MATHEMATICS

P. I. PETROV

THE FUNDAMENTAL PROBLEM OF NON-RIEMANNIAN GEOMETRY IN THE BINARY DOMAIN

(Presented by Academician P. S. Aleksandrov on 18 V 1961)

1. Eisenhart and Veblen posed the problem of finding conditions under which a parallel displacement given by its (\Gamma^k_{ij}(x)) is Riemannian ((^1)). This integral problem of tensor analysis, also called the fundamental problem of non-Riemannian geometry, as well as the question of the reducibility of a space (A_n) to a Weyl space ((W_n)), served as a topic of investigation for the Princeton school of geometers. However, their results, set forth in Thomas’s monograph ((^2)), do not exhaust the problem of metrizability of manifolds with symmetric affine connection (A_n). Moreover, attempts to overcome it even for (n=2) are erroneous: the conditions for the reducibility of (A_2) to (W_2) in article ((^3)), contrary to the assertion of its author, are only sufficient; in articles ((^{4,5})), conversely, requirements are listed which are only necessary for the reducibility of (A_2) to a Riemannian space (V_2).

In the present note a solution is proposed for the fundamental problem of non-Riemannian geometry in the binary domain.

1. As a preparatory step toward the problem of the present communication, we classify the spaces (A_2) by their differential invariants, understanding the question of classification itself in the sense of partitioning the set of all possible objects of symmetric connection (\Gamma^k_{ij}(x)) into nonequivalent nonempty classes without common elements. The latter reduces to finding the types of the affine-curvature tensor (B^\alpha_{\beta\gamma\delta}), since for (n=2) the quantities (\Gamma^k_{ij}(x)) and (B^i_{jkl}) determine one another (((^2), \S 101)). Putting

[
B^\alpha_\beta = \tfrac12 B^\alpha_{\beta\lambda\mu} e^{\lambda\mu},
\tag{1}
]

where (e^{\lambda\mu}=-e^{\mu\lambda}), (B^\alpha_{\beta\gamma\delta}) may be identified with a linear vector-function of the first kind

[
B=\begin{pmatrix}
B^1_1 & B^1_2\
B^2_1 & B^2_2
\end{pmatrix}.
\tag{2}
]

In view of this fact, with any (A_2) there is associated an arithmetic invariant ([e_1 e_2])—the Weierstrass characteristic of the vector-function (B), corresponding to the curvature tensor of the space, which we shall call the characteristic of the manifold (A_2). We agree to assign to one class spaces with equal characteristics, although they are not necessarily equivalent to one another.

Thus we obtain:

Theorem 1. The spaces (A_2), according to their characteristics ([11]), ([(11)]), ([2]), split into three nonempty nonequivalent classes.

1. If in a space (A_2) with coefficients of parallel displacement (\Gamma^k_{ij}(x)) ((-\Gamma^k_{ji})) there exists a nonsingular symmetric tensor (g_{ij})

which is an integral of the differential system

[
\frac{\partial g_{ij}}{\partial x^k}-g_{\sigma j}\Gamma^\sigma_{ik}-g_{i\sigma}\Gamma^\sigma_{jk}=\omega_k g_{ij},
\tag{3}
]

where (\omega_k) is a nonzero vector, then (A_2) is said to be reducible to Weyl geometry ((W_2)). Consideration of the integrability conditions of system (3),

[
g_{\mu j}B^\mu{}{ikl}+g}B^\mu{{jkl}-g=0,}B^\sigma{}_{\sigma kl
\tag{4}
]

where (B^\lambda{}{\beta\gamma\delta}) are the components of the tensor of affine curvature of the object (\Gamma^k(x)), for spaces (A_2) of nonsimple type immediately leads to the conclusion:

Lemma 1. An (A_2) of nonsimple type is irreducible to (W_2).

