| Title:
|
Constant Jacobi osculating rank of $\mathbf{U(3)/(U(1) \times U(1) \times U(1))}$ (English) |
| Author:
|
Arias-Marco, Teresa |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
45 |
| Issue:
|
4 |
| Year:
|
2009 |
| Pages:
|
241-254 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we obtain an interesting relation between the covariant derivatives of the Jacobi operator valid for all geodesic on the flag manifold $M^6=U(3)/(U(1) \times U(1) \times U(1))$. As a consequence, an explicit expression of the Jacobi operator independent of the geodesic can be obtained on such a manifold. Moreover, we show the way to calculate the Jacobi vector fields on this manifold by a new formula valid on every g.o. space. (English) |
| Keyword:
|
naturally reductive space |
| Keyword:
|
g.o. space |
| Keyword:
|
Jacobi operator |
| Keyword:
|
Jacobi osculating rank |
| MSC:
|
53C20 |
| MSC:
|
53C21 |
| MSC:
|
53C22 |
| MSC:
|
53C25 |
| MSC:
|
53C30 |
| idZBL:
|
Zbl 1212.53059 |
| idMR:
|
MR2591679 |
| . |
| Date available:
|
2009-12-22T07:52:37Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/137457 |
| . |
| Reference:
|
[1] Arias-Marco, T.: Constant Jacobi osculating rank of $U(3)/(U(1) \times U(1) \times U(1))$ -Appendix-.ArXiv:0906.2890v1. MR 2591679 |
| Reference:
|
[2] Arias-Marco, T.: Study of homogeneous D’Atri spaces of the Jacobi operator on g.o. spaces and the locally homogeneous connections on 2-dimensional manifolds with the help of Mathematica$^{\scriptstyle {\bf ©}}$.thematica$^{\scriptstyle {\mathbf ©}}$, Universitat de València, Valencia, Spain, 2007, ISBN: 978-84-370-6838-1, http://www.tdx.cat/TDX-0911108-110640. |
| Reference:
|
[3] Arias-Marco, T.: Methods for solving the Jacobi equation. Constant osculating rank vs. constant Jacobi osculating rank.Differential Geometry Proceedings of the VIII International Colloquium, 2009, pp. 207–216. Zbl 1180.53042, MR 2523506 |
| Reference:
|
[4] Arias-Marco, T., Naveira, A. M.: Constant Jacobi osculating rank of a g.o. space. A method to obtain explicitly the Jacobi operator.Publ. Math. Debrecen 74 (2009), 135–157. Zbl 1199.53111, MR 2490427 |
| Reference:
|
[5] Chavel, I.: Isotropic Jacobi fields, and Jacobi’s equations on Riemannian homogeneous spaces.Comment. Math. Helvetici 42 (1967), 237–248. Zbl 0166.17501, MR 0221426, 10.1007/BF02564419 |
| Reference:
|
[6] Kaplan, A.: On the geometry of groups of Heisenberg type.Bull. London Math. Soc. 15 (1983), 35–42. Zbl 0521.53048, MR 0686346, 10.1112/blms/15.1.35 |
| Reference:
|
[7] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry I, II.Wiley-Interscience, New York, 1996. |
| Reference:
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[8] Kowalski, O., Prüfer, F., Vanhecke, L.: D’Atri spaces.Progr. Nonlinear Differential Equations Appl. 20 (1996), 241–284. MR 1390318 |
| Reference:
|
[9] Macías-Virgós, E., Naveira, A. M., Tarrío, A.: The constant osculating rank of the Wilking manifold $V_3$.C. R. Acad. Sci. Paris, Ser. I. Math. 346 (2008), 67–70. Zbl 1134.53025, MR 2385057, 10.1016/j.crma.2007.11.009 |
| Reference:
|
[10] Naveira, A. M., Tarrío, A.: A method for the resolution of the Jacobi equation $Y^{\prime \prime } + R Y = 0$ on the manifold $Sp(2)/SU(2)$.Monatsh. Math. 158 (3) (2008), 231–246. Zbl 1152.53039, 10.1007/s00605-008-0551-3 |
| Reference:
|
[11] Tsukada, K.: Totally geodesic submanifolds of Riemannian manifolds and curvature invariant subspaces.Kodai Math. J. 19 (1996), 395–437. Zbl 0871.53017, MR 1418571, 10.2996/kmj/1138043656 |
| . |
