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Reports of the Academy of Sciences of the USSR
- Volume 131, No. 1
MATHEMATICS
M. A. Ulanovskii
ON STATIONARY GROUPS OF MOTIONS OF SPACES OF AFFINE CONNECTION
(Presented by Academician I. G. Petrovskii, 13 XI 1959)
Let \(A_n\) be a space of affine connection with base manifold \(M_n\); let \(S\) be a group of its motions leaving fixed a certain point \(O \in M_n\); and let \(E_n\) be the tangent vector space at the point \(O\).
Join an arbitrary point \(x \in M_n\) by a geodesic to the point \(O\) and consider the development \((^1)\) of this geodesic in \(E_n\)—the segment of the straight line \(Ox'\). By assigning to the point \(x \in M_n\) the point \(x' \in E_n\), we define a one-to-one correspondence between certain neighborhoods of the base manifold and of the tangent space \(E_n\). For convenience of exposition, in what follows we shall identify the corresponding points \(x\) and \(x'\); lower-case Latin letters \(x, y, u, \ldots\) will denote simultaneously vectors of \(E_n\) and the corresponding points of \(M_n\). The components \(x^i\) of these vectors with respect to an arbitrary basis in \(E_n\) are the well-known normal coordinates in \(M_n\). This same convention determines the meaning of the terms linear subspace, the point \(y - x\) (the point of \(M_n\) corresponding to the difference of the vectors \(y - x\) in \(E_n\)), etc., used below as applied to the base manifold. It is important to note that, by virtue of this same convention, one may regard the group \(S\) as a group of nonsingular real matrices of order \(n\).
We shall also introduce the following definitions. If \(S\) is a linear (matrix) group, then by \(\overline{S}\) we shall denote the semigroup—the closure of \(S\) in the space of all matrices of the given order* . Further, let \(E_n\) be a vector space in which a certain linear group \(S\) acts, and let \(E_n^k\) be the space whose points are systems of \(k\) linearly independent vectors from \(E_n\) (i.e., the direct product of \(k\) copies of \(E_n\), from which the cone corresponding to systems of linearly dependent vectors has been removed). We shall call the linear group \(S\) \(k\)-transitive if it is transitive \((^2)\) as a group of transformations of \(E_n^k\) (of each connected component of \(E_n^k\) in the case \(k = n\)). We shall call a system of \(k\) vectors (or a point of \(E_n^k\)) an exact frame if there exists no nontrivial transformation of the group \(S\) leaving this system fixed, and if no part of this system satisfies the latter condition. If almost all systems of \(k\) vectors from \(E_n\) (i.e., all points of \(E_n^k\) with the exception, possibly, of a set of measure zero—surfaces in \(E_n^k\)) are exact frames, then we shall say that a frame of the group contains \(k\) vectors.
Consider arbitrary points \(u, x\) of \(M_n\) (or vectors of \(E_n\)). Construct the development of the geodesic broken line \(Oux\), consisting of the geodesic arcs \(Ou\) and \(ux\). This development is a broken line, one of whose segments coincides with \(Ou\) (by virtue of the identification of the points of \(M_n\) and \(E_n\) adopted above), while the other is a certain segment \(uy\). This construction makes it possible to define a vector-function of two vector arguments \(f(u,x)\) by means of the formula
\[ f(u,x) = y - x. \tag{1} \]
* \(\overline{S}\), generally speaking, also contains singular matrices.
It is easy to see that \(f(u,x)\) is uniquely determined for any pair \(u,x\) from some neighborhood of the point \(O\) containing no geodesic triangles. Further, if \(A_n\) belongs to the class \(C_{p+1}\), then \(f(u,x)\) is at least \(p\) times continuously differentiable in all arguments; we shall assume that \(p \geq 2\).
Below we use the following properties of \(f(u,x)\):
1) If \(u=\lambda x\) (\(\lambda\) is a scalar), then \(f(u,x)=0\).
