MATHEMATICS

Ya. Shapiro

GEODESIC FIELDS OF DIRECTIONS AND GROUPS OF HOMOTHETIES IN SPACES OF AFFINE CONNECTION

(Presented by Academician P. S. Aleksandrov, 13 XII 1957)

I. The concept of a geodesic field of directions in \(A_n\) (an \(n\)-dimensional space of affine connection) was introduced by the author \({}^{1}\) and then by Hantzes \({}^{2}\). Closely related to it is the concept of a torsion-forming field of directions, introduced by Yano \({}^{3}\).

Let in \(A_{n+1}\) \((n \ge 2)\), referred to coordinates \(x^\beta\), a field of directions be given by a vector field

\[ A^\alpha(x^\beta);\qquad \alpha,\beta=0,1,\ldots,n, \]

and let a two-dimensional surface \((S)\) be formed by the trajectories (“\(A\)-lines”) of the vector field \(A^\alpha\), intersecting a geodesic line of \(A_{n+1}\).

If the surface \(S\), constructed for any geodesic, is completely geodesic, then the field of directions determined by \(A^\alpha\) will be called geodesic.

The necessary and sufficient conditions for the field of directions represented by \(A^\alpha\) to be geodesic consist in the equalities \({}^{1}\)

\[ A^\alpha{}_{,\beta}=T\delta^\alpha_\beta+B_\beta A^\alpha, \tag{1a} \]

\[ R_{\mu(\beta\sigma)}{}^\alpha A^\mu=H^\alpha{}_{(\beta\sigma)}+F_{\beta\sigma}A^\alpha, \tag{1b} \]

where \(T\) is a certain scalar function; \(B_\beta, H^\alpha{}_\beta\), and \(F_{\beta\sigma}\) are certain tensors; \(A^\alpha{}_{,\beta}\) denotes the covariant derivative of \(A^\alpha\), and parentheses denote symmetrization over the indices contained in them.

If only (1a) holds, then the field of directions is called torsion-forming.

In a space of constant curvature, any geodesic field of directions is formed by the directions of the straight lines of any connection. A more general example of spaces containing geodesic fields of directions is given by subprojective spaces.

Closely connected with a geodesic field of directions is the concept of a system of paths included in \(A_n\). If, in some coordinate system \(u^\alpha\) \((\alpha=0,1,\ldots,n)\), \(n-1\) equations of the system of geodesics of \(A_{n+1}\) can be written in the form

\[ f_B(C_\theta,u^i)=0;\qquad B=1,\ldots,n-1;\qquad i=1,\ldots,n, \tag{2} \]

containing \(2(n-1)\) parameters \(C_\theta\) in such a way that the latter are uniquely determined by the equalities

\[ f_B(C_\theta,u^i_1)=0,\qquad f_B(C_\theta,u^i_2)=0, \]

then \(A_{n+1}\) includes the system of paths (2).

A subprojective space \((A_{n+1}\) or \(V_{n+1})\) may be characterized as a space containing a system of straight lines of an \(n\)-dimensional Euclidean space.

It has been proved that a system of paths included in \(A_{n+1}\) is a system of geodesics of some \(A_n\), and that a necessary and sufficient condition for the inclusion in \(A_{n+1}\) of an \(n\)-dimensional system of paths is the presence in it of a geodesic field of directions; the included system of paths is isomorphic to the set of lines obtained by projecting—by means of the trajectories of the geodesic field of directions—the geodesic lines of \(A_{n+1}\) onto some one of its hypersurfaces \((^4)\).

II. Let \(A_{m+n}\) contain \(m\) geodesic fields of directions, represented by the vectors \(\overset{a}{A}{}^\alpha\), satisfying (1a), (1b):

\[ \overset{a}{A}{}^\alpha{}_{,\beta} = \overset{a}{T}\,\delta^\alpha_\beta + \overset{a}{B}_{\beta}\,\overset{a}{A}{}^\alpha, \tag{2a} \]

\[ R_{\mu(\beta\sigma)}{}^\alpha \overset{a}{A}{}^\mu = \overset{a}{H}_{(\beta\sigma)}{}^\alpha + \overset{a}{F}_{\beta\sigma}\,\overset{a}{A}{}^\alpha . \tag{2б} \]

In these equalities and below, the indices from \(a\) to \(h\) take the values \(1,\ldots,m\), the indices from \(i\) onward take the values \(m+1,\ldots,m+n\), and the indices of the Greek alphabet take all the values \(1,\ldots,m+n\).

