Rigidity of the Ebin metric

Abstract: In 1970, Ebin introduced a natural L2-type metric on the infinite-dimensional space of Riemannian metrics over a given manifold. Though the infinite dimensional geometry of this space has been extensively-studied, a new metric perspective emerged in 2013 when Clarke showed that the completion with respect to the Ebin metric turns out to be a CAT(0) space.

Recently, Cavallucci provided a shorter and more conceptual proof of a strengthened result that in addition to being CAT(0) establishes the completion of the space of Riemannian metrics to depend only on the dimension of the underlying manifold.

After reviewing this recent progress, I will present new results providing a complete characterization of the Ebin metric's self-isometries. Furthermore, I will show that—in contrast to Cavallucci's findings on the completion—the isometry class of the uncompleted space recovers the underlying manifold in the strongest plausible way.

differential geometrygeometric topologymetric geometry

Audience: researchers in the discipline

( paper )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Differential geometry.

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