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A characterization of isometries between Riemannian manifolds by using development along geodesic triangles
Archivum mathematicum, Tome 48 (2012) no. 3, pp. 207-231 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2012-3-207
Classification : 53B05, 53B21, 53C05
Keywords: Cartan-Ambrose-Hicks theorem; development; linear and affine connections; rolling of manifolds 
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     author = {Kokkonen, Petri},
     title = {A characterization of isometries between {Riemannian} manifolds by using development along geodesic triangles},
     journal = {Archivum mathematicum},
     pages = {207--231},
     year = {2012},
     volume = {48},
     number = {3},
     doi = {10.5817/AM2012-3-207},
     mrnumber = {2995873},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2012-3-207/}
}
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Kokkonen, Petri. A characterization of isometries between Riemannian manifolds by using development along geodesic triangles. Archivum mathematicum, Tome 48 (2012) no. 3, pp. 207-231. doi: 10.5817/AM2012-3-207

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