Indeed, in this case the arcs \(Ou\) and \(ux\) form one smooth geodesic arc, whose development coincides with it, whence \(f(u,x)=x-x=0\). In particular, \(f(u,0)=f(0,x)=0\).
2) If \(\alpha\in S\), then
\[ f(\alpha u,\alpha x)=\alpha f(u,x). \tag{2} \]
Indeed, let \(\gamma\) be an arbitrary arc with initial point at \(O\), and let \(R(\gamma)\) be its development; let the transformation \(\alpha\) carry the arcs \(\gamma\) and \(R(\gamma)\), respectively, into \(\alpha\gamma\) and \(\alpha R(\gamma)\). If \(\alpha\in S\), then the transformation \(\alpha\) (a motion) preserves the development operation, i.e. \(\alpha R(\gamma)=R(\alpha\gamma)\), where \(R(\alpha\gamma)\) is the development of the arc \(\alpha\gamma\). If, in particular, \(\gamma\) is the broken line \(O,u,x\), \(R(\gamma)\) is \(O,u,(f(u,x)+x)\) (according to (1)), then \(\alpha\gamma\) coincides with the broken line \(O,\alpha u,\alpha x\), and \(R(\alpha\gamma)\) with the broken line \(O,\alpha u,(f(\alpha u,\alpha x)+\alpha x)\); finally, \(\alpha R(\gamma)\) with the broken line \(O,\alpha u,\alpha(f(u,x)+x)\), whence we obtain (2).
3) If \(H\) is a linear subspace and \(f(h_1,h_2)\in H\) for any \(h_1\) and \(h_2\) from \(H\), then \(H\) (as a submanifold in \(M_n\)) is a totally geodesic surface.
4) If \(f(u,x)\) is linear in \(x\) and \(A_n\) is a torsion-free connection, then \(A_n\) is a locally affine connection.
Theorem 1. Let \(\alpha\in \bar S\); let \(U_k, V_{n-k}\) be complementary linear subspaces such that \(\alpha^p U_k=0\), \(\alpha^p E_n=V_{n-k}\) for some integer \(p\). Then \(U_k, V_{n-k}\) are totally geodesic surfaces. Every linear subspace of dimension \(k+1\) containing \(U_k\) is also a totally geodesic surface.
We prove one of the assertions of the theorem. Let \(v_1\in V_{n-k}\), \(v_2\in V_{n-k}\). Then \(v_1=\alpha^p x_1\), \(v_2=\alpha^p x_2\), where \(x_1,x_2\) are some vectors from \(E_n\). Further,
\[
f(v_1,v_2)=f(\alpha^p x_1,\alpha^p x_2)=\alpha^p f(x_1,x_2)\in V_{n-k}.
\]
From property 3) it now follows that \(V_{n-k}\) is a totally geodesic surface. The remaining assertions of the theorem are proved in a similar way.
As an example of the application of the same method, let us examine in detail one special case. Namely, let \(S\) contain the group of all linear transformations leaving fixed \((n-1)\) certain vectors. Let \(V_{n-1}\) be the linear subspace spanned by these vectors. For any \(x\) not belonging to \(V_{n-1}\), one can find a matrix \(\alpha_x\in\bar S\) such that \(\alpha_x x=0\), \(\alpha_x E_n=V_{n-1}\). If \(x,y\) are arbitrary vectors from \(E_n\), and \(\alpha_x,\alpha_y\) are the corresponding matrices from \(\bar S\), then
\[
\alpha_x f(x,y)=f(\alpha_x x,\alpha_x y)=f(0,\alpha_x y)=0;
\]
in exactly the same way, \(\alpha_y f(x,y)=0\). As is easy to see, it follows that \(f(x,y)=0\). Thus, if \(S\) contains the indicated subgroup and \(A_n\) is a torsion-free connection, then \(A_n\) is a locally affine connection.
Theorems 2 and 3 are proved by this same method.
