I am trying understand the arguments for short time existence for curvature flows in the book "Curvature Problems" by Claus Gerhardt and there he assumes that $M$ is a compact Riemannian manifold and in some theorems he assumes that $M \in H^{2+l}$ and $M \in C^{m+2,\alpha}$, but this does not make sense for me since $M$ is not a function, but it is a set endowed with an atlas and a Riemannian metric. Thus, I would like to know what means a Riemannian manifold be in a function space.
I put some definitions and theorems of the book below to give more details.
Thanks in advance!
