UDC 513.838

MATHEMATICS

A. S. SOLODOVNIKOV

ON THE GLOBAL STRUCTURE OF ANALYTIC SEMI-REDUCIBLE RIEMANNIAN SPACES

(Presented by Academician I. G. Petrovskii, February 6, 1967)

1°. Various questions, relating mainly to mappings of Riemannian spaces, lead to the so-called semi-reducible spaces, i.e., Riemannian spaces with a metric of the form

\[ ds^2 = d\rho^2 + f\,d\theta^2 . \tag{1} \]

In this notation it is assumed that the coordinates \(x_1,\ldots,x_n\) are divided into two groups, say \(x_1,\ldots,x_k\) and \(x_{k+1},\ldots,x_n\), with the metric \(d\rho^2\) and the function \(f\) depending on the coordinates of the first group, while \(d\theta^2\) depends on the coordinates of the second.

In the works known to us, semi-reducible spaces are studied mainly locally. In the present article the spaces are considered globally, and the metric is assumed to be analytic. More precisely, we consider a space \(V_n\) satisfying the following condition:

Condition A. \(V_n\) is a connected complete analytic Riemannian space whose metric, in some coordinate domain \(U\),

\[ |x_i - x_i^0| < \varepsilon \qquad (i=1,\ldots,n) \]

is represented in the form (1), where

\[ d\rho^2 = \sum_{\alpha,\beta=1}^{k} a_{\alpha\beta}(x_1,\ldots,x_k)\,dx_\alpha dx_\beta, \qquad d\theta^2 = \sum_{\gamma,\delta=k+1}^{n} b_{\gamma\delta}(x_{k+1},\ldots,x_n)\,dx_\gamma dx_\delta \]

are positive definite quadratic forms with analytic coefficients, and \(f(x_1,\ldots,x_k)\) is a positive analytic function.

The question that will interest us in this article is to what extent the decomposition (1) extends to the whole space \(V_n\), and how \(V_n\) is structured globally.

2°. We indicate two types of examples of spaces \(V_n\) satisfying condition A.

1) Let \(V_k\) and \(V_{n-k}\) be two connected complete analytic Riemannian spaces, and let \(F\) be a strictly positive analytic function on \(V_k\). Then one may construct a new space of dimension \(n\) by defining it as the Cartesian product of \(V_k\) and \(V_{n-k}\), endowed with the metric

\[ d\rho^2 + F d\theta^2, \tag{2} \]

where \(d\rho^2\) is the metric in \(V_k\), and \(d\theta^2\) is the metric in \(V_{n-k}\). We shall denote this space in what follows by \(V_k \times F V_{n-k}\). Obviously, it satisfies condition A.

2) In the \((n+1)\)-dimensional Euclidean space \(E_{n+1}\), fix two subspaces, \(E_k\) and \(E_{k+1}\), with \(E_k \subset E_{k+1}\). Let \(\sigma\) be a \(k\)-dimensional surface in \(E_{k+1}\), analytic, regular, connected, and complete (with respect to the induced metric); assume also that this surface is symmetric with respect to \(E_k\). Rotating \(\sigma\) in \(E_{n+1}\) about the subspace \(E_k\), we obtain some \(n\)-dimensional surface \(S\). From obvious

from geometric considerations it follows that the metric on \(S\) has the form (2), where \(d\rho^2\) is the metric in \(\sigma\), \(d\theta^2\) is the metric of the sphere \(S_{n-k}\) of unit radius, and \(F=\varphi^2\), where \(\varphi\) is the distance from an arbitrary point \(p\in S\) to \(E_k\). The space \(S\) satisfies condition A. We also note the inequality \(d\rho\ge |d\varphi|\), which follows from considering the surface \(\sigma\).

Fig. 1.

The construction of the surface \(S\) admits the following generalization.

