Heterogeneous Riemannian manifolds
A Riemannian metric on a manifold \(M\) is called heterogeneous if no two distinct points of \(M\) have isometric neighborhoods. Intuitively, such metrics are as far as possible from being homogeneous. The author proves that the set of heterogeneous metrics on a smooth manifold \(M\) of dimension \(n \geq 2\) is generic, more precisely, it is residual in the space of Riemannian metrics on \(M\) with the strong \(C^\infty\) topology. (This result was known before for compact manifolds \(M\) by \textit{T. Sunada}, [Ann. Math. (2) 121, 169--186 (1985; Zbl 0585.58047)]). The second result of the paper is a solution of Ambrose's problem for complete, connected, simply connected, heterogeneous manifolds, which shows that such a manifold is determined up to a global isometry by the behavior of the curvature tensor along geodesics emanating from a given point.
scientific article; zbMATH DE number 5695240