Preserve one, preserve all
Meera Mainkar, Benjamin Schmidt

TL;DR
The paper investigates conditions under which functions preserving a small level set of the distance in a complete Riemannian manifold must be isometries, extending understanding of metric space symmetries.
Contribution
It formulates and proves cases of a conjecture linking level set preservation to isometries in complete Riemannian manifolds.
Findings
Functions preserving small distance level sets are isometries in certain cases
Preservation of a single small level set implies strong geometric constraints
The results extend known properties of metric space isometries
Abstract
Isometries of metric spaces preserve all level sets of . We formulate and prove cases of a conjecture asserting if is a complete Riemannian manifold, then a function preserving at least one level set , with small enough, is an isometry.
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Mathematical Dynamics and Fractals