Let \(V_k\) be a connected complete analytic Riemannian space and let \(F\) be an analytic function on it having the following properties:

a) \(F\ge 0\), and the set \(\{x\in V_k\mid F(x)=0\}\) is a complete geodesic surface\(^*\) \(V_{k-1}\) (generally speaking, not connected), dividing the space into two half-spaces \(V'\) and \(V''\).

b) There exists an isometry \(\tau\) of the space \(V_k\), the set of fixed points of which is \(V_{k-1}\); hence it follows that the points \(p\) and \(\tau(p)\) \((p\notin V_{k-1})\) belong to different half-spaces.

c) \(F(p)=F(\tau(p))\) for all \(p\in V_k\).

d)
\[ \lim_{p\to p^*}\frac{F(p)}{r^2(p)}=1, \]
where \(p^*\) is any point of \(V_{k-1}\), and \(r(p)\) is the distance from \(p\) to \(V_{k-1}\).

In the construction of the surface \(S\), the role of \(V_k\) was played by the surface \(\sigma\), and the role of \(V_{k-1}\) by the intersection of \(\sigma\) with \(E_k\).

Any pair \(V_k,F\) with the properties listed allows one to define a new space \(V_n\), where \(n\) is any integer greater than \(k\). The space \(V_n\) is defined by the following construction.

Onto each pair of points \(p,\tau(p)\), where \(p\in V'\), a sphere of radius \(F(p)\) and of dimension \(n-k\) is “strung,” in such a way that the points \(p\) and \(\tau(p)\) are its diametrically opposite points. In the resulting set \(V_n\) a metric (2) is introduced, where \(d\rho^2\) is the metric in \(V_k\), and \(d\theta^2\) is the metric on the sphere \(S_{n-k}\) of unit radius. As a result \(V_n\) becomes a connected complete analytic Riemannian space. This space satisfies condition A. Figure 1 depicts the case \(n-k=1\).

The space \(V_n\) defined in the above manner will be called a space of rotation and denoted \(V_k\circ FS_{n-k}\).

\(3^\circ\). We present some facts and definitions. In each of them the space is assumed to be analytic.

Two spaces \(V_n\) and \(V'_n\) are called locally isometric if some domain in \(V_n\) is isometric to some domain in \(V'_n\).

Let \(K\) be the class of all connected complete spaces locally isometric to a given space \(V_n\). In the class \(K\) there always exists a unique, up to isometry, simply connected space \(\widetilde V_n\). Any other space of this class is isometric to the quotient space \(\widetilde V_n/\Gamma\), where \(\Gamma\) is a discrete group of motions of \(\widetilde V_n\) without fixed points.

A \(k\)-dimensional analytic sheet in \(V_n\) is a pair \((E_k,\psi)\), where \(E_k\) is a \(k\)-dimensional Euclidean space, and \(\psi\) is its analytic regular mapping into \(V_n\); it is also required that \(\psi(x)\ne\psi(y)\) for \(x\ne y\) \((x,y\in E_k)\). In what follows we identify the sheet \((E_k,\psi)\) with the set \(T=\psi(E_k)\subset V_n\).

The sheet \(T\) is called complete geodesic if every geodesic in \(T\) is also a geodesic in \(V_n\).

A \(k\)-dimensional analytic surface in \(V_n\) is a pair \((M,\Psi)\), where \(M\) is a \(k\)-dimensional analytic manifold, and \(\Psi\) is its regular analytic mapping into \(V_n\).

\(^*\) The necessary definitions are given below in Section \(3^\circ\).

For each analytic sheet \(T\) in \(V_n\) there exists a uniquely determined analytic surface \(\widetilde T\), which it is natural to regard as the maximal analytic continuation of the sheet \(T\). We shall indicate how this surface is defined.

Two \(k\)-dimensional analytic sheets will be considered as belonging to the same \(k\)-dimensional germ at a point \(p\), if both sheets pass through \(p\) and some of their subsheets*, containing \(p\), coincide (as subsets of \(V_n\)). The germ to which the sheet \(T\) belongs at the point \(p \in T\) will be denoted by \(\tau_p(T)\); the set \(\{\tau_p(T)\mid p\in T\}\) will be denoted by \(g(T)\).

