Optimal Inequalities, Contact delta-Invariants and Their Applications

Associated with a k-tuple (n1,...,nk) element of g(2n+1) with n ≥1, we define a contact delta-invariant, delta^c(n1,...,nk), on an almost contact metric (2n+1)-manifold M. For an arbitrary isometric immersion of M into a Riemannian manifold, we establish an optimal inequality involving delta^c(n1,..., nk) and the squared mean curvature of the immersion. Furthermore, we investigate isometric immersions of contact metric and K-contact manifolds into Riemannian space forms which verify the equality case of the inequality for some k-tuple.