×

Unit tangent sphere bundles with constant scalar curvature. (English) Zbl 1079.53063

Summary: As a first step in the search for curvature homogeneous unit tangent sphere bundles we derive necessary and sufficient conditions for a manifold to have a unit tangent sphere bundle with constant scalar curvature. We give complete classifications for low dimensions and for conformally flat manifolds. Further, we determine when the unit tangent sphere bundle is Einstein or Ricci-parallel.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] J. Berndt, F. Tricerri and L. Vanhecke: Generalized Heisenberg groups and Damek-Ricci harmonic spaces. Lecture Notes in Math. 1598, Springer-Verlag, Berlin, Heidelberg, New York, 1995. · Zbl 0818.53067 · doi:10.1007/BFb0076902
[2] A. L. Besse: Manifolds all of whose geodesics are closed. Ergeb. Math. Grenzgeb. 93, Springer-Verlag, Berlin, Heidelberg, New York, 1978. · Zbl 0387.53010
[3] A. L. Besse: Einstein manifolds. Ergeb. Math. Grenzgeb. 3. Folge 10, Springer-Verlag, Berlin, Heidelberg, New York, 1987. · Zbl 0613.53001
[4] D. E. Blair: Contact manifolds in Riemannian geometry. Lecture Notes in Math. 509, Springer-Verlag, Berlin, Heidelberg, New York, 1976. · Zbl 0319.53026 · doi:10.1007/BFb0079307
[5] D. E. Blair: When is the tangent sphere bundle locally symmetric? Geometry and Topology. World Scientific, Singapore, 1989, pp. 15-30.
[6] D. E. Blair and T. Koufogiorgos: When is the tangent sphere bundle conformally flat? J. Geom. 49 (1994), 55-66. · Zbl 0815.53045 · doi:10.1007/BF01228050
[7] E. Boeckx, O. Kowalski and L. Vanhecke: Riemannian manifolds of conullity two. World Scientific, Singapore, 1996. · Zbl 0904.53006
[8] E. Boeckx and L. Vanhecke: Characteristic reflections on unit tangent sphere bundles. Houston J. Math. 23 (1997), 427-448. · Zbl 0897.53010
[9] E. Boeckx and L. Vanhecke: Curvature homogeneous unit tangent sphere bundles. Publ. Math. Debrecen 53 (1998), 389-413. · Zbl 0910.53008
[10] P. Bueken: Three-dimensional Riemannian manifolds with constant principal Ricci curvatures \(\rho _1=\rho _2\neq \rho _3\). J. Math. Phys. 37 (1996), 4062-4075. · Zbl 0866.53026 · doi:10.1063/1.531626
[11] B.-Y. Chen and L. Vanhecke: Differential geometry of geodesic spheres. J. Reine Angew. Math. 325 (1981), 28-67. · Zbl 0503.53013 · doi:10.1515/crll.1981.325.28
[12] P. Gilkey, A. Swann and L. Vanhecke: Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator. Quart. J. Math. Oxford 46 (1995), 299-320. · Zbl 0848.53023 · doi:10.1093/qmath/46.3.299
[13] A. Gray: Einstein-like manifolds which are not Einstein. Geom. Dedicata 7 (1978), 259-280. · Zbl 0378.53018 · doi:10.1007/BF00151525
[14] A. Gray and L. Vanhecke: Riemannian geometry as determined by the volumes of small geodesic balls. Acta Math. 142 (1979), 157-198. · Zbl 0428.53017 · doi:10.1007/BF02395060
[15] A. Gray and T. J. Willmore: Mean-value theorems for Riemannian manifolds. Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), 343-364. · Zbl 0495.53040 · doi:10.1017/S0308210500032571
[16] S. Ivanov and I. Petrova: Riemannian manifolds in which certain curvature operator has constant eigenvalues along each circle. Ann. Global Anal. Geom. 15 (1997), 157-171. · Zbl 0873.53029 · doi:10.1023/A:1006548328030
[17] O. Kowalski: A note to a theorem by K. Sekigawa. Comment. Math. Univ. Carolin. 30 (1989), 85-88. · Zbl 0679.53043
[18] O. Kowalski: A classification of Riemannian 3-manifolds with constant principal Ricci curvatures \(\rho _1=\rho _2\neq \rho _3\). Nagoya Math. J. 132 (1993), 1-36.
[19] O. Kowalski: An explicit classification of 3-dimensional Riemannian spaces satisfying \(R(X,Y)\cdot R=0\). Czechoslovak Math. J. 46 (1996), 427-474. · Zbl 0879.53014
[20] J. W. Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21 (1976), 293-329. · Zbl 0341.53030 · doi:10.1016/S0001-8708(76)80002-3
[21] E. Musso and F. Tricerri: Riemannian metrics on tangent bundles. Ann. Mat. Pura Appl. 150 (1988), 1-20. · Zbl 0658.53045 · doi:10.1007/BF01761461
[22] K. Sekigawa and L. Vanhecke: Volume preserving geodesic symmetries on four-dimensional Kähler manifolds. Differential Geometry Peñiscola, 1985, Proceedings, A. M. Naveira, A. Ferrández and F. Mascaró (eds.), Lecture Notes in Math. 1209, Springer, pp. 275-290.
[23] I. M. Singer: Infinitesimally homogeneous spaces. Comm. Pure Appl. Math. 13 (1960), 685-697. · Zbl 0171.42503 · doi:10.1002/cpa.3160130408
[24] I. M. Singer and J. A. Thorpe: The curvature of 4-dimensional Einstein spaces. Global Analysis. Papers in honor of K. Kodaira, Princeton University Press, Princeton, 1969, pp. 355-365.
[25] Z. I. Szabó: Structure theorems on Riemannian manifolds satisfying \(R(X,Y)\cdot R=0\), I, Local version. J. Differential Geom. 17 (1982), 531-582. · Zbl 0508.53025
[26] H. Takagi: Conformally flat Riemannian manifolds admitting a transitive group of isometries. Tôhoku Math. J. 27 (1975), 103-110. · Zbl 0323.53037 · doi:10.2748/tmj/1203529254
[27] A. Tomassini: Curvature homogeneous metrics on principal fibre bundles. Ann. Mat. Pura Appl. 172 (1997), 287-295. · Zbl 0933.53020 · doi:10.1007/BF01782616
[28] A. L. Yampol’skii: The curvature of the Sasaki metric of tangent sphere bundles (Russian). Ukrain. Geom. Sb. 28 (1985), 132-145.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.