Notes on the Sasaki metric

TL;DR

This paper surveys the geometry of the tangent bundle of a Riemannian manifold with Sasaki's metric, exploring invariants, lifts of vector fields, and new equations related to the mirror map.

Contribution

It revisits Sasaki's classical results using different methods and introduces new equations of the mirror map, enhancing understanding of tangent bundle geometry.

Findings

Analysis of invariants of the classical metric

Relations between vector fields and geometric structures

Introduction of new equations of the mirror map

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.