Volume 36 • Number 2 • 2013   • Optimal Inequalities, Contact $\delta$-Invariants and Their Applications Bang-Yen Chen and Veronica Martin-Molina Abstract. Associated with a $k$-tuple $(n_1,\ldots,n_k)\in \mathcal S(2n+1)$ with $n\geq 1$, we define a contact $\delta$-invariant, $\delta^c(n_1,\ldots,n_k)$, 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 $\delta^c(n_1,\ldots,n_k)$ 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. 2010 Mathematics Subject Classification: 51M16, 53C40, 53C25, 53D15 Full text: PDF