Abstract.
We construct a three-parameter family of contact metric structures on the unit tangent sphere bundle T 1 M of a Riemannian manifold M and we study some of their special properties related to the Levi-Civita connection. More precisely, we give the necessary and sufficient conditions for a constructed contact metric structure to be K-contact, Sasakian, to satisfy some variational conditions or to define a strongly pseudo-convex CR-structure. The obtained results generalize classical theorems on the standard contact metric structure of T 1 M.
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References
KMT Abbassi (2004) ArticleTitleNote on the classification Theorems of g-natural metrics on the tangent bundle of a Riemannian manifold (M, g) Comment Math Univ Carolinae 45 591–596 Occurrence Handle1097.53013 Occurrence Handle2103077
Abbassi KMT, Kowalski O (2005) On g-natural metrics on unit tangent sphere bundles of Riemannian manifolds. Preprint
Abbassi KMT, Kowalski O (2005) On g-natural metrics with constant scalar curvature on unit tangent sphere bundles. In: Matsushita Y et al (eds) Topics in Almost Hermitian Geometry and Related Fields, Proc in Honor of K. Sekigawa’s 60th birthday, Hackensack, NJ: World Scientific, pp 1–29
Abbassi KMT, Kowalski O (2005) Some Einstein g-natural metrics on unit tangent sphere bundles over space forms. Preprint
KMT Abbassi M Sarih (2005) ArticleTitleOn natural metrics on tangent bundles of Riemannian manifolds Arch Math (Brno) 41 71–92 Occurrence Handle1114.53015 Occurrence Handle2142144
KMT Abbassi M Sarih (2005) ArticleTitleOn some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds Diff Geometry and Appl 22 19–47 Occurrence Handle1068.53016 Occurrence Handle10.1016/j.difgeo.2004.07.003 Occurrence Handle2106375
DE Blair (1984) ArticleTitleCritical associated metrics on contact manifolds J Austral Math Soc 37 IssueIDSeries A 82–88 Occurrence Handle0552.53014 Occurrence Handle742245
DE Blair (1991) ArticleTitleCritical associated metrics on contact manifolds III J Austral Math Soc 50 IssueIDSeries A 189–196 Occurrence Handle0733.53015 Occurrence Handle1094916 Occurrence Handle10.1017/S1446788700032675
DE Blair (2002) Riemannian geometry of contact and sympletic manifolds Birkäuser Basel
E Boeckx L Vanhecke (2000) ArticleTitleHarmonic and minimal vector fields on tangent and unit tangent bundles Diff Geom Appl 13 77–93 Occurrence Handle0973.53053 Occurrence Handle10.1016/S0926-2245(00)00021-8 Occurrence Handle1775222
E Boeckx D Perrone L Vanhecke (1998) ArticleTitleUnit tangent sphere bundles and two-point homogeneous spaces Period Math Hungar 36 79–95 Occurrence Handle0924.53014 Occurrence Handle10.1023/A:1004629423529 Occurrence Handle1694613
G Calvaruso (2005) Contact metric geometry of the unit tangent sphere bundle O Kowalski (Eds) et al. Complex, Contact and Symmetric Manifolds Birkäuser Boston 41–57
Calvaruso G, Perrone D (2006) H-contact unit tangent sphere bundles. Rocky Mount J Math to appear
E Cartan (1983) Geometry of Riemannian Spaces Math Sci Press Brookline, MA Occurrence Handle0603.53001
QS Chi (1988) ArticleTitleA curvature characterization of certain locally rank-one symmetric spaces J Differ Geom 28 187–202 Occurrence Handle0654.53053 Occurrence Handle961513
I Kolář PW Michor J Slovák (1993) Natural Operations in Differential Geometry Springer Berlin Heidelberg New York Occurrence Handle0782.53013
O Kowalski M Sekizawa (1988) ArticleTitleNatural transformations of Riemannian metrics on manifolds to metrics on tangent bundles – a classification Bull Tokyo Gakugei Univ 40 1–29 Occurrence Handle974641
Y Nikolayevsky (2005) ArticleTitleOsserman conjecture in dimension n ≠ 8,16 Math Ann 331 505–522 Occurrence Handle1075.53016 Occurrence Handle10.1007/s00208-004-0580-8 Occurrence Handle2122538
Y Nikolayevsky (2004) ArticleTitleOsserman manifolds of dimension 8 Manuscripta Math 115 31–53 Occurrence Handle1065.53034 Occurrence Handle10.1007/s00229-004-0480-y Occurrence Handle2092775
Nikolayevsky Y (2005) On Osserman conjecture in dimension 16. Proceedings of the “Conference on contemporary geometry and related topics”, Belgrade, June 27–July 1
D Perrone (1992) ArticleTitleTorsion and critical metrics on contact (2n + 1)-manifolds Monatsh Math 114 245–259 Occurrence Handle0765.58007 Occurrence Handle10.1007/BF01299383 Occurrence Handle1203974
D Perrone (1994) ArticleTitleTangent sphere bundles satisfying ∇ξ τ = 0 J Geom 49 178–188 Occurrence Handle0794.53030 Occurrence Handle10.1007/BF01228060 Occurrence Handle1261117
S Tanno (1989) ArticleTitleVariational problems on contact Riemannian manifolds Trans Amer Math Soc 314 349–379 Occurrence Handle0677.53043 Occurrence Handle10.2307/2001446 Occurrence Handle1000553
S Tanno (1992) ArticleTitleThe standard CR structure on the unit tangent sphere bundle Tôhoku Math J 44 535–543 Occurrence Handle0779.53024 Occurrence Handle1190919
Y Tashiro (1969) ArticleTitleOn contact structures on tangent sphere bundles Tôhoku Math J 21 117–143 Occurrence Handle0182.55501 Occurrence Handle244885
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Author supported by funds of the University of Lecce.
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Abbassi, M., Calvaruso, G. g-Natural Contact Metrics on Unit Tangent Sphere Bundles. Mh Math 151, 89–109 (2007). https://doi.org/10.1007/s00605-006-0421-9
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DOI: https://doi.org/10.1007/s00605-006-0421-9