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 $A_0colon T|_{x_0} Mightarrow T|_{hat{x}_0}hat{M}$ 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 $fcolon Mightarrow hat{M}$ whose differential at $x_0$ is precisely $A_0$. The converse of this result is also true.
展开▼
机译：在本文中，我们通过发展测地线来描述从完全简单连接的黎曼流形$（M，g）$到完整黎曼流形$（ hat {M}， hat {g}）$的黎曼覆盖图的存在$ M $到$ hat {M} $的三角形。更确切地说，我们证明如果$ A_0 冒号T | _ {x_0} M rightarrow T | _ { hat {x} _0} hat {M} $是切线空间之间的等距映射，并且对于任何两个测地三角形$ gamma $，$ omega $ of $ M $基于$ x_0 $通过复合路径$ gamma cdot omega $到$ hat {M} $的$ A_0 $的展开导致封闭路径以$ hat {x} _0 $为基础，则存在一个黎曼覆盖图$ f 冒号M rightarrow hat {M} $，其在$ x_0 $的差值恰好是$ A_0 $。反之亦然。
展开▼
