Constancy of Jacobi osculating rank of g.o. spaces of compact and non-compact type

• Autores: Teresa Arias Marco , Andreas Arvanitoyeorgos, Antonio Martínez Naveira
• Localización: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas ( RACSAM ), ISSN-e 1578-7303, Vol. 105, Nº. 1, 2011, págs. 207-221
• Idioma: inglés
• DOI: 10.1007/s13398-011-0019-5
• Texto completo no disponible (Saber más ...)
• Resumen

  • We prove that if M is either a compact g. o. space which is not naturally reductive, or a g. o. space which admits a transitive non-compact semisimple Lie group of isometries and is not naturally reductive, then its Jacobi osculating rank is not always constant. © 2011 Springer-Verlag.

• Referencias bibliográficas
  • Alekseevsky, D., Arvanitoyeorgos, A., Riemannian flag manifolds with homogeneous geodesics (2007) Trans. Am. Math. Soc, 359, pp. 3769-3789
  • Arias-Marco, T., (2009) Some Curvature Conditions of Riemannian Manifolds, Ledger's Conditions and Jacobi Osculating Rank, , Saarbrücken:...
  • Arias-Marco, T., Constant Jacobi osculating rank of U(3)/(U(1) × U(1) × U(1)) (2009) Arch. Math. (Brno), 45, pp. 273-286
  • Arias-Marco, T., (2009) Methods for Solving the Jacobi Equation. Constant Osculating Rank Vs. Constant Jacobi Osculating Rank. Differential...
  • Arias-Marco, T., Naveira, A.M., Constant Jacobi osculating rank of a g.o. space. A method to obtain explicitly the Jacobi operator (2009)...
  • Arnold, V.I., (1978) Mathematical Methods of Classical Mechanics, , Berlin: Springer
  • Berndt, J., Tricerri, F., Vanhecke, L., (1995) Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces. Lecture Notes in Mathematics,...
  • Dušek, Z., Survey on homogeneous geodesics (2009) Note Math., 28 (SUPPL. 1), pp. 147-168
  • Dušek, Z., The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds (2010) J. Geom. Phys., 60, pp. 687-689
  • Dušek, Z., Kowalski, O., Nikčević, S.Z., New examples of Riemannian g. o. manifolds in dimension 7 (2004) Diff. Geom. Appl., 21, pp. 65-78
  • Gagnon, L., Hussin, V., Winternitz, P., Nonlinear equations with superposition formulas and exceptional group G2. III. The superposition...
  • González-Dávila, J.C., Jacobi osculating rank and isotropic geodesics on naturally reductive 3-manifolds (2009) Diff. Geom. Appl., 27, pp....
  • González-Dávila, J.C., Naveira, A.M., Jacobi fields and osculating rank of the Jacobi operator in some special classes of homogeneous Riemannian...
  • Gordon, C.S., Homogeneous Riemannian manifolds whose geodesics are orbits (1996) Prog. Nonlinear Diff. Equ. Appl., 20, pp. 155-174
  • Kajzer, V.V., Conjugate point of left-invariant metrics on Lie Groups (1990) J. Soviet Math., 34, pp. 32-44
  • Kaplan, A., On the geometry of groups of Heisenberg type (1983) Bull. Lond Math. Soc., 15, pp. 35-42
  • Kobayashi, S., Nomizu, K., (1963) Foundations of Differential Geometry, I-II. , Interscience, New York
  • Kostant, B., On differential geometry and homogeneous spaces II (1956) Proc. Natl. Acad. Sci. USA, 42, pp. 354-357
  • Kowalski, O., Nikčević, S.Z., On geodesic graphs of Riemannian g.o. spaces (1999) Arch. Math., 73, pp. 223-234
  • Kowalski, O., Szenthe, J., On the existence of homogeneous geodesics in homogeneous Riemannian manifolds (2000) Geom. Ded., 81, pp. 209-214....
  • Kowalski, O., Vanhecke, L., Riemannian manifolds with homogeneosus geodesics (1991) Boll. Un. Math. Ital. B, 7 (5), pp. 189-246
  • Macías-Virgós, E., Naveira, A.M., Tarrío, A., The constant osculating rank of the Wilking manifold V3 (2008) C. R. Acad. Sci. Paris, Ser....
  • Marinosci, R.A., Homogeneous geodesics in a three-dimensional Lie group (2002) Comment. Math. Univ. Carolinae, 43, pp. 261-270
  • Naveira, A.M., Tarrío, A., A method for the resolution of the Jacobi equation Y′′ + RY = 0 on the manifold Sp(2)/SU(2) (2008) Monasth....
  • Riehm, C., Explicit spin representations and Lie algebras of Heisenberg type (1984) J. Lond. Math. Soc, 29, pp. 46-62
  • Szenthe, J., Homogeneous geodesics of left-invariant metrics (2000) Univ. Iagellonicae Acta Math. Fasc, XXXVIII, pp. 99-103
  • Tamaru, H., Riemannian g.o. spaces fibered over irreducible symmetric spaces (1999) Osaka J. Math., 36, pp. 835-851
  • Tsukada, K., Totally geodesic submanifolds of Riemannian manifolds and curvature invariant subspaces (1996) Kodai Math. J., 19, pp. 395-437
  • Vingerg, E.B., Invariant linear connections in a homogeneous manifold (1960) Trudy MMO, 6, pp. 191-210