Notes on the Sasaki metric

Abstract

We survey on the geometry of the tangent bundle of a Riemannian manifold, endowed with the classical metric established by S. Sasaki 60 years ago. Following the results of Sasaki, we try to write and deduce them by different means. Questions of vector fields, mainly those arising from the base, are related as invariants of the classical metric, contact and Hermitian structures. Attention is given to the natural notion of extension or complete lift of a vector field, from the base to the tangent manifold. Few results are original, but finally new equations of the mirror map are considered.

A blow-up method to prescribed mean curvature graphs with fixed boundaries

“…Thus our assumption is false and Ω + is also a Λ-perimeter minimizer in Ω δ . We obtain the conclusion (1).…”

“…Since ∂Ω is connected, this means curvature conclusion holds on the whole ∂Ω. However, this is a contradiction to condition (1). No matter which case we will obtain the contradiction.…”

“…Let M be a m(≥ 2)-dimensional manifold with a metric σ. Let T M be the tangent bundle with the natural Sasaki metric σ s (see [1,39]) from σ. Therefore we have a C 1,α norm for C 1,α functions on T M .…”

On the geometry of the tangent bundle with gradient Sasaki metric

“…The sectional curvatures and the scalar curvature of this metric have been obtained in Refs. [8][9][10][11][12][13][14][15][16]. These results are completed in 2002 by S. Gudmundson and E. Kappos in Ref.…”

“…V.Oproiu and his collaborators constructed a family of Riemannian metrics on the tangent bundles of Riemannian manifolds which possess interesting geometric properties (see Refs. [19, 20]). In particular, the scalar curvature of TM can be constant also for a non-flat base manifold with constant sectional curvature.…”