Notion of manifold curvature?

Question

Consider a particular embedding of a $C^2$ manifold $\mathcal{M}\subseteq\mathbb{R}^m$. Given $p\in\mathcal{M}$, suppose $\epsilon>0$ is small enough that the portion $U$ of $\mathcal{M}$ which is within $\epsilon$ (in $\ell_2^m$) of $p$ is diffeomorphic to some neighborhood of $0$ in the tangent space $T_p\mathcal{M}\cong\mathbb{R}^n$. Intuitively, there is some $B=B(p,\epsilon)$ with the property that there exists a chart $\varphi\colon U\rightarrow\mathbb{R}^n$ such that the second Fréchet derivative of $\varphi^{-1}$ is smaller than $B$ (in some sense) at every point in $\varphi(U)$.

Is there a well-known notion of manifold curvature that I can use to derive an explicit bound $B(p,\epsilon)$? References which are readable by non-experts are welcome.

Answer 1

The thing you are looking for is called the shape operator. Denote by $n$ the dimension of your manifold $\mathcal{M}$. $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\Gr}{\mathbf{Gr}}$ Denote by $\Gr_n(\bR^m)$ the Grassmannian of $k$-dimensional subspaces of $\bR^m$. The submanifold $\newcommand{\M}{\mathcal{M}}$ determines a Gauss map

$$\Gamma:\M\to\Gr_n(\bR^m),\;\;\M\ni p\mapsto T_p\M\in\Gr_n(\bR^m). $$

The shape operator of $M$ at $p$ is the differential of the Gauss map at $p\in\M$. It is thus a map $\DeclareMathOperator{\Hom}{Hom}$ $\newcommand{\bS}{\boldsymbol{S}}$

$$\bS_p: T_p\M\to T_{T_p\M} \Gr_n(\bR^m)=\Hom\bigl( T_p\M, (T_p\M)^\perp\,\bigr).$$

Thus the shape operator measures the rate of change of the tangent space $T_p\M$ inside $\bR^m$ as $p$ moves around $M$. This operator is intimately related to the curvature of $\M$ via the celebrated Teorema Egregium.

As for your question, the only reference I know that could be accessible to non-experts is Whitney's marvelous monograph Geometric Integration Theory, now available as a Dover edition as well. You will find the answer to your question hidden in Section A of Chapter 4.

Answer 2

I'm not sure what you mean by $\ell_2^m$, but I'm assuming you are using the normal Euclidean distance. Also, since you want to estimate the size of $\varphi^{-1}$, you need to have an intrinsic metric on $\mathcal{M}$, so let's assume that $(\mathcal{M},g)$ is a Riemannian manifold.

If you take $\varphi$ to be a normal coordinate chart around $p$, then the Fréchet derivative of $\varphi$ is the identity at $p$, and thus also for $\varphi^{-1}$, hence the bound $B = 1$ at $p$. By the implicit function theorem, a larger bound, say $B = 2$ holds on a sufficiently small neighborhood around $p$. If you want to get more explicit estimates, you need a bound on the Riemannian curvature $R$, basically $\epsilon$ will scale as $\epsilon \propto \|R\|^{-1}$. For some ideas in this direction, see e.g. section 2.3 in my book (a free PDF here).