Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9500463 | Differential Geometry and its Applications | 2005 | 29 Pages |
Abstract
It is well known that if the tangent bundle TM of a Riemannian manifold (M,g) is endowed with the Sasaki metric gs, then the flatness property on TM is inherited by the base manifold [Kowalski, J. Reine Angew. Math. 250 (1971) 124-129]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace gs by the most general Riemannian “g-natural metric” on TM (see [Kowalski and Sekizawa, Bull. Tokyo Gakugei Univ. (4) 40 (1988) 1-29; Abbassi and Sarih, Arch. Math. (Brno), submitted for publication]). In this direction, we prove that if (TM,G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M,g) possesses the same property, respectively. We also give explicit examples of g-natural metrics of arbitrary constant scalar curvature on TM.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Mohamed Tahar Kadaoui Abbassi, Maâti Sarih,