Weak Riemannian Hilbert manifold

This is a variation of Riemannian manifold

Definition

A weak Riemannian Hilbert manifold is the following data:

• A Hilbert manifold modelled on a smooth infinite-dimensional Hilbert space $H$
• A smooth assignment of a continuous, positive-definite, symmetric bilinear form to the tangent space at each point (each tangent space is isomorphic to $H$ as a vector space)

Note that the tangent space at each point need not be complete as a metric space with respect to the bilinear form defined at it.

Facts

Levi-Civita connection

A weak Riemannian Hilbert manifold may not, in general, have a smooth Levi-Civita connection. However, it is true that if a smooth Levi-Civita connection exists, then it is unique. The proof for this is the same as in the finite-dimensional case (viz for an ordinary Riemannian manifold).

References

• Exponential map of a weak Riemannian Hilbert manifold by Leonardo Bibliotti,