Editing Riemannian manifold (section)

=== Definitions ===
Riemannian metrics are defined in a way similar to the finite-dimensional case. However, there is a distinction between two types of Riemannian metrics:
* A ''weak Riemannian metric'' on <math>M</math> is a smooth function <math>g : TM \times TM \to \R,</math> such that for any <math>x \in M</math> the restriction <math>g_x : T_xM \times T_xM \to \R</math> is an inner product on <math>T_xM.</math>{{citation needed|date=July 2024}}
* A ''strong Riemannian metric'' on <math>M</math> is a weak Riemannian metric such that <math>g_x</math> induces the topology on <math>T_xM</math>. If <math>g</math> is a strong Riemannian metric, then <math>M</math> must be a Hilbert manifold.{{citation needed|date=July 2024}}