Some examples of manifolds of nonnegative curvature

Abstract

The purpose of this note is to describe some examples of manifolds of nonnegative curvature and positive Ricci curvature.Apart from homogeneous spaces, no such examples appear in the literature.Our main tool is the formula of O'Neill [8] for riemannian submersions.Recall that the map π\ M n+k -> N n of riemannian manifolds is called a riemannian submersion if 1. π is a differentiate submersion, i.e., for all m e M, rank dπ m = n, 2. dπ\H m is an isometry for all m e M.Here H m is the orthogonal complement of the kernelbe orthonormal, x, y their horizontal lifts at m, and K, K denote sectional curvature.Then the formula of O'Neill says Let G X M -» M be an action of a Lie group on M such that all orbits are closed and of the same type.Then π: M -> G/M is a submersion, and any G-invariant riemannian structure on M induces in an obvious way a riemannian structure on G\M such that π becomes a riemannian submersion.If M has nonnegative curvature, then so does G\M. 1 If G acts on N 19 M ί freely and properly discontinuously on iV l5 then it acts freely and properly discontinuously on N λ X M 1 by the diagonal action.Hence further examples arise by taking products.Example 1 (Associated bundles).Let M = G λ X M l5 where G λ is a Lie group with bi-invariant metric, and M 1 has nonnegative curvature.Suppose G C G x is a closed subgroup which acts on M λ by isometries.Then (g 1? m) -> (gi 8~\8m) defines a free properly discontinuous action of G onM.As above, G\M inherits a metric of nonnegative curvature.Topologically, G\M is of course the bundle with fibre M 1 associated to the principal fibration G-^ G 1 -R

Locations

• Journal of Differential Geometry - View - PDF