Consider a manifold $\mathcal{M}$ embedded in $\mathbb{R}^d$ and let $V$ and $W$ be two vector fields on $\mathcal{M}$.
As usual, we define the second fundamental form to be $ \mathrm{I\!I}(V, W) = (\tilde{\nabla}_V (W))^\perp $, where $\tilde{\nabla}$ is the ambient covariant derivative and $X^\perp$ represents the normal projection of $X$.
I am interested in bounding $\| \mathrm{I\!I}(V, W) \|$ in terms of the bounds on the norms of $V$ and $W$.
For example, when the manifold is the sphere with the usual inherited metric, assuming that for all $x$ it holds that $\| V(x) \| \leq \alpha $ and $\| W(X) \| \leq \beta $, I believe we have $\| \mathrm{I\!I}(V, W) \| \leq \alpha \beta $.
In general, I would expect such a bound to also depend on the curvature of the manifold.
Later edit: when $\mathcal{M}$ is a hypersurface in $\mathbb{R}^d$, letting $N$ be a smooth normal vector field, I know that
$ \| \mathrm{I\!I}(V, W) \| = \lvert \langle N, \mathrm{I\!I}(V, W) \rangle \rvert = \lvert h(V, W) \rvert $, where $h$ is the scalar second fundamental form. In this case, the principal curvatures $\mathcal{k}_i$ are defined as the eigenalues of the shape operator $s : T_x \mathcal{M} \rightarrow T_x \mathcal{M}$, $s = W_N$, where $W_N$ is the Weingarten map in direction $N$. We can write $h(V, W)$ in the basis of $T_x \mathcal{M}$ given by the eigenvectors $b_i$ of $s$ (with corresponding eigenvalue $\mathcal{k}_i$) to obtain
$$ h(V, W) = \langle W, sV \rangle = \langle W, s \sum_i \langle V, b_i \rangle b_i \rangle = \sum_i \mathcal{k}_i \langle V, b_i \rangle \langle W, b_i \rangle $$
From here, it's easy to obtain a bound on $\lvert h(V, W) \rvert$ in terms of the maximum principal curvature.
The question would be how to extend this for the case when $\mathcal{M}$ is an arbitrary smooth manifold. I know, for example, that the second fundamental form is related to the Riemann curvature tensor through the Gauss Equation (in Lee's Riemannian manifolds, Theorem 8.5), but it's not clear how to proceed from there. I am willing to assume I have bounds on (any sort of) curvature that comes up.
Any pointers on how I could proceed for obtaining such a bound in the general case are appreciated!
Best Answer
Even in the case of surfaces, it is hopeless to obtain bounds purely in terms of the Gaussian curvature. This can be seen from the example of a thin tube. The Gaussian curvature is zero, but the second fundamental form can be arbitrarily large. You mention that one can obtain bounds in terms of the principal curvatures. Note that no intrinsic invariants would capture those, and therefore looking for bounds in terms of intrinsic invariants is hopeless.