WHEN IS THE TANGENT SPHERE BUNDLE LOCALLY SYMMETRIC?

1989 ◽  
pp. 15-30 ◽  
Author(s):  
David B. Blair
Keyword(s):  
Sphere Bundle ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere

Geodesics on the unit tangent bundle

2003 ◽  
Vol 133 (6) ◽  
pp. 1209-1229 ◽  
Author(s):  
J. Berndt ◽  
E. Boeckx ◽  
P. T. Nagy ◽  
L. Vanhecke
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Unit Vector ◽  
Sphere Bundle ◽  
Sasaki Metric ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
Unit Tangent Bundle ◽  
Unit Tangent Sphere Bundle

A geodesic γ on the unit tangent sphere bundle T1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it. We study the curves x. In particular, we investigate for which manifolds (M, g) all these curves have constant first curvature κ1 or have vanishing curvature κi for some i = 1, 2 or 3.

AN INTEGRAL FORMULA AND ITS APPLICATION TO NORMAL EULER NUMBER

2008 ◽  
Vol 19 (08) ◽  
pp. 891-897 ◽  
Author(s):  
JIANQUAN GE ◽  
ZIZHOU TANG
Keyword(s):  
Euclidean Space ◽  
Euler Number ◽  
Integral Formula ◽  
Closed Manifold ◽  
Sphere Bundle ◽  
Geometrical Proof ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
General Integral ◽  
Topological Theorem

We establish a general integral formula over sphere, and then apply it to give a geometrical proof of the celebrated topological theorem of Lashof and Smale, which asserts that the tangential degree of the tangent sphere bundle coincides with the normal Euler number for an immersion Mn → E2n of an oriented closed manifold into Euclidean space of twice dimension.

scholarly journals Geodesics and Killing vector fields on the tangent sphere bundle

1998 ◽  
Vol 151 ◽  
pp. 91-97 ◽  
Author(s):  
Tatsuo Konno ◽  
Shukichi Tanno
Keyword(s):  
Vector Field ◽  
Constant Curvature ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Sphere Bundle ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
Space Of Constant Curvature ◽  
Unit Tangent Sphere Bundle

Abstract.We show that any Killing vector field on the unit tangent sphere bundle with Sasaki metric of a space of constant curvature k ≠ 1 is fiber preserving by studying some property of geodesies on the bundle. As a consequence, any Killing vector field on the unit tangent sphere bundle of a space of constant curvature k ≠ 1 can be extended to a Killing vector field on the tangent bundle.

scholarly journals H-CONTACT UNIT TANGENT SPHERE BUNDLES OF FOUR-DIMENSIONAL RIEMANNIAN MANIFOLDS

2011 ◽  
Vol 91 (2) ◽  
pp. 243-256 ◽  
Author(s):  
SUN HYANG CHUN ◽  
JEONGHYEONG PARK ◽  
KOUEI SEKIGAWA
Keyword(s):  
Sphere Bundle ◽  
Base Manifold ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
Contact Metric Structure ◽  
Standard Contact ◽  
Unit Tangent Sphere Bundles ◽  
Contact Unit ◽  
Necessary And Sufficient ◽  
Unit Tangent Sphere Bundle

AbstractWe study the geometric properties of a base manifold whose unit tangent sphere bundle, equipped with the standard contact metric structure, is H-contact. We prove that a necessary and sufficient condition for the unit tangent sphere bundle of a four-dimensional Riemannian manifold to be H-contact is that the base manifold is 2-stein.

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