Abstract
Minimal compact hypersurface in the unit sphere S-n +/-(1) having squared length of shape operator IIAII(2) < n are totally geodesic and with IIAII(2) = n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in S-n +/-(1). One finds a naturally induced vector field w called the associated vector field and a smooth function? called support function on the hypersurface M of S-n +/-(1). It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S-5 to be totally geodesic is that the support function ? is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in S-n +/-(1), (n > 2), provided the scalar curvature t is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in S-n +/-(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.