Let $M$ a riemannian manifold such that the secctional curvature of $M$ is greater than or equal $-k$, i.e, $Sec_{M}\geq -k$ where $k$ is a non-negative real number. There exists a hope of an estimative of the ricci curvature of $M$ as $Ric_{M}\geq something$?
Edit: Let $p \in M$ , and $\{z_{1},z_{2},\dots, z_{n}\}$ an orthonormal basis in $T_{p}M$. Now, if $X=z_{1}$ , the Ricci curvature in the direction $X$ is defined by
$Ric(X)=\sum\limits_{i=1}^{n} <R(X,z_{i})X,z_{i}>$