Basic Constructions and Examples

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


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.


Sectional Curvature Ricci Curvature Principal Bundle Warped Product Holonomy Group 
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© Birkhäuser Verlag AG 2009

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