We shall call a coordinate net the principal net of the space (A_2) if the directions tangent to the lines of each of its families at each point coincide with the vectors of the principal direction of the vector function (B). Let the manifold of affine connection of simple type under consideration be referred to its principal net. Then the matrix (B), as is known, assumes diagonal form, and relations (4) reduce to the following two equations:

[
g_{11}\cdot\bigl(B^1{}{112}-B^2{}\bigr)=0,\qquad
g_{22}\cdot\bigl(B^2{}{212}-B^1{}\bigr)=0.
\tag{5}
]

Equations (5) can be satisfied in two ways:

I. (B^1{}{112}-B^2{}), functionally independent with respect to the three arbitrary constants contained in them ((^6)). Consequently, we have:}=0). In this case the characteristic of the space is expressed by the symbol ([(11)]), and system (3), being completely integrable, admits three solutions: (g_{11}, g_{12}, g_{22

Lemma 2. A space (A_2) with characteristic ([(11)]) is Weyl.

II. (g_{11}=g_{22}=0). Substituting these values into (3), we obtain a mixed system for (g_{12}):

[
g_{11}\Gamma^2_{11}=0,\qquad
g_{12}\Gamma^2_{12}=0,\qquad
g_{12}\Gamma^1_{12}=0,\qquad
g_{12}\Gamma^1_{22}=0,
]

[
\frac{\partial g_{12}}{\partial x^1}-g_{12}(\Gamma^1_{11}+\Gamma^2_{12})=\omega_1 g_{12},
\tag{6}
]

[
\frac{\partial g_{12}}{\partial x^2}-g_{12}(\Gamma^1_{12}+\Gamma^2_{22})=\omega_2 g_{12}.
]

Thus, in order that the mixed system (3), II have a solution (g_{ij}) subject to the restriction (|g_{ij}|\ne0), it is necessary that

[
\Gamma^2_{11}=\Gamma^1_{22}=\Gamma^1_{12}=\Gamma^2_{12}=0.
\tag{7}
]

Conversely, if conditions (7) are fulfilled, then (g_{12}) is determined by solving a completely integrable system. Its integral (g_{12}), by the theorem on a mixed system of differential equations, depends on one arbitrary constant.

In terms of the theory of nets our result may be stated as follows:

Lemma 3. A space (A_2) with characteristic ([11]) is Weyl if and only if its principal net is simultaneously also a Descartes net.

If the functions (\Gamma^k_{ij}(x)) have first partial derivatives of order (r), then (A_2) is called of class (C^{(r)}). Put (\varphi(\lambda)=|B-\lambda E|). The rank of the matrix (\varphi'(B)) will be denoted by (\rho). Using the terms and notations introduced, the criterion for Weyl metrizability of two-dimensional spaces of symmetric affine connection may be formulated as follows:

Theorem 2. In order that a space (A_2) of class (C^{(1)}) be a Weyl space (W_2), it is necessary and sufficient that one of the following two conditions hold:

a) (\varphi'(B)=0),

b) ([\varphi'(B)]^2 \ne 0,\ \rho=2), the principal net is Descartes.

1. As is known, the special case of Weyl spaces for which the complementary vectors (\omega_i) are gradients is called Riemannian and is denoted by (V_2). Taking this definition into account, from Theorem 2 we derive a criterion for Riemannian connectedness:

Theorem 3. In order that a space (A_2) be Riemannian (V_2), it is necessary and sufficient that either of the following two conditions hold:

a) (\varphi'(B)=0,\ B^\sigma_{\sigma}=0);

b) ([\varphi'(B)]^2 \ne 0,\ \rho=2,\ B^\sigma_{\sigma}=0), the principal net is Descartes.

Theorems 1 and 2 together exhaust the question of the metrizability of manifolds (A_2) and provide a means of surveying the varieties of spaces of affine connection of two dimensions.

Kazan State University
named after V. I. Ulyanov-Lenin

Received
17 V 1961

REFERENCES

¹ L. P. Eisenhart, O. Veblen, Proc. Nat. Acad. USA, 8, No. 2, 19 (1922).
² T. Y. Thomas, The Differential Invariants of Generalized Spaces, Cambridge, 1934, p. 95.
³ A. P. Norden, Spaces of Affine Connection, Moscow, 1950, p. 329.
⁴ A. Moor, J. f. reine u. angew. Math., 199, H. 1/2, S. 94, Satz 2 (1958).
⁵ S. Golab, Tensor, New Ser., 9, No. 1, p. 7 (Satz) (1959).
⁶ L. P. Eisenhart, Continuous Groups of Transformations, Moscow, 1947, ch. I, pp. 7–12.