Tangent bundles with Sasaki metric

1975 ◽  
pp. 89-95
Author(s):  
Mu Chou Liu
Keyword(s):  
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.

scholarly journals Chern-Simons invariants and heterotic superpotentials

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Lara B. Anderson ◽  
James Gray ◽  
Andre Lukas ◽  
Juntao Wang
Keyword(s):  
Heterotic String ◽  
New Techniques ◽  
Vacuum Stability ◽  
Large Classes ◽  
Moduli Stabilization ◽  
String Compactifications ◽  
Key Features ◽  
Effective Theories ◽  
Chern Simons ◽  
Tangent Bundles

Abstract The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technical ingredient in our computations are real bundle morphisms between the gauge and tangent bundles. We find that there are large classes of examples, beyond the standard embedding, where the Chern-Simons superpotential vanishes. We also provide explicit examples for non-flat bundles where it is non-vanishing and non-integer quantized, generalizing previous results for Wilson lines.

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