I have found an answer to my question on Mathoverflow.net here. @Raziel gave the reference
H. L. Royden, MR 948079 Comparison theorems for the matrix Riccati equation, Comm. Pure Appl. Math. 41 (1988), no. 5, 739--746.
where I can find a comparison theorem for eigenvalues of $S$ but not for eigenvalues of $\nabla\nabla \eta_\Sigma$.
I keep the question and the answer on Mathoverflow since it might be useful for someone else.
