Partial Scalar Curvatures and Topological Obstructions for Submanifolds

We investigate specific intrinsic curvatures ₖ (where 1 k n) that interpolate between the minimum Ricci curvature ₁ and the normalized scalar curvature ₙ= 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 ^, 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 ₖ, H and ^. 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.