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Rigidity of minimal submanifolds in space forms. (English) Zbl 0531.53047

Let c be a real number and \(\bar M^{n+q}(c)\) an n-dimensional space form of sectional curvature c. Let \(M^ n\) (\(n\geq 3)\) be a connected Riemannian manifold and p be a point of M, where the nullity \(\mu\) (p) of the curvature tensor of M \[ \mu(p)=\dim \{X\in T_ pM;R(X,Y)=c<Y,\cdot>X-c<X,\cdot>Y\quad for\quad all\quad Y\in T_ pM\}, \] satisfies \(\mu\) (p)\(\leq n-3\). We show that if \(f:M^ n\to \bar M^{n+1}(c)\) and \(g:M^ n\to \bar M^{n+k}(c)\) are isometric minimal immersions then there is a rigid motion T of \(\bar M^{n+k}\) such that \(g=T{\mathbb{O}}f\), \(\bar M^{n+1}(c)\) being considered as a totally geodesic submanifold of \(\bar M^{n+k}(c)\).

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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References:

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