My question involves an infinite dimensional Hilbert manifold with a Riemannian metric.
My question is: What is the form of the curvature tensor for a infinite dimensional Hilbert manifold with a Riemannian metric?
Note- please try not suggesting books or long reads. I'd like an equation for the curvature tensor and I'll probably ask questions abort terms. Please and thank you.
I've never more than looked lightly into infinite-dimensional manifolds, but much of the finite theory carries over, provided things converge.
The Lie bracket is definable by $[X, Y](f) = X(Y(f)) - Y(X(f))$ for smooth functions $f$.
The Levi-Civita connection $D_VW$ is definable by the Koszul formula: $$\begin{align}2\langle D_VW, X\rangle &= V\langle W, X\rangle + W\langle V, X\rangle -X\langle V, W\rangle\\ &- \langle V, [W, X]\rangle + \langle W, [X, V]\rangle + \langle X, [V, W]\rangle\end{align}$$
The curvature tensor is definable as $$R_{XY}Z = D_{[X,Y]}Z - [D_X, D_Y]Z$$
(I am using the conventions of Barrett O'Neil's Semi-Riemannian Geometry - some Authors have the opposite definition for $R$)
As far as I can see, all of those definitions work in infinite dimensions.