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Basic Constructions and Examples

Part of the Progress in Mathematics book series (PM, volume 268)

Abstract

Any Riemannian submersion can be used to generate new ones by deforming the metric in the vertical direction. To be specific, let π : (M, 〈, 〉 → B be a Riemannian submersion. Given φ : M → ℝ, define a new metric 〈, 〉φ on M by
$$\left\langle {e,f} \right\rangle _\phi = e^{2\phi (p)} \left\langle {e^v ,f^v } \right\rangle + \left\langle {e^h ,f^h } \right\rangle , e,f \in M_p , p \in M. $$
Since the horizontal metric is unchanged, π : (M, 〈, 〉φ) → B is still a Riemannian submersion. X, Y, Z will denote basic fields, Ti vertical ones, and \( \tilde \nabla \), \( \tilde R \) the Levi-Civita connection and curvature tensor, respectively, of 〈, 〉φ. We will assume that the deformation is constant along fibers, or equivalently, that the gradient of φ is basic.

Keywords

Sectional Curvature Ricci Curvature Principal Bundle Warped Product Holonomy Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Birkhäuser Verlag AG 2009

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