Let $M^n$ be an $n$-dimensional smooth compact Riemannian manifold with boundary. Assume that the sectional curvature of $M$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form of the boundary is at least $\lambda$.
Question. Does there exist a lower bound on the sectional curvature of the boundary $\partial M$ in terms of $n,\kappa,\lambda$ and (possibly) $D$?
Even partial results under extra assumptions might be helpful.
I assume that convex surfaces have nonnegative second fundamental forms.
If $\lambda\geqslant 0$ then the answer is yes, and it follows from the Gauss formula.
On the other hand, if $\lambda<0$, then it implies no bound on sectional curvature. This can be seen already for surfaces in Euclidean space.
However, for $\lambda<0$, the Gauss formula implies the following inequality, which is helpful sometimes: $$\mathrm{sec}_{\partial M}\geqslant \kappa + \lambda\cdot (H-n\cdot\lambda);$$ here $H$ denotes the mean curvature of the boundary. Again, we assume that $H\geqslant0$ for convex hypersurfaces.
