Partial Scalar Curvatures and Topological Obstructions for Submanifolds

We investigate specific intrinsic curvatures ρₖ (where 1≤ k≤ n) that interpolate between the minimum Ricci curvature ρ₁ and the normalized scalar curvature ρₙ=ρ 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 ρ^⊥, 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 Ln/2-norms of certain functions involving ρₖ,H and ρ^⊥. These obstructions are expressed in terms of the Betti numbers. Our main result applies for any 1≤ k ≤ 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 S⁶ with arbitrarily large first Betti number.