Partial Scalar Curvatures and Topological Obstructions for Submanifolds

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.