Constant Jacobi osculating rank of 𝐔 ( 3 ) / ( 𝐔 ( 1 ) × 𝐔 ( 1 ) × 𝐔 ( 1 ) )

Teresa Arias-Marco

Archivum Mathematicum (2009)

  • Volume: 045, Issue: 4, page 241-254
  • ISSN: 0044-8753

Abstract

top
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 ) × U ( 1 ) × 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.

How to cite

top

Arias-Marco, Teresa. "Constant Jacobi osculating rank of $\mathbf {U(3)/(U(1) \times U(1) \times U(1))}$." Archivum Mathematicum 045.4 (2009): 241-254. <http://eudml.org/doc/261052>.

@article{Arias2009,
abstract = {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.},
author = {Arias-Marco, Teresa},
journal = {Archivum Mathematicum},
keywords = {naturally reductive space; g.o. space; Jacobi operator; Jacobi osculating rank; naturally reductive space; g.o. space; Jacobi operator; Jacobi osculating rank},
language = {eng},
number = {4},
pages = {241-254},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Constant Jacobi osculating rank of $\mathbf \{U(3)/(U(1) \times U(1) \times U(1))\}$},
url = {http://eudml.org/doc/261052},
volume = {045},
year = {2009},
}

TY - JOUR
AU - Arias-Marco, Teresa
TI - Constant Jacobi osculating rank of $\mathbf {U(3)/(U(1) \times U(1) \times U(1))}$
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 4
SP - 241
EP - 254
AB - 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.
LA - eng
KW - naturally reductive space; g.o. space; Jacobi operator; Jacobi osculating rank; naturally reductive space; g.o. space; Jacobi operator; Jacobi osculating rank
UR - http://eudml.org/doc/261052
ER -

References

top
  1. Arias-Marco, T., Constant Jacobi osculating rank of U ( 3 ) / ( U ( 1 ) × U ( 1 ) × U ( 1 ) ) -Appendix-, ArXiv:0906.2890v1. MR2591679
  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 © , thematica, Universitat de València, Valencia, Spain, 2007, ISBN: 978-84-370-6838-1, http://www.tdx.cat/TDX-0911108-110640. (2007) 
  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. (2009) Zbl1180.53042MR2523506
  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. (2009) Zbl1199.53111MR2490427
  5. Chavel, I., 10.1007/BF02564419, Comment. Math. Helvetici 42 (1967), 237–248. (1967) Zbl0166.17501MR0221426DOI10.1007/BF02564419
  6. Kaplan, A., 10.1112/blms/15.1.35, Bull. London Math. Soc. 15 (1983), 35–42. (1983) Zbl0521.53048MR0686346DOI10.1112/blms/15.1.35
  7. Kobayashi, S., Nomizu, K., Foundations of Differential Geometry I, II, Wiley-Interscience, New York, 1996. (1996) 
  8. Kowalski, O., Prüfer, F., Vanhecke, L., D’Atri spaces, Progr. Nonlinear Differential Equations Appl. 20 (1996), 241–284. (1996) MR1390318
  9. Macías-Virgós, E., Naveira, A. M., Tarrío, A., 10.1016/j.crma.2007.11.009, C. R. Acad. Sci. Paris, Ser. I. Math. 346 (2008), 67–70. (2008) Zbl1134.53025MR2385057DOI10.1016/j.crma.2007.11.009
  10. Naveira, A. M., Tarrío, A., 10.1007/s00605-008-0551-3, Monatsh. Math. 158 (3) (2008), 231–246. (2008) Zbl1152.53039DOI10.1007/s00605-008-0551-3
  11. Tsukada, K., 10.2996/kmj/1138043656, Kodai Math. J. 19 (1996), 395–437. (1996) Zbl0871.53017MR1418571DOI10.2996/kmj/1138043656

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.