Integral Inequalities for Compact Hypersurfaces with Constant Scalar Curvature in the Euclidean Sphere

We study the rigidity of compact-oriented hypersurfaces with constant scalar curvature isometrically immersed into the unit Euclidean sphere S n + 1 . In particular, we establish a sharp integral inequality for the behavior of the norm of the total umbilicity tensor, equality characterizing the totally umbilical hypersurfaces, and a certain family of standard tori of the form S 1 ( 1 - r 2 ) × S n - 1 ( r ) . Moreover, under an appropriate constraint on the total umbilicity tensor, we are able to extend this result for any integer k , with 2 ≤ k ≤ n - 1 , equality characterizing the totally umbilical hypersurfaces and a certain family of standard product of spheres of the form S k ( 1 - r 2 ) × S n - k ( r ) .