Weak Riemannian Hilbert manifold
From Diffgeom
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
- A smooth assignment of a continuous, positive-definite, symmetric bilinear form to the tangent space at each point (each tangent space is isomorphic to 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,