Theorem 2. For any sequence \(\{\alpha_k\}\in S\) \((k=1,2,\ldots,\infty)\), the following linear subspaces are totally geodesic surfaces: a) the maximal \(U\) possessing the property that \(\alpha_k U\to 0\); b) the maximal \(V\) such that the sequence \(\{\alpha_k x\}\) is bounded for any \(x\in V\).
For the proof it suffices to consider the sequence \(\{\alpha_k f(x_1,x_2)\}\) \((k=1,2,\ldots,\infty)\), where \(x_1\) and \(x_2\) are vectors from the subspace \(U\) (or \(V\)). By properties 1), 2), this sequence of the vector-function \(f(x_1,x_2)\) converges to \(0\) (or is bounded), and therefore \(f(x_1,x_2)\) belongs-
belong to the subspace \(U\) (respectively, \(V\)). From property 3) it now follows that \(U\) (\(V\)) is a totally geodesic surface.
Applying Theorem 2 (with minor additions) to the sequence of powers of the matrix, we obtain Theorem 3.
Theorem 3. For any real number \(c\) \((0<c<1)\), the invariant subspace corresponding to all eigenvalues of matrices \(a\in S\) whose absolute value does not exceed \(c\) is a totally geodesic surface.
Of course, the theorem is also valid for eigenvalues not less than some \(c>1\).
Theorem 4. If \(S\) is doubly transitive and \(A_n\) is a connection without torsion, then \(A_n\) is a locally affine connection.
For the proof we represent \(f(u,x)\) in the form
\[ f(u,x)=l(u,x)+m(u,x), \]
where \(l(u,x)\) is linear with respect to \(x\), and
\[ \frac{|m(u,x)|}{|x|}\to 0 \qquad \text{as } x\to 0. \]
(By the symbol \(|x|\) we denote the norm of the vector, defined, for example, as follows:
\[ |x|=\left(\sum_i x^{i2}\right)^{1/2}, \]
where \(x^i\) are the components of \(x\) in some fixed basis. As the norm of the matrix \(a\) we shall, as usual, take
\[ |a|=\max \frac{|ax|}{|x|}. \]
If \(a\in S\), then \(m(au,ax)=a m(u,x)\). On the other hand, for any \(u\) and \(x\) from \(M_n\) one can find a sequence \(\{a_k\}\in S\) such that \(a_k u\to 0\), \(a_k x\to 0\), and \(|a_k x|\cdot |a_k^{-1}|<c\); then
\[ |m(u,x)|\le |a_k x|\cdot |a_k^{-1}|\cdot \frac{|m(a_k u,a_k x)|}{|a_k x|}\to 0, \]
i.e. \(f(u,x)\) is linear with respect to \(x\).
Let us also note two simple theorems.
Theorem 5. If the frame of the group \(S\) contains \(n\) vectors, then \(A_n\) is a subprojective connection.
Theorem 6. If the frame of the group \(S\) contains more than two vectors, then every two-dimensional linear manifold (“plane” passing through the point \(O\)) belongs to some proper linear subspace \(\bar H_k\) \((k<n)\), which is a totally geodesic surface.
Indeed, let \(U_2\) be an arbitrary plane in \(E_n\); consider \(U_{2+k}\)—the linear span of \(U_2\) and all values of the vector-function \(f\) from all possible pairs of vectors in \(U_2\), then \(U_{2+k+l}\)—the linear span of \(U_{2+k}\) and the values of \(f\) from pairs of vectors in \(U_{2+k}\), and so on. If this process terminates in some proper linear subspace, then it is a totally geodesic manifold (according to 3)), containing \(U_2\). Otherwise, as is easy to see, any pair of vectors from \(U_2\) is a frame of the group \(S\).
Theorem 5 is proved in the same way.
Voroshilovgrad Mining and Metallurgical Institute
Received
27 X 1959
CITED LITERATURE
- É. Cartan, The Geometry of Riemannian Spaces, Moscow, 1948.
- N. G. Chebotarev, Theory of Lie Groups, Moscow—Leningrad, 1940.