arXiv subjects

Gerard Walschap

Publications and source records attributed to Gerard Walschap.

4 recordsLinked to original sources

arXiv2013

Twisted submersions in nonnegative sectional curvature

B. Wilking introduced the dual foliation associated to a metric foliation in a Riemannian manifold with nonnegative sectional curvature, and proved that when the curvature is strictly positive, the dual foliation contains a single leaf, so that any two points in the ambient space can be joined by a horizontal curve. We show that the same phenomenon often occurs for nonnegatively curved Riemannian submersions even without the strict positive curvature condition, and irrespective of the particular metric.

math.DG↗

arXiv2011

The curvature operator at the soul

We prove two splitting theorems, one topological, the other metric, for open manifolds with nonnegative sectional curvature.

math.DG↗

arXiv2009

Submetries vs. submersions

We study submetries between Alexandrov spaces and show how some of the usual features of Riemannian submersions fail due to the lack of smoothness.

math.DG↗

arXiv1997

Parallel Connections Over Symmetric Spaces

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces to finding representations of the structure group of the canonical principal bundle.

dg-ga↗