The contact Whitney sphere

Abstract

In this paper, we introduce the contact Whitney sphere as an imbedding of the $n$--dimensional unit sphere as an integral submanifold of the standard contact structure on $\R^{2n+1}$. We obtain a general inequality for integral submanifolds in $\R^{2n+1}$, involving both the scalar curvature and the mean curvature, and we use the equality case in order to characterize the contact Whitney sphere. We also study a similar problem foranti-invariant submanifolds of $\R^{2n+1}$, tangent to the structure vector field.