Let \(G\) be the set of all \(k\)-dimensional germs in \(V_n\). Introduce in \(G\) the topology in which the subsets \(g(T)\) (where \(T\) is an arbitrary \(k\)-dimensional sheet) form, by definition, a basis of open sets. This topology is Hausdorff.

Associating with each \(k\)-dimensional germ at a point \(p\) the point \(p\) itself, we obtain a mapping \(\pi:G\to V_n\). Since \(\pi\) maps each \(g(T)\) homeomorphically onto \(T\), an analytic structure arises on the space \(G\), as well as an analytic metric borrowed from \(V_n\). Thus \(G\) is an analytic Riemannian space.

For each sheet \(T\) we now define \(\widetilde T\) as the connected component of the space \(G\) containing \(g(T)\). It is easy to see that \((\widetilde T,\pi)\) is an analytic surface containing, in the natural sense, the sheet \(T\).

The question of the completeness of the surface \(\widetilde T\) (from the point of view of the induced metric) is very important. In this connection we shall confine ourselves only to the following propositions.

If \(V_n\) is a complete space and \(T\) is a completely geodesic sheet in it, then the surface \(\widetilde T\) is complete.

4°. Let \(V_n\) satisfy condition A. Through a point \(p^0(x_1^0,\ldots,x_n^0)\in U\) pass two mutually orthogonal coordinate sheets: the \(k\)-dimensional sheet
\(T^{(\rho)}: x_{k+1}=x_{k+1}^0,\ldots,x_n=x_n^0\), carrying the metric \(d\rho^2\), and the \((n-k)\)-dimensional sheet
\(T^{(\theta)}: x_1=x_1^0,\ldots,x_k=x_k^0\), whose metric is \(f(x_1^0,\ldots,x_k^0)\,d\theta^2\). In view of the obvious inequality \(ds^2\ge d\rho^2\), valid throughout the domain \(U\), the sheet \(T^{(\rho)}\) is completely geodesic.

Lemma 1 (fundamental). The surfaces \(\widetilde T^{(\rho)}\) and \(\widetilde T^{(\theta)}\) (the maximal analytic continuations of the sheets \(T^{(\rho)}\) and \(T^{(\theta)}\)) are complete.

Let us note that the completeness of \(\widetilde T^{(\rho)}\) follows directly from the last proposition of item \(3^\circ\). For the surface \(\widetilde T^{(\theta)}\) the reasoning is more delicate.

Lemma 2. If the space \(V_n\) is simply connected, then the function \(f\) admits a single-valued analytic continuation \(F\) to the whole space \(\widetilde T^{(\rho)}\); moreover \(F\ge 0\).

Theorem 1. If \(V_n\) is simply connected and \(F\ne 0\) everywhere in \(\widetilde T^{(\rho)}\), then \(V_n\) is isometric to
\[ \widetilde T^{(\rho)} \times \frac{F}{c}\,\widetilde T^{(\theta)}, \]
where \(c=f(x_1^0,\ldots,x_k^0)\).

Theorem 2. If \(V_n\) is simply connected and the set of zeros of the function \(F\) is nonempty, then \(\widetilde T^{(\theta)}\) is isometric to a sphere of some radius \(c\) (of dimension \(n-k\)), and \(V_n\) is isometric to the space of rotation
\[ \widetilde T^{(\rho)} \circ c^2 F S_{n-k}. \]

From Theorems 1 and 2 it follows, in particular, that if \(k=1\), then \(V_n\) is homeomorphic (in the simply connected case) to one of the spaces
\[ E_1\times \widetilde T^{(\theta)},\qquad E_n,\qquad S_n. \]

In conclusion, the author expresses profound gratitude to P. K. Rashevskii, in whose seminar this work arose.

Moscow State Correspondence
Pedagogical Institute

Received
1 XII 1966

* If a sheet is defined by a pair \((E_k,\Psi)\), then a subsheet is a pair \((D,\Psi')\), where \(D\) is a domain in \(E_k\), and \(\psi'\) is the restriction of \(\psi\) to \(D\).