As follows from equality (2a), for the commutator vector

\[ (\overset{a}{A},\overset{b}{A})^\alpha = \overset{a}{A}{}^\alpha{}_{,\beta}\,\overset{b}{A}{}^\beta - \overset{b}{A}{}^\alpha{}_{,\beta}\,\overset{a}{A}{}^\beta \]

we have

\[ (\overset{a}{A},\overset{b}{A})^\alpha = \left(\overset{a}{T}-\overset{a}{B}_{\beta}\overset{b}{A}{}^\beta\right)\overset{b}{A}{}^\alpha - \left(\overset{b}{T}-\overset{b}{B}_{\beta}\overset{a}{A}{}^\beta\right)\overset{a}{A}{}^\alpha . \tag{I} \]

Hence follows the existence of coordinate systems (forming, we shall say, the class \(\Sigma'\)) for which the trajectories of the vector fields \(\overset{a}{A}\) are the coordinate \(u^a\)-lines.

Theorem 1. If \(A_{n+m}\) contains \(m\) geodesic fields of directions, then in coordinates \(u^\alpha\) of the class \(\Sigma'\) for its connection coefficients we have:

\[ \Gamma^\sigma_{\alpha\beta} = \Pi^\sigma_{\alpha\beta}(u^i)+\psi_{(\alpha\delta\beta)}, \]

\[ \Pi^b_{\alpha\beta} = \Pi^b_{\alpha\beta}(u^b,u^i), \qquad \Pi^i_{\alpha\beta} = \Pi^i_{\alpha\beta}(u^k), \tag{3} \]

\[ \Pi^{P_a}_{a\beta}=0, \qquad \Pi^{P_{a_1}}_{\alpha\beta}=0, \]

where \(\psi_\alpha\) are certain functions, \(\Pi^b_{\alpha\beta}\) and \(\Pi^i_{\alpha\beta}\) are certain functions of the indicated arguments, and the index \(P_a\) takes all values \(1,\ldots,m+n\), except one, namely \(a\). Conversely, from conditions (3) it follows that the coordinate \(u^a\)-lines are trajectories of geodesic fields of directions.

As follows from (I), \(A_{n+m}\), containing \(m\) geodesic fields of directions, is decomposed into \(\infty^n\) subspaces of \(m\) dimensions carrying the trajectories of these fields.

In a special coordinate system of the class \(\Sigma'\), these subspaces coincide with coordinate manifolds on which only the coordinates \(u^a\) vary; they turn out to be completely geodesic manifolds; moreover, each of them is mapped onto an \(m\)-dimensional Euclidean space in such a way that geodesics go over into straight lines. In connection with what has been said, there exist such coordinates \(u^\alpha\) of the class \(\Sigma'\) that, in the equations of a geodesic belonging to the manifold \(u^i=\mathrm{const}\), the current coordinates

\(u^\alpha\) are connected by linear relations; we shall call the class of such coordinate systems \(\Sigma''\). For coordinates \(u^\alpha\) of class \(\Sigma''\), of course, (3) holds; moreover, one may set (by normalizing the quantities \(\psi_a\) in the corresponding way)

\[ \Pi^a_{bc}=0. \tag{3a} \]

Conversely, if (3), (3a) hold, then the coordinate system is of class \(\Sigma''\).

Let \({}^{a}A^\alpha\) be \(m\) vectors \((m>1)\), representing \(m\) geodesic fields of directions in \(A_{n+m}\), and suppose it is possible, by normalization—by multiplying \({}^{a}A^\alpha\) by certain functional factors—to achieve that: 1) for any constant values \(\lambda_a\), the vector \(\sum_a \lambda_a\,{}^{a}A^\alpha\) determines a geodesic field of directions; 2) among the vectors \({}^{a}A^\alpha\) there is no linear dependence with constant coefficients. The set of geodesic fields of directions determined in this case by the vectors \(\sum_a \lambda_a\,{}^{a}A^\alpha\) will be called a linear manifold of geodesic fields of directions of dimension \(m-1\).

The vector fields \({}^{a}A^\alpha\), normalized in the described manner, will be called \(L\)-normalized.

