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The metric in the superspace of Riemannian metrics and its relation to gravity

H. -J. Schmidt
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Abstract

The space of all Riemannian metrics is infinite-dimensional. Nevertheless a great deal of usual Riemannian geometry can be carried over. The superspace of all Riemannian metrics shall be endowed with a class of Riemannian metrics; their curvature and invariance properties are discussed. Just one of this class has the property to bring the lagrangian of General Relativity into the form of a classical particle's motion. The signature of the superspace metric depends in a non-trivial manner on the signature of the original metric, we derive the corresponding formula. Our approach is a local one: the essence is a metric in the space of all symmetric rank-two tensors, and then the space becomes a warped product of the real line with an Einstein space.

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H. -J. Schmidt. 2001-09-01. The metric in the superspace of Riemannian metrics and its relation to gravity. https://arxiv.org/abs/gr-qc/0109001

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