scholarly journals Infinitesimal affine transformations of the tangent bundles with Sasaki metric

1974 ◽  
Vol 26 (3) ◽  
pp. 353-361 ◽  
Author(s):  
Kōjirō Satō
Keyword(s):  
Affine Transformations ◽  
Sasaki Metric ◽  
Tangent Bundles

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 Generalized Sasaki Metrics on Tangent Bundles

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Izu Vaisman
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Riemannian Metrics ◽  
Sasaki Metric ◽  
Canonical Metric ◽  
Generalized Complex ◽  
Tangent Manifold ◽  
Metric Connection ◽  
Kähler Structures ◽  
Tangent Bundles

Abstract We define a class of metrics that extend the Sasaki metric of the tangent manifold of a Riemannian manifold. The new metrics are obtained by the transfer of the generalized (pseudo-)Riemannian metrics of the pullback bundle π−1(TM⊕T*M), where π : T M → M is the natural projection. We obtain the expression of the transferred metric and define a canonical metric connection with torsion. We calculate the torsion, curvature and Ricci curvature of this connection and give a few applications of the results. We also discuss the transfer of generalized complex and generalized Kähler structures from the pullback bundle to the tangent manifold.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

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