Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles - a classification. (English) Zbl 0656.53021

Kowalski, Oldřich; Sekizawa, Masami

Bull. Tokyo Gakugei Univ., Sect. IV, Ser. Math. Nat. Sci. 40, 1-29 (1988).

Let (M,g) be a Riemannian manifold and TM its tangent bundle. As it is well-known there are three metrics on TM obtained from g, namely the Sasaki metric g, the horizontal lift \(g^ H\) and the vertical lift \(g^ V\). They are examples of natural constructions of second order. In this paper the authors describe explicitly all second order natural transformations of the form (M,g)\(\to (TM,G)\), where G is a pseudo- Riemannian metric on TM. Using some results of D. Krupka the problem is reduced to a classification problem of second order differential invariants which leads to solve a system of partial differential equations. Thus the authors prove that each naturally transformed metric G is a module over real functions generated by some generalizations of the classical lifts described above. Let us finally remark that a nonclassical example of a natural metric, namely the Cheeger-Gromoll metric is obtained by this way.

Reviewer: M.de Leon

MSC:

53B20	Local Riemannian geometry
53B30	Local differential geometry of Lorentz metrics, indefinite metrics
53C20	Global Riemannian geometry, including pinching

Keywords:

Sasaki metric; horizontal lift; vertical lift; second order natural transformations; pseudo-Riemannian metric; second order differential invariants; lifts; Cheeger-Gromoll metric