| Title:
|
Geodesically equivalent metrics on homogenous spaces (English) |
| Author:
|
Bokan, Neda |
| Author:
|
Šukilović, Tijana |
| Author:
|
Vukmirović, Srdjan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
945-954 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two $G$-invariant metrics of arbitrary signature on homogenous space $G/H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, $G$-invariant metrics on homogenous space $G/H$ implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metrics of any signature on sphere $S^3$ are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics. (English) |
| Keyword:
|
invariant metric |
| Keyword:
|
geodesically equivalent metric |
| Keyword:
|
affinely equivalent metric |
| MSC:
|
22E15 |
| MSC:
|
53C22 |
| MSC:
|
53C30 |
| idZBL:
|
07144866 |
| idMR:
|
MR4039611 |
| DOI:
|
10.21136/CMJ.2018.0557-17 |
| . |
| Date available:
|
2019-11-28T08:46:59Z |
| Last updated:
|
2022-01-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147905 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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[4] Hall, G. S., Lonie, D. P.: Holonomy groups and spacetimes.Classical Quantum Gravity 17 (2000), 1369-1382. Zbl 0957.83014, MR 1752641, 10.1088/0264-9381/17/6/304 |
| Reference:
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| Reference:
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| Reference:
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[7] Kiosak, V., Matveev, V. S.: Proof of the projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two.Commun. Math. Phys. 297 (2010), 401-426. Zbl 1197.53055, MR 2651904, 10.1007/s00220-010-1037-4 |
| Reference:
|
[8] Levi-Civita, T.: Sulle trasformazioni dello equazioni dinamiche.Annali di Mat. 24 Italian (1896), 255-300 \99999JFM99999 27.0603.04. MR 2551879, 10.1007/BF02419530 |
| Reference:
|
[9] Sinyukov, N. S.: On geodesic mappings of Riemannian spaces onto symmetric Riemannian spaces.Dokl. Akad. Nauk SSSR, n. Ser. 98 (1954), 21-23 Russian. Zbl 0056.15301, MR 0065994 |
| Reference:
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| Reference:
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| . |
