On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds

It is well known that if the tangent bundle TM of a Riemannian manifold ( M , g ) is endowed with the Sasaki metric g s , 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...

Full description

Saved in:
Bibliographic Details
Published in:Differential geometry and its applications 2005, Vol.22 (1), p.19-47
Main Authors: Abbassi, Mohamed Tahar Kadaoui, Sarih, Maâti
Format: Article
Language:English
Subjects:
Curvatures
F-tensor field
g-natural metric
Natural operation
Riemannian manifold
Tangent bundle
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:It is well known that if the tangent bundle TM of a Riemannian manifold ( M , g ) is endowed with the Sasaki metric g s , 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 g s 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.
ISSN:0926-2245
1872-6984
DOI:10.1016/j.difgeo.2004.07.003