There is a theorem of Reilly concerning the first Dirichlet eigenvalue $\lambda$ of the Laplacian on a n-dimensional compact Riemannian Manifold $(M,g)$ whose Ricci curvature $Ric$ has a positive lower bound $(n-1)K$ for some constant $K>0$. Namely, it states that $\lambda \geq nK$. (Note, that the manifold has non-empty boundary.)
I would like to know, what does the condition $Ric\geq (n-1)K$ precisely means, because the left hand side is a rank 2 tensor and the righthand side is a real number. How can I check that condition?
Remark: Sometimes the conditions is also writen as $Ric(M)\geq(n-1)K$.
It means ${\rm Ric} \ge (n-1)Kg$ in the sense of quadratic forms; i.e. that the 2-tensor $${\rm Ric} - (n-1)Kg$$ is non-negative definite. More explicitly, $${\rm Ric}(v,v) \ge (n-1) K|v|^2$$ for all $v \in TM$.