Let $N$ be a compact oriented smooth manifold, and let $M$ be a Riemannian manifold. Denote by $Imm(N,M)$ the space of smooth immersions $N\to M$. As explained for example here , $Imm(N,M)$ is an open set of $C^\infty (N,M)$ with the $C^\infty$ topology, and so it is an infinite-dimensional Frechet manifold.
We can also define on $Imm(N,M)$ a distance defined by $d(f,g)=\sup_{p\in N} d(f(p),g(p))$ where the distance on the right hand term is the geodesic distance of $M$.
My question is: is there a natural way to define on $ Imm(N,M) $ an "infinite dimensional Riemannian metric"? And how are the $C^\infty $ topology, the possible infinite dimensional Riemannian metric and the sup distance defined above related? References treating this subject are welcome.
Thank you.
The tangent space $T_f(Imm(N,M))$ is $\Gamma(f^*TM)$, the space of vector fields along $f$. The Riemannian metric on $M$ provides a natural inner product on $\Gamma(f^*TM)$, for instance, $$ \int_N g(X, Y) dVol_{f^*(g)} $$ where $g$ is the Riemannian metric on $M$, $dVol$ is the volume form, $X, Y$ are vector fields along $f$. I think, but did not check, that the corresponding distance functions yields the uniform topology on connected components of $Imm(N,M)$.