Abstract
Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2<n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+1. One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in Sn+1, (n>2), provided the scalar curvature τ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in Sn+1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field Aw.
Highlights
It is interesting to note that this differential equation plays an important role in characterizing totally geodesic hypersurfaces in Sn+1 as observed in this paper
Let v be the concircular vector field on Sn+1 considered in the introduction, which satisfies Equation (3), where ρ is the function defined on Sn+1 by ρ = Z, N
We call w the associated vector field on M and call the functions ρ, f the support function and the associated function, respectively, of M
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Sn+1 is the totally geodesic sphere Sn. important minimal hypersurfaces of of. It is interesting to note that this differential equation plays an important role in characterizing totally geodesic hypersurfaces in Sn+1 as observed in this paper. As Sn is totally geodesic, for a vector field U on Sn , on using Equation (3), we find U ( f ) = Ug(v, N ) = g(−ρU, N ) = 0, that is, f is a constant c. Corresponding to eigenvector ξ if ∆ξ = −λξ) These raise two questions: (i) Given a minimal compact hypersurface M of Sn+1 that has support function ρ a non-trivial solution of static perfect fluid equation necessarily totally geodesic? Mathematics 2021, 9, 3161 w an eigenvector of the Laplace operator corresponding to eigenvalue 1, is this hypersurface necessarily totally geodesic?
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