## First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature.(English)Zbl 1155.53030

For an $$n$$-dimensional compact hypersurface $$M$$ with constant scalar curvature $$n(n-1)r$$, $$r>1$$, in a unit sphere $$S^{n+1}(r)$$, the author studies the first eigenvalue $$\lambda _{1}$$ of the Jacobi operator $$J_{S}$$, following the definition given by H. Alencar, M. do Carmo and A. G. Colares [Math. Z. 213, No. 1, 117–131 (1993; Zbl 0792.53057)].
An inequality involving $$\lambda _{1}$$, the mean curvature $$H$$, the radius $$r$$ and the dimension $$n$$ of the hypersurface $$M$$ is proved. Equality holds if and only if either $$M$$ is totally umbilical and non-totally geodesic or $$M$$ is a Riemannian product $$S^{m}(c)\times S^{n-m}(\sqrt{1-c^{2}})$$, $$1\leq m\leq n-1$$, $$r>1$$.

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Zbl 0792.53057
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### References:

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