# Invariants and flow geometry

J. González-Dávila; L. Vanhecke

Colloquium Mathematicae (1999)

- Volume: 81, Issue: 1, page 33-50
- ISSN: 0010-1354

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topGonzález-Dávila, J., and Vanhecke, L.. "Invariants and flow geometry." Colloquium Mathematicae 81.1 (1999): 33-50. <http://eudml.org/doc/210728>.

@article{González1999,

abstract = {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.},

author = {González-Dávila, J., Vanhecke, L.},

journal = {Colloquium Mathematicae},

keywords = {flow model spaces; normal, contact and curvature homogeneous flows; invariants and characterizations of special Riemannian manifolds; flows generated by a unit Killing vector field; curvature homogeneous flow; invariants; special Riemannian manifolds; Sasakian space; flow geometry},

language = {eng},

number = {1},

pages = {33-50},

title = {Invariants and flow geometry},

url = {http://eudml.org/doc/210728},

volume = {81},

year = {1999},

}

TY - JOUR

AU - González-Dávila, J.

AU - Vanhecke, L.

TI - Invariants and flow geometry

JO - Colloquium Mathematicae

PY - 1999

VL - 81

IS - 1

SP - 33

EP - 50

AB - 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.

LA - eng

KW - flow model spaces; normal, contact and curvature homogeneous flows; invariants and characterizations of special Riemannian manifolds; flows generated by a unit Killing vector field; curvature homogeneous flow; invariants; special Riemannian manifolds; Sasakian space; flow geometry

UR - http://eudml.org/doc/210728

ER -

## References

top- [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. Zbl0319.53026
- [3] D. E. Blair and L. Vanhecke, Symmetries and φ-symmetric spaces, Tôhoku Math. J. 39 (1987), 373-383. Zbl0632.53039
- [4] E. Boeckx, O. Kowalski and L. Vanhecke, Riemannian Manifolds of Conullity Two, World Scientific, Singapore, 1996. Zbl0904.53006
- [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. Zbl0689.53021
- [7] P. Bueken and L. Vanhecke, Curvature characterizations in contact geometry, Riv. Mat. Univ. Parma 14 (1988), 303-313. Zbl0689.53021
- [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. Zbl0839.53017
- [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. Zbl0890.53037
- [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. Zbl0859.53016
- [12] J. C. González-Dávila and L. Vanhecke, Geodesic spheres and isometric flows, Colloq. Math. 67 (1994), 223-240. Zbl0827.53040
- [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. Zbl0912.53023
- [14] A. Gray and L. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math. 142 (1979), 157-198. Zbl0428.53017
- [15] D. Janssens and L. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. 4 (1981), 1-27. Zbl0472.53043
- [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. Zbl0862.53039
- [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. Zbl0436.53048
- [19] Y. Shibuya, Some isospectral problems, ibid. 5 (1982), 1-12. Zbl0488.53034
- [20] I. M. Singer, Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13 (1960), 685-697. Zbl0171.42503
- [21] T T. Takahashi, Sasakian φ-symmetric spaces, Tôhoku Math. J. 29 (1977), 91-113. Zbl0343.53030
- [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. Zbl0718.53022
- [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. Zbl0409.53034
- [25] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398. Zbl0484.53014
- [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. Zbl0585.53043
- [27] L. Vanhecke, Scalar curvature invariants and local homogeneity, Rend. Circ. Mat. Palermo (2) Suppl. 49 (1997), 275-287. Zbl0894.53046
- [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. Zbl0156.42501
- [30] K. Yano and M. Kon, Structures on Manifolds, Ser. Pure Math. 3, World Scientific, Singapore, 1984.

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