A characterization of isometries between Riemannian manifolds by using development along geodesic triangles

Petri Kokkonen

Address: L2S, Université Paris-Sud XI, CNRS and Supélec, Gif-sur-Yvette, 91192, France and University of Eastern Finland, Department of Applied Physics, 70211, Kuopio, Finland

E-mail: petri.kokkonen@lss.supelec.fr

Abstract: In this paper we characterize the existence of Riemannian covering maps from a complete simply connected Riemannian manifold $(M,g)$ onto a complete Riemannian manifold $(\hat{M},\hat{g})$ in terms of developing geodesic triangles of $M$ onto $\hat{M}$. More precisely, we show that if $A_0\colon T|_{x_0} M\rightarrow T|_{\hat{x}_0}\hat{M}$ is some isometric map between the tangent spaces and if for any two geodesic triangles $\gamma $, $\omega $ of $M$ based at $x_0$ the development through $A_0$ of the composite path $\gamma \cdot \omega $ onto $\hat{M}$ results in a closed path based at $\hat{x}_0$, then there exists a Riemannian covering map $f\colon M\rightarrow \hat{M}$ whose differential at $x_0$ is precisely $A_0$. The converse of this result is also true.

AMSclassification: primary 53B05; secondary 53C05, 53B21.

Keywords: Cartan-Ambrose-Hicks theorem, development, linear and affine connections, rolling of manifolds.

DOI: 10.5817/AM2012-3-207