The Setting: Let $H$ be the Hilbert space of all class $C^k$-curves into $\mathbb{R}$ with inner product: \begin{equation} <f,g>:=\int_{\mathbb{R}} f'(x)g'(x) e^{-x}dx \end{equation}
Furthermore let $U$ be a Domain (simply connected and open subset) in $\mathbb{R}^d$ and let $F:U \rightarrow H$ (be a family of finitely parametrized $C^k$-curves into $\mathbb{R}$).
Let $M:=Im(F)\subseteq H$, be a finitely parametrized manifold.
My question is: (Christoffel symbols)
Is there a general form for the Christoffel symbols of such a manifold; where the inner product is induced by the embedding of $M$ into $H$?
