Junior Geometry Seminar – David Lenze (KIT)

Abstract: In 1970, Ebin introduced a natural L2-type Riemannian structure on the infinite-dimensional space of Riemannian metrics. While early research focused on the infinite-dimensional Riemannian geometry of this space, Clarke (2013) shifted the focus into a more metric direction, by showing that the Ebin metric induces a distance — a non-trivial result in the infinite-dimensional setting — and that the completion of the resulting metric space is CAT(0).

Recently, Cavallucci provided a more conceptual approach to this metric perspective which yielded a surprising result: in addition to being CAT(0), the isometry class of the completion depends solely on the dimension of the underlying manifold.

In this talk, I will survey these developments and present a full characterization of the Ebin metric’s self-isometries. I will demonstrate that—despite Cavallucci’s results on the completion—the isometry class of the uncompleted space recovers the underlying manifold in the strongest plausible way.