A topological characterization of compact \(n\)-manifolds with \(C^ 2\)- boundary in \(R^ n\) (Q1190689)

In this paper, an \(n\)-manifold \(M\) with \(C^ 2\)-boundary \(\partial M\) in \(\mathbb{R}^ n\) is called \(p^*\)-convex if each point on \(\partial M\) has at most \(p\) negative principal normal curvatures with respect to the inward normal vector and if the homology groups \(H_ i(M)\) of \(M\) of dimension \(i\geq p\) vanish. The author's main result is that a compact \(n\)-manifold \(M\) with \(C^ 2\)-boundary in \(\mathbb{R}^ n\) is \(p^*\)-convex if and only if for almost all \(x\) in \(\mathbb{R}^ n\), all \(r>0\) and all \(i\geq p\) it satisfies \(H_ i(M\cap B(x,r))=0\), where \(B(x,r)\) denotes the ball with centre \(x\) and radius \(r\). The proof uses Morse theory.