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Rigidity theorems for compact manifolds with boundary and positive Ricci curvature. (English) Zbl 1175.53056
In the paper under review the authors prove two very interesting theorems motivated by a conjecture of Min-Oo: Let \((M^n, g)\) \((n\geq 2)\) be a compact Riemannian manifold with nonempty boundary \(\Sigma =\partial M\). Suppose 4mm
{\(\bullet\)}
\(\text{Ric}\geq (n-1)g\),
{\(\bullet\)}
\((\Sigma, g|_\Sigma)\) is isometric to the standard sphere \(\mathbb S^{n-1}\subset \mathbb R^n\),
{\(\bullet\)}
\(\Sigma\) is convex in \(M\) in the sense that its second fundamental form is nonnegative. Then \((M^n, g)\) is isometric to the hemisphere \(\mathbb S^{n-1}=\{x\in\mathbb R^{n+1}: |x|=1, x_{n+1}\geq 0\}\subset \mathbb R^{n+1}\).
and
Let \((M^n, g)\) be a compact Riemannian manifold with nonempty boundary \(\Sigma =\partial M\) and \(\overline\Omega\subset \mathbb S^n_+\) is a compact domain with smooth boundary in the open hemisphere. Suppose 4mm
{\(\bullet\)}
\(\text{Ric}\geq (n-1)g\),
{\(\bullet\)}
there is an isometric embedding \(\iota: (\Sigma, g_\Sigma) \to \partial \overline\Omega\)
{\(\bullet\)}
\(\Pi \geq \Pi_0\circ \iota\), where \(\Pi\) is the second fundamental form of \(\Sigma\) in \(M\) and \(\Pi_0\) is the second fundamental form of \(\partial \overline\Omega\) in \(\mathbb S^n_+\).

MSC:
53C24 Rigidity results
58J32 Boundary value problems on manifolds
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References:
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