Let $(M,g)$ be a Riemannian manifold, $\Omega\subset M$ be a closed set with smooth boundary $\partial\Omega$ and $\nu$ be the unit normal of $\partial\Omega$ pointing into $\Omega$.
$\Omega$ is said to be geodesically convex iff $\forall x_0, x_1\in\Omega$ $\exists c:[0,1]\stackrel{\text{geodesic}}\to(M,g)$ s.t. $c(0)=x_0, c(1)=x_1$, $c([0,1])\subset\Omega$, $\mathrm{Length}[c]=d_g(x_0,x_1)$.
Suppose $\Omega$ is geodesically convex. Then...
[Q.1] Does it hold that the 2nd fundamental form of $\partial\Omega$ toward $\nu$ is nonnegative definite at each point on $\partial\Omega$?
[Q.2] Let $\psi_r(x):=\mathrm{exp}^g_x [r\nu(x)]\in N$ $(x\in\partial\Omega)$. Then for small $|r|$, $\psi_r$ is an embedding. Here, does it hold that the inner 2nd fundamental form of $\psi_r$ is nonnegative definite at each point on $\partial\Omega$ when $r>0$ and is sufficinetly small?
Thank you.