A class of metrics on tangent bundles of pseudo-Riemannian manifolds. (English) Zbl 1249.53020

Dida, Mohammed Hamou; Ikemakhen, Aziz

Arch. Math., Brno 47, No. 4, 293-308 (2011).

Let \(u\) be a section of the tangent bundle \(TM \to M\). This determines, for every vector field \(X\) on a manifold \(M\), the vector field (called vertical lift) \(X^v\) in the vertical subbundle \(VTM \subseteq TTM\) in an obvious way. Given an affine connection \(\nabla \) on \(M\), we obtain by the standard theory also the horizontal lift \(X^h\) which is a section of the horizontal subbundle \(HTM \subseteq TTM\). Assuming \(\nabla \) is the Levi-Civita connection of the pseudometric \(g\) on \(M\), horizontal and vertical lifts offer several ways how to define a pseudometric on \(TM\). The authors study the case of the Sasaki pseudometric \(g^s\) and a certain neutral signature pseudometric \(g^n\) on \(TM\). Specifically, the cases of \((TM,g^s)\) and \((TM,g^n)\) being locally symmetric, Einstein or Kählerian are characterized in terms of \((M,g)\). It is also argued that \((TM,g^s)\) and \((TM,g^n)\) provide examples of indecomposable reducible manifolds hence should contribute to the holonomy classification in general signature.

Reviewer: Josef Šilhan (Brno)

MSC:

53B30	Local differential geometry of Lorentz metrics, indefinite metrics
53C07	Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
53C29	Issues of holonomy in differential geometry
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

pseudo-Riemannian manifold; Sasaki metric; neutral metric; indecomposable-reducible manifold; Einstein manifold