The classification of pseudo-Riemannian spaces $V^n$ with poles for~$n\geqslant3$

N. R. Kamyshanskii

Sbornik. Mathematics, Tome 34 (1978) no. 1, pp. 25-54

Résumé

The goal of this article is the description of all complete, simply-connected, analytic pseudo-Riemannian spaces $V^n$ of dimension $n\geqslant3$ and index $k$ which contain at least one pole. Recall that a point $p$ in $V^n$ is called a pole if the group of motions of $V^n$ which fix $p$ has dimension $n(n-1)/2$. To each complete space $V^n$ ($n\geqslant3$) with poles there corresponds a class $\chi(V^n)$ of real analytic functions on $\mathbf R$, the characteristic functions for the space $V^n$; the group of affine transformations of the line $\mathbf R$ acts transitively on $\chi(V^n)$. A necessary and sufficient condition is stated for a given real analytic function $a(\tau)$ on $\mathbf R$ to be a characteristic function for an analytic pseudo-Riemannian space $V^n$ ($n\geqslant3$) which contains a pole. A simply-connected space $V^n$ of index $k$ is uniquely determined (up to isometry) by its characteristic function. In the article is an example of a complete, simply-connected, analytic pseudo-Riemannian space $\widetilde V^n_0$ of dimension $n\geqslant3$ and index $k$ for which the set of poles is infinite. It is shown that every complete, simply-connected, analytic pseudo-Riemannian space of dimension $n\geqslant3$ and index $k$ which has poles is conformally equivalent to a region in $\widetilde V^n_0$. Figures: 2. Bibliography: 3 titles.