Theorem 2. In order that the vectors \({}^{a}A^\alpha\) satisfying (2a) and (2b) be \(L\)-normalized, it is necessary and sufficient that the equalities

\[ {}^{1}B_\beta=\cdots={}^{m}B_\beta;\qquad {}^{1}F_{\alpha\beta}=\cdots={}^{m}F_{\alpha\beta}. \]

hold.

Theorem 3. In order that the geodesic fields of directions determined by \({}^{a}A^\alpha\) belong to a linear manifold of similar fields, it is necessary and sufficient that: 1) the difference of any pair of vectors \({}^{a}B_\beta\) from (2a) be a vector-gradient; 2) \({}^{1}F_{\alpha\beta}={}^{m}F_{\alpha\beta}\).

Theorem 4. If \(A_{n+m}\) admits a linear manifold of geodesic fields of directions of dimension \(m-1\), then there exist coordinates \(u^\alpha\) of class \(\Sigma''\), for which the vectors \({}^{a}A^\alpha=\delta^\alpha_a\) (representing \(m\) geodesic fields of directions) are \(L\)-normalized.

We shall call them (i.e. the coordinates \(u^\alpha\)) coordinates of class \(\Sigma\).

Theorem 5. The connection coefficients of \(A_{n+m}\) with a linear manifold of geodesic fields of directions of dimension \(m-1\) in a coordinate system \((u^\alpha)\) of class \(\Sigma\) have the form

\[ \Gamma^\alpha_{\beta\sigma} = \Pi^\alpha_{\beta\sigma} + \psi_{(\beta}\delta^\alpha_{\sigma)}, \]

\[ \Pi^{P_a}_{a\beta}=0,\quad \Pi^a_{bc}=0,\quad \Pi^i_{jk}=\Pi^i_{jk}(u^l),\quad \Pi^a_{ai}=\Pi_i(u^k), \tag{4} \]

\[ \Pi^a_{ij}=u^a C_{ij}(u^k)+{}^{a}D_{ij}(u^k), \]

where \(\varphi_\beta\) are certain functions; \(\Pi^i_{jk}, \Pi_i, C_{ij}, {}^{a}D_{ij}\) are certain functions of the indicated arguments; the index \(P_a\) runs through all values \(1,\ldots,m+n\), except \(a\).

The converse is also valid.

III. Let \({}^{a}A^\alpha \dfrac{\partial f}{\partial x^\alpha}\) be the operators of a group (of order \(m\)) of automorphic transformations of \(A_{n+m}\).

If the field of directions determined by the vector \(\sum_a \lambda_a A^a\) of any one-parameter subgroup is geodesic, then we shall call the transformations of the group homotheties of the space \(A_{n+m}\).

The maximal order of the group of homotheties of \(A_n\) is realized in an affine space, for example; in this case the transformations reduce to homotheties in the usual sense, which in Cartesian coordinates are given by

\[ x^{\prime i}=b x^i+a^i \qquad (i=1,\ldots,n), \]

where \(b, a^i\) are parameters.

Theorem 6. The necessary and sufficient conditions that the vectors \(A^a\) \((a=1,\ldots,m)\) determine in \(A_n\) a group of homotheties of order \(m\) consist in the equalities

\[ A^a_{\alpha,\beta}=T^a \delta^\alpha_\beta+B_\beta A^{a\alpha}, \]

\[ A^a_{\alpha,\beta\sigma}=A^{a\lambda}R_{\lambda\sigma\beta}{}^\alpha, \]

where no restrictions are imposed on \(T^a, B_\beta\) (not following from the written equalities).

Theorem 7. The group of homotheties of order \(m\) of the space \(A_n\) is locally isomorphic to the group of homotheties of an \((m-1)\)-dimensional affine space.

Theorem 8. Geodesic fields of directions determined by vectors \(A^a\) that generate in \(A_n\) a group of homotheties of order \(m\) belong to an \((m-1)\)-dimensional linear manifold of such fields.

Theorem 9. The systems of paths included in \(A_n\) \((^3)\) by means of the vectors of the group of homotheties are isomorphic.

Gorky State University
named after N. I. Lobachevsky

Received
23 III 1957

CITED LITERATURE

\(^1\) Ya. Shapiro, DAN, 32, 237 (1941).
\(^2\) J. Haantjes, Nieuw Arch. Wiskunde, 2, No. 2–3, 97 (1954).
\(^3\) K. Yano, Proc. Imp. Acad. Tokyo, 20, 340 (1944).
\(^4\) Ya. Shapiro, Matem. sborn., 36 (78), 1 (1955).