Let $\Omega$ be an open subset of a Riemannian manifold $M$. Assume that $\Sigma:=\partial \Omega$ is $C^2$.
Let $U$ be a neighborhood of $\Omega$ such that $\exp\_p(t\nu(p))$ is diffieomorphism, where $p\in\Sigma$ and $\nu$ is the outward unit normal vector field along $\Sigma$. We now consider the distance function $d\_\Sigma(x):=d(x,\Sigma)$, $x\in U\setminus\Omega$ and the squared distance function $\eta\_\Sigma(x):=d^2\_\Sigma(x)$. Let denote the Hessian of $d\_\Sigma$ by $S=\nabla\nabla d\_\Sigma$. We have the Riccati euquation 
$$\dot{S}=-R(J(t),\nu(\gamma(t)))\nu(\gamma\_t)-S^2.$$ If $M$ is a space form (e.g. $\mathbb R^n$, $S^n$, $\mathbb H^n$) and we know the eigenvalues of $S$ at time 0 (i.e. the principal curvature of $\Sigma$), then using the Riccati equation we can compare the eigenvalues of $S(s)$ (i.e. the principal curvature of $\Sigma\_s=\{\exp\_p(s\nu(p)):p\in \Sigma\}$) with the eigenvalues of $S(0)$. For example when $M=\mathbb R^n$, if denote the eigenvalues of $S(s)$ by $\kappa\_i(s)$, then
$$\kappa\_i(s)=\frac{\kappa\_i(0)}{1+s\kappa\_i(0)}$$


**My question is**, what if we only know a lower sectional curvature bound $\alpha$? For example, can we have any explicit comparison between $\kappa\_i(s)$ and $\bar{\kappa}\_i(s)$ where $\bar{\kappa}\_i(s)$ is the eigenvalues of $\bar{S}(s)$? Here $\bar{S}(s)$ is the solution to the Riccati equation in the space form of curvature $\alpha$ with the initial value $\bar{S}(0)=S(0)$. I have the same question for eigenvalues of $\nabla^2\eta\_\Sigma$.



Riccati equation and principal curvatures
I have found an answer to my question on Mathoverflow.net [here](https://mathoverflow.net/questions/249184/comparison-of-eigenvalue-of-solution-of-riccati-equation?rq=1). @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.](http://onlinelibrary.wiley.com/doi/10.1002/cpa.3160410512/abstract)


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.


