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

Petri Kokkonen

Archivum Mathematicum (2012)

  • Volume: 048, Issue: 3, page 207-231
  • ISSN: 0044-8753

Abstract

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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 ( M ^ , g ^ ) in terms of developing geodesic triangles of M onto M ^ . More precisely, we show that if A 0 : T | x 0 M → T | x ^ 0 M ^ is some isometric map between the tangent spaces and if for any two geodesic triangles γ , ω of M based at x 0 the development through A 0 of the composite path γ · ω onto M ^ results in a closed path based at x ^ 0 , then there exists a Riemannian covering map f : M → M ^ whose differential at x 0 is precisely A 0 . The converse of this result is also true.

How to cite

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Kokkonen, Petri. "A characterization of isometries between Riemannian manifolds by using development along geodesic triangles." Archivum Mathematicum 048.3 (2012): 207-231. <http://eudml.org/doc/246323>.

@article{Kokkonen2012,
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.},
author = {Kokkonen, Petri},
journal = {Archivum Mathematicum},
keywords = {Cartan-Ambrose-Hicks theorem; development; linear and affine connections; rolling of manifolds; Cartan-Ambrose-Hicks theorem; development; linear and affine connections; rolling of manifolds},
language = {eng},
number = {3},
pages = {207-231},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A characterization of isometries between Riemannian manifolds by using development along geodesic triangles},
url = {http://eudml.org/doc/246323},
volume = {048},
year = {2012},
}

TY - JOUR
AU - Kokkonen, Petri
TI - A characterization of isometries between Riemannian manifolds by using development along geodesic triangles
JO - Archivum Mathematicum
PY - 2012
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 048
IS - 3
SP - 207
EP - 231
AB - 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.
LA - eng
KW - Cartan-Ambrose-Hicks theorem; development; linear and affine connections; rolling of manifolds; Cartan-Ambrose-Hicks theorem; development; linear and affine connections; rolling of manifolds
UR - http://eudml.org/doc/246323
ER -

References

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  10. Pawel, K., Reckziegel, H., 10.2996/kmj/1071674466, Kodai Math. J. 25 (3) (2002), 341–356. (2002) Zbl1065.53021MR1942783DOI10.2996/kmj/1071674466
  11. Robbin, J. W., Salamon, D. A., Introduction to Differential Geometry, ETH, Lecture Notes, preliminary version, January 2011, http://www.math.ethz.ch/~salamon/PREPRINTS/diffgeo2011.pdf. 
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  13. Sharpe, R. W., Differential Geometry: Cartan’s Generalization of Klein’s Erlangen Program, Graduate Texts in Mathematics, vol. 166, Springer–Verlag, New York, 1997. (1997) MR1453120

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