Tokyo Journal of Mathematics

The Geometry of Generalised Cheeger-Gromoll Metrics

Michèle BENYOUNES, Eric LOUBEAU, and Chris M. WOOD

Full-text: Open access

Abstract

We study the geometry of the tangent bundle equipped with a two-parameter family of metrics, deforming the Sasaki and Cheeger-Gromoll metrics. After deriving the expression for the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. We identify all metrics whose restrictions to the fibres have positive sectional curvature. When the base manifold is a space form, we characterise metrics with non-negative sectional curvature and show that one can always find parameters ensuring positive scalar curvature. This extends to compact manifolds and, under some curvature conditions, to general manifolds.

Article information

Source
Tokyo J. Math., Volume 32, Number 2 (2009), 287-312.

Dates
First available in Project Euclid: 22 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1264170234

Digital Object Identifier
doi:10.3836/tjm/1264170234

Mathematical Reviews number (MathSciNet)
MR2589947

Zentralblatt MATH identifier
1200.53025

Subjects
Primary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]

Citation

BENYOUNES, Michèle; LOUBEAU, Eric; WOOD, Chris M. The Geometry of Generalised Cheeger-Gromoll Metrics. Tokyo J. Math. 32 (2009), no. 2, 287--312. doi:10.3836/tjm/1264170234. https://projecteuclid.org/euclid.tjm/1264170234


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