volume and curvature of submanifolds

Question

Suppose an $m$-dimensional manifold in an $n$-dimensional euclidean space, choose some point on this manifold and take an $n$-dimensional ball of certain radius $R$ centred in this point. If the volume of the manifold "enclosed" in this ball is $V$, what can we tell about the bounds for the different types of curvatures for the manifold inside this ball? Thanks very much in advance!

Answer 1

Not a complete answer by any means, but a pretty and germane elementary theorem:

If $S$ is a sphere in $\mathbf{R}^3$ having radius $R_0$ and passing through the origin $O$, $R$ is a real number such that $0 < R \leq 2R_0$, and $\Sigma_R$ is the sphere of radius $R$ centered at $O$, then the area of the portion of $S$ inside $\Sigma_R$ is $\pi R^2$. (!)

Philosophically, the volume (here, area) of $S$ inside $\Sigma_R$ cannot distinguish $S$ (with constant positive curvature) from a flat plane until $R > 2R_0$.