Consider $M$ to be a compact manifold and consider the based loop space $\Omega(M,q_0,q_1)$ of loops of class $W^{1,2}$. It can be shown that that this has a structure of an hilbert manifold. It's also known that a metric $\langle \cdot,\cdot \rangle $ on $M$ will induce a metric $g$ on $\Omega(M,q_0,q_1)$ by setting $g(\eta_t,\xi_t):=\int_{0}^{1}\langle \eta_t,\xi_t\rangle_{c(t)}+ \langle \nabla_t \eta_t , \nabla_t \xi_t\rangle_{c(t)} dt. $
Now we that we have a metric $g$ on $\Omega(M,q_0,q_1)$ we could try and find it's levi civita connection. My question, and what I have been thinking about without getting somewhere conclusive, is that if we consider the levi civita connection $\nabla$ on $M$ of $\langle \cdot , \cdot \rangle $ and consider the connection this induces on $\Omega(M,q_0,q_1)$ will we have that this is the levi civita connection of $g$? Therefore we could have that for $c\in \Omega(M,q_0,q_1)$ and $Y(t)$ a vector field along $c, \exp_c(Y)= \exp_{c(t)}(Y(t))$, where $\exp_cY$ is the exponential with respect to $g$ and $\exp_{c(t)}Y(t)$ is the exponential with respect to $\langle \cdot,\cdot \rangle$ ?
I have tried using the Koszul formula to check this but I couldn't get anywhere conclusive. Any insight is appreciated, thanks in advance.
