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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
##### Keywords:
hemisphere; rigidity; Ricci curvature; conformal change
Full Text:
##### References:
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