) at each point is linearly

depends on the direction of the tangent. In this case one may assume that
\(\mu^i=\bar\mu^i+\mu_\alpha^i \xi^\alpha,\ \lambda^i=\xi^i\), where \(\bar\mu^i\) and \(\mu_j^i\) depend only on the point. From conditions (1), under analytic dependence of the coefficients on \(\xi^i\), it follows that there exist only two types of linear almost geodesic mappings. For one of them \(\mu^i=\mu_\alpha^i\xi^\alpha\),

\[ P_{ij}^{k}=\varphi_i\delta_j^{k}+\varphi_j\delta_i^{k}+\psi_i\mu_j^{k}+\psi_j\mu_i^{k}, \]

\[ \mu_{j,k}^{i}+\mu_{k,j}^{i}+2(\psi_j\mu_\alpha^i\mu_k^\alpha+\psi_k\mu_\alpha^i\mu_j^\alpha) =\rho_j\delta_k^i+\rho_k\delta_j^i+\sigma_j\mu_k^i+\sigma_k\mu_j^i, \tag{4} \]

and for the other \(\mu^i=\bar\mu^i\),

\[ P_{ij}^{k}=\varphi_i\delta_j^{k}+\varphi_j\delta_i^{k}+\psi_{ij}\mu^{k},\qquad \mu^i{}_{,j}=\rho\delta_j^i+\sigma_j\mu^i. \tag{5} \]

Here \(\varphi_i,\psi_j,\sigma_k,\rho_j,\mu^i\) are vectors, \(\mu_j^i\) is an affinor, and \(\psi_{ij}\) is a symmetric tensor. Equations (4) also give the additional relations

\[ \psi_{j,k}+\psi_{k,j}=\nu_j\psi_k+\nu_k\psi_j \tag{6} \]

for \(\mu_j^i\ne\tau\delta_j^i+\xi^i\theta_j\) for the first type, and

\[ \psi_{(jkl)}=0,\qquad \psi_{jkl}=\psi_{jk,l}-\nu_l\psi_{jk} \tag{7} \]

for \(\mu^i\ne0\) for the second.

If, in item 3, the requirement of analyticity is somewhat weakened, then conditions (6) and (7) disappear.

1. Finally, let us consider linear almost geodesic mappings of type II, distinct from geodesic mappings, for Riemannian spaces \(\bar V_n\) and \(V_n\) (with metric tensors \(\bar g_{ij}\) and \(g_{ij}\)). In this case, in \(V_n\) there must exist a vector \(\mu^i\ne0\) satisfying conditions (5). If it is nonisotropic, by normalization one can pass to the gradient vector \(\tilde\mu_i=\partial\mu/\partial x^i\) and, in a special coordinate system, reduce the metric form of the space to the form \(ds^2=e\,dx^{1\,2}+F\,d\tilde s^2\), where \(e=\pm1\), \(F\) is an arbitrary function of \(x^1,x^2,\ldots,x^n\), and \(d\tilde s^2\) is an arbitrary metric of \(\tilde V_{n-1}\) in the manifold \(x^2,x^3,\ldots,x^n\).

These conditions are also sufficient, since a conformal mapping of \(V_n\) by means of any function of \(\tilde\mu\) turns out to be a linear almost geodesic mapping of type II (and conditions (7) are satisfied automatically). If \(\mu^i\) in (5) is isotropic, then \(\rho=0\), and it is absolutely parallel. When \(\sigma_j\) is a gradient, analogously to the preceding case we arrive at the gradient vector \(\tilde\mu_i\), at a conformal mapping that is a linear almost geodesic mapping of type II, and at the well-known canonical form of the metric form of the space. When one of the spaces in a linear almost geodesic correspondence of type II is flat, the spaces (4) correspond to the solution found.

If, similarly to the preceding, one considers curves of affinely connected spaces for which the minimal dimension of the field of parallel planes \(E_k\) passing through the tangent vector is \(k>2\) (\(<n\)), one obtains an analogous generalization of the theory of \(n-k\) projective spaces.

In conclusion, taking this opportunity, I express my deep gratitude to Professors S. P. Finikov, I. P. Egorov, and A. M. Vasiliev for their great attention to this work.

Odessa State University
named after I. I. Mechnikov

Received
22 I 1963

References

1. I. A. Schouten, D. J. Struik, Introduction to the New Methods of Differential Geometry, 2, Moscow, 1948.
2. G. Vrănceanu, Bull. Math. Soc. Roumaine Sci., 48, No. 1–2 (1947).
3. D. V. Vedenyapin, Scientific Reports of Higher School, Physico-Mathematical Sciences, No. 6 (1958).
4. V. F. Kagan, Subprojective Spaces, Moscow, 1961.