We can find many examples of smooth Riemannian manifolds with boundaries whose boundaries are convex. But it seems to me I know no any example of smooth Riemannian manifold with concave boundary. So my question is What kinds of smooth Riemnnian manifolds admit concave boundaries? Do they exist? Thank you in advance!



What kinds of manifolds admit concave boundary?
I think the answer is yes and in a strong way. 


Precisely, let $M$ be a compact manifold, and put *any* Riemann metric on $\partial M$ Then I claim there is a Riemann metric on $M$, extending the Riemann metric on $\partial M$ with $\partial M$ convex in the sense above -- that the geodesic curvature of boundary curves point out. Moreover, you can do the same if you replace convex with concave. 


The idea is fairly simple. Take a collar neighbourhood of $\partial M$ in $M$. The collar neighbourhood is diffeomorphic to $[0,1] \times \partial M$. The idea is to pull-back the metric on $\partial M$ to $[0,1] \times \partial M$, and then linearly re-scale the metric but only in the $\partial M$ direction. If you re-scale so that $\{t\} \times \partial M$ has its Riemann metric $(t+1)$ times the Riemann metric for $\{0\} \times \partial M$, this does the job. You then use a partition of unity to extend the metric to all $M$. 


