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1999 | 81 | 1 | 33-50
Tytuł artykułu

Invariants and flow geometry

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We continue the study of Riemannian manifolds (M,g) equipped with an isometric flow $ℱ_ξ$ generated by a unit Killing vector field ξ. We derive some new results for normal and contact flows and use invariants with respect to the group of ξ-preserving isometries to charaterize special (M,g,$ℱ_ξ$), in particular Einstein, η-Einstein, η-parallel and locally Killing-transversally symmetric spaces. Furthermore, we introduce curvature homogeneous flows and flow model spaces and derive an algebraic characterization of Killing-transversally symmetric spaces by using the curvature tensor of special flow model spaces. All these results extend the corresponding theory in Sasakian geometry to flow geometry.
Rocznik
Tom
81
Numer
1
Strony
33-50
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-12-17
Twórcy
  • Departamento de Matemática Fundamental, Sección de Geometría y Topología, Universidad de La Laguna, La Laguna, Spain
autor
  • Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium
Bibliografia
  • [1] A. L. Besse, Einstein Manifolds, Ergeb. Math. Grenzgeb. (3) 10, Springer, Berlin, 1987.
  • [2] B D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer, Berlin, 1976.
  • [3] D. E. Blair and L. Vanhecke, Symmetries and φ-symmetric spaces, Tôhoku Math. J. 39 (1987), 373-383.
  • [4] E. Boeckx, O. Kowalski and L. Vanhecke, Riemannian Manifolds of Conullity Two, World Scientific, Singapore, 1996.
  • [5] P. Bueken, Reflections and rotations in contact geometry, doctoral dissertation, Katholieke Universiteit Leuven, 1992.
  • [6] P. Bueken and L. Vanhecke, Algebraic characterizations by means of the curvature in contact geometry, in: Proc. III Internat. Sympos. Diff. Geom., Pe níscola, Lecture Notes in Math. 1410, Springer, Berlin, 1988, 77-86.
  • [7] P. Bueken and L. Vanhecke, Curvature characterizations in contact geometry, Riv. Mat. Univ. Parma 14 (1988), 303-313.
  • [8] B. Y. Chen and L. Vanhecke, Differential geometry of geodesic spheres, J. Reine Angew. Math. 325 (1981), 28-67.
  • [9] J. C. González-Dávila, M. C. González-Dávila and L. Vanhecke, Reflections and isometric flows, Kyungpook Math. J. 35 (1995), 113-144.
  • [10] J. C. González-Dávila, M. C. González-Dávila and L. Vanhecke, Classification of Killing-transversally symmetric spaces, Tsukuba J. Math. 20 (1996), 321-347.
  • [11] J. C. González-Dávila, M. C. González-Dávila and L. Vanhecke, Normal flow space forms and their classification, Publ. Math. Debrecen 48 (1996), 151-173.
  • [12] J. C. González-Dávila and L. Vanhecke, Geodesic spheres and isometric flows, Colloq. Math. 67 (1994), 223-240.
  • [13] J. C. González-Dávila and L. Vanhecke, D'Atri spaces and C-spaces in flow geometry, Indian J. Pure Appl. Math. 29 (1998), 487-499.
  • [14] A. Gray and L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math. 142 (1979), 157-198.
  • [15] D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), 1-27.
  • [16] O. Kowalski, F. Prüfer and L. Vanhecke, D'Atri spaces, in: Topics in Geometry: In Memory of Joseph D'Atri, S. Gindikin (ed.), Progr. Nonlinear Differential Equations, 20, Birkhäuser, Boston, 1996, 241-284.
  • [17] O B. O'Neill, The fundamental equation of a submersion, Michigan Math. J. 13 (1966), 459-469.
  • [18] Y. Shibuya, The spectrum of Sasakian manifolds, Kodai Math. J. 3 (1980), 197-211.
  • [19] Y. Shibuya, Some isospectral problems, ibid. 5 (1982), 1-12.
  • [20] I. M. Singer, Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685-697.
  • [21] T T. Takahashi, Sasakian φ-symmetric spaces, Tôhoku Math. J. 29 (1977), 91-113.
  • [22] Ph. Tondeur, Foliations on Riemannian Manifolds, Universitext, Springer, Berlin, 1988.
  • [23] Ph. Tondeur and L. Vanhecke, Transversally symmetric Riemannian foliations, Tôhoku Math. J. 42 (1990), 307-317.
  • [24] F. Tricerri and L. Vanhecke, Decomposition of a space of curvature tensors on a quaternionic Kähler manifold and spectrum theory, Simon Stevin 53 (1979), 163-173.
  • [25] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398.
  • [26] F. Tricerri and L. Vanhecke, Variétés riemanniennes dont le tenseur de courbure est celui d'un espace symétrique irréductible, C. R. Acad. Sci. Paris Sér. I 302 (1986), 233-235.
  • [27] L. Vanhecke, Scalar curvature invariants and local homogeneity, Rend. Circ. Mat. Palermo (2) Suppl. 49 (1997), 275-287.
  • [28] Y. Watanabe, Geodesic symmetries in Sasakian locally φ-symmetric spaces, Kodai Math. J. 3 (1980), 48-55.
  • [29] K. Yano and S. Ishihara, Fibred spaces with invariant Riemannian metric, Kōdai Math. Sem. Rep. 19 (1967), 317-360.
  • [30] K. Yano and M. Kon, Structures on Manifolds, Ser. Pure Math. 3, World Scientific, Singapore, 1984.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-cmv81i1p33bwm
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