Symmetry breaking in geometry
Andreas Fuchs, Udo Hertrich-Jeromin, Mason Pember
TL;DR
The paper explores the concept of symmetry breaking in geometry, illustrating how it occurs spontaneously or explicitly in differential geometry, and proposes a systematic approach for further research in this area.
Contribution
It introduces a systematic framework for studying symmetry breaking in geometry, drawing analogies from physics and discussing potential applications in solving geometric problems.
Findings
Spontaneous symmetry breaking can occur in differential geometry.
Explicit symmetry breaking can be used to address geometric problems.
A preliminary framework for systematic study of symmetry breaking in geometry is proposed.
Abstract
A geometric mechanism that may, in analogy to similar notions in physics, be considered as "symmetry breaking" in geometry is described, and several instances of this mechanism in differential geometry are discussed: it is shown how spontaneous symmetry breaking may occur, and it is discussed how explicit symmetry breaking may be used to tackle certain geometric problems. A systematic study of symmetry breaking in geometry is proposed, and some preliminary thoughts on further research are formulated.
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Taxonomy
TopicsMolecular spectroscopy and chirality · Geometric and Algebraic Topology · Nonlinear Waves and Solitons