Abstract
We prove that the L 2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L 2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.
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Communicated by J. Jost.
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Clarke, B. The metric geometry of the manifold of Riemannian metrics over a closed manifold. Calc. Var. 39, 533–545 (2010). https://doi.org/10.1007/s00526-010-0323-5
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DOI: https://doi.org/10.1007/s00526-010-0323-5