Natural Tensors

GUILLERMO HENRY

La Falda, Cordoba, Argentina

Congreso; II Encuentro de Geometria Diferencial; 2005

Famaf, Universidad de Cordoba

The aim of this work is the study of Natural Metrics on the tangent bundle TM of Riemannian manifolds (M, g), and the relationship between the geometry of the base manifolds and its tangent bundle, equipped with a natural metric. By a Natural Metric G we mean a natural tensor of type (0,2) on TM which make the canonical projection from the tangent bundle (TM, G) onto (M, g) a Riemannian submersion. Examples of these met- rics are the well known Sasaki and Cheeger-Gromoll metrics, which were studied by Kowalski, Sekizawa, Aso, Muso, Triceri and several others. Con- sidering the manifold N:O(M)xRn and the O´Neill formula for submersions, we calculated, in an appropriate basis, the curvature tensor of (TM ,G). As a consequence of the curvature equations, in which one can see the relation between the tangent bundle curvature and the base manifold curvature, we obtain expressions for the sectional curvature of (TM, G), that generalize the known results for the Sasaki and Cheeger-Gromoll metrics. Let G be a natural metric. If (TM, G) is at then (M, g) is also at. For the Sasaki met- ric the converse holds, but it´s not true in general (Cheeger-Gromoll metric it´s a counterexample). We give a characterization of the natural metrics which verify the following statement: (TM, G) is at if and only if (M, g) is flat.