On the geometry of the tangent bundle with gradient Sasaki metric

PurposeLet (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.Design/methodology/approachIn this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.FindingsThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.Originality/valueThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.

Lifts of connections to the bundle of (1,1) type tensor frames

In this paper we consider the bundle of (1,1) type tensor frames over a smooth manifold, define the horizontal and complete lifts of symmetric linear connection into this bundle. Also we study the properties of the geodesic lines corresponding to the complete lift of the linear connection and investigate the relations between Sasaki metric and lifted connections on the bundle of (1,1) type tensor frames.

The super-Sasaki metric on the antitangent bundle

We show how to lift a Riemannian metric and almost symplectic form on a manifold to a Riemannian structure on a canonically associated supermanifold known as the antitangent or shifted tangent bundle. We view this construction as a generalization of Sasaki’s construction of a Riemannian metric on the tangent bundle of a Riemannian manifold.