Partial scalar curvatures and topological obstructions for submanifolds

We investigate specific intrinsic curvatures \rho_{k} , 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.