| Title:
|
A characterization of isometries between Riemannian manifolds by using development along geodesic triangles (English) |
| Author:
|
Kokkonen, Petri |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
48 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
207-231 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Cartan-Ambrose-Hicks theorem |
| Keyword:
|
development |
| Keyword:
|
linear and affine connections |
| Keyword:
|
rolling of manifolds |
| MSC:
|
53B05 |
| MSC:
|
53B21 |
| MSC:
|
53C05 |
| idMR:
|
MR2995873 |
| DOI:
|
10.5817/AM2012-3-207 |
| . |
| Date available:
|
2012-10-03T15:04:30Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142990 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
