Abstract
A hypersurface M n immersed in a space form is r-minimal if its (r + 1)th-curvature (the (r + 1)th elementary symmetric function of its principal curvatures) vanishes identically. Let W be the set of points which are omitted by the totally geodesic hypersurfaces tangent to M. We will prove that if an orientable hypersurface M n is r-minimal and its r th-curvature is nonzero everywhere, and the set W is nonempty and open, then M n has relative nullity n − r. Also we will prove that if an orientable hypersurface M n is r-minimal and its r th-curvature is nonzero everywhere, and the ambient space is euclidean or hyperbolic and W is nonempty, then M n is r-stable.
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The author is partially supported by CNPq.
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Alencar, H., Batista, M. Hypersurfaces with null higher order mean curvature. Bull Braz Math Soc, New Series 41, 481–493 (2010). https://doi.org/10.1007/s00574-010-0022-z
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DOI: https://doi.org/10.1007/s00574-010-0022-z