Partial Scalar Curvatures and Topological Obstructions for Submanifolds

TL;DR

This paper explores intrinsic curvatures of submanifolds, establishing inequalities and topological obstructions, and constructs examples of minimal Wintgen ideal submanifolds with large Betti numbers.

Contribution

It introduces new inequalities involving intrinsic curvatures and derives topological obstructions for submanifolds, also constructing examples with large Betti numbers.

Findings

Inequalities relating intrinsic curvatures, mean curvature, and normal scalar curvature.

Topological obstructions based on $L^{n/2}$-norms and Betti numbers.

Existence of minimal Wintgen ideal submanifolds with arbitrarily large first Betti number.

Abstract

We investigate specific intrinsic curvatures $\rho_k$ (where $1\leq k\leq n$) 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 $1\leq k \leq n-1$, 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.