On the Structure of Equidistant Foliations of Euclidean Space
Christian Boltner
TL;DR
This thesis studies equidistant foliations of Euclidean space, revealing their structure, existence of affine leaves, conditions for homogeneity, and providing new examples of inhomogeneous foliations.
Contribution
It generalizes Gromoll and Walschap's result, proves the existence of affine leaves, and constructs new inhomogeneous examples of equidistant foliations.
Findings
Equidistant foliations always have an affine leaf.
Under certain conditions, foliations are homogeneous.
New noncompact inhomogeneous examples are constructed.
Abstract
This thesis is concerned with equidistant foliations of Euclidean space, i.e. partitions into complete, connected, properly embedded smooth submanifolds. The space of leaves is an Alexandrov space of nonnegative curvature and the canonical projection is a submetry. Generalizing a result of Gromoll and Walschap we show that an equidistant foliation always has an affine leaf and we prove homogeneneity of the foliation under certain additional assumptions. Moreover, we give several reducibility results and construct new (noncompact) inhomogeneous examples of equidistant foliations.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Point processes and geometric inequalities · Geometric and Algebraic Topology