Partial scalar curvatures and topological obstructions for submanifolds

Abstract

We investigate specific intrinsic curvatures ${\rho }_k$, $1\le k\le n$, that interpolate between the minimum Ricci curvature ${\rho }_1$ and the normalized scalar curvature ${\rho }_n=\rho$ of $n$-dimensional Riemannian manifolds. For $n$-dimensional submanifolds in space forms, these curvatures satisfy an inequality involving the mean curvature $H$ and the normal scalar curvature ${\rho }^{\perp }$, which reduces to the well-known DDVV inequality when $k=n$. We derive topological obstructions for compact $n$-dimensional submanifolds based on universal lower bounds of the $L^{n/2}$-norms of certain functions involving ${\rho }_k$, $H$ and ${\rho }^{\perp }$. These obstructions are expressed in terms of the Betti numbers. Our main result applies for any $1\le k\le n-1$, but it generally fails for $k=n$, where the involved norm vanishes precisely for Wintgen ideal submanifolds. We demonstrate this by providing a method of constructing new compact 3-dimensional minimal Wintgen ideal submanifolds in even-dimensional spheres. Specifically, we prove that such submanifolds exist in ${\mathbb{S}}^6$ with arbitrarily large first Betti number.