Abstract
We obtain a sharp estimate to the norm of the traceless second fundamental form of complete hypersurfaces with constant scalar curvature immersed into a locally symmetric Riemannian manifold obeying standard curvature constraints (which includes, in particular, the Riemannian space forms with constant sectional curvature). When the equality holds, we prove that these hypersurfaces must be isoparametric with two distinct principal curvatures. Our approach involves a suitable Okumura type inequality which was introduced by Meléndez (Bull Braz Math Soc 45:385–404, 2014) , corresponding to a weaker hypothesis when compared with to the assumption that these hypersurfaces have a priori at most two distinct principal curvatures.
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Acknowledgements
The second author is partially supported by CNPq, Brazil, Grant 301970/2019-0. The authors would like to thank the referee for reading the manuscript in great detail and for his/her valuable suggestions and useful comments which improved the paper.
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de Lima, E.L., de Lima, H.F. A gap theorem for constant scalar curvature hypersurfaces. Collect. Math. 73, 1–11 (2022). https://doi.org/10.1007/s13348-020-00304-3
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DOI: https://doi.org/10.1007/s13348-020-00304-3
Keywords
- Locally symmetric Riemannian manifolds
- Riemannian space forms
- Complete hypersurfaces
- Constant scalar curvature
- Okumura type inequality