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.

scholarly journals Slant Curves in the Unit Tangent Bundles of Surfaces

ISRN Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Zhong Hua Hou ◽  
Lei Sun
Keyword(s):  
Vector Field ◽  
Tangent Bundle ◽  
Unit Vector ◽  
Sasaki Metric ◽  
Geometric Properties ◽  
Unit Vector Field ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Slant Curves

Let (M,g) be a surface and let (U(TM),G) be the unit tangent bundle of M endowed with the Sasaki metric. We know that any curve Γ(s) in U(TM) consist of a curve γ(s) in M and as unit vector field X(s) along γ(s). In this paper we study the geometric properties γ(s) and X(s) satisfying when Γ(s) is a slant geodesic.

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.

ALMOST PARACONTACT STRUCTURES ON TANGENT SPHERE BUNDLE

2013 ◽  
Vol 10 (09) ◽  
pp. 1320015 ◽  
Author(s):  
E. PEYGHAN ◽  
A. NADERIFARD ◽  
A. TAYEBI
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Product Structure ◽  
Sphere Bundle ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
Almost Product Structure

Using the almost product structure given by Druta, we introduce a metrical framed f(3, -1)-structure on the tangent bundle of a Riemannian manifold. Then by restricting this metrical framed f(3, -1)-structure to the tangent sphere bundle, we obtain an almost metrical paracontact structure on the tangent sphere bundle.

scholarly journals Totally geodesic property of the unit tangent sphere bundle with g-natural metrics

2018 ◽  
Vol 10 (1) ◽  
pp. 152-166
Author(s):  
Esmaeil Peyghan ◽  
Farshad Firuzi
Keyword(s):  
Riemannian Manifold ◽  
Sphere Bundle ◽  
Totally Geodesic ◽  
Base Manifold ◽  
Tangent Sphere Bundle ◽  
Tangent Sphere ◽  
Totally Geodesic Submanifolds ◽  
Negative Constant ◽  
Metric Structures ◽  
Unit Tangent Sphere Bundle

Abstract In this paper, we consider the tangent bundle of a Riemannian manifold (M, g) with g-natural metrics and among all of these metrics, we specify those with respect to which the unit tangent sphere bundle with induced g-natural metric is totally geodesic. Also, we equip the unit tangent sphere bundle T1M with g-natural contact (paracontact) metric structures, and we show that such structures are totally geodesic K-contact (K-paracontact) submanifolds of TM, if and only if the base manifold (M, g) has positive (negative) constant sectional curvature. Moreover, we establish a condition for g-natural almost contact B-metric structures on T1Msuch that these structures be totally geodesic submanifolds of TM.

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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