# Questions about proving laplacian comparison

For $$(p,m)\in M$$,a complete Riemannian manifold of dim n.Let $$r(x)=d(x,p)$$ be the distance function,$$c:[0,a]\to M$$ is a geodisc joining $$p,x$$ parametrized by arclength.

In usual proof of Laplacian comparison,we choose orthonormal basis $$\{e_{1},\cdots,e_{n-1},c'(0)\}$$ in $$T_{p}M$$ and use their parallel transport $$\{e_{i}(s),c'(s)\}$$ as basis of $$T_{c(s)}M$$.

Then we construct Jacobi fields $$X_{i}(s)$$ with $$X_{i}(0)=0,X_{i}(a)=e_{i}(a).$$So we can compute $$\Delta r(x)$$.

$$\Delta r(x)=\Sigma_{i=1}^{n-1} \int_{0}^{d(x, p)}\left(\left|X_{i}^{\prime}\right|^{2}+R\left(c^{\prime}(s), X_{i}, c^{\prime}(s), X_{i}\right) d s=\Sigma_{i=1}^{n-1} I\left(X_{i}, X_{i}\right)\right.$$

However ,in Petersen's Riemannian Geometry.He use parallel fields instead of Jacobi fields and proved the following inequality $$\partial_{r} \Delta r+\frac{(\Delta r)^{2}}{n-1} \leq \partial_{r} \Delta r+|\operatorname{Hess} r|^{2}=-\operatorname{Ric}\left(\partial_{r}, \partial_{r}\right).$$

Using this inequality and Riccati comparison we can easily get Laplacian comparison.（I know this inequality comes from Bochner technique.)

So I want to use jacobi fields to prove this inequality since it's natural when you have the formula of $$\Delta r(x)$$ and compute its derivative but I met some trouble with computing $$\partial_{r}\Delta r$$.

So can I prove this inequality with Jacobi fields or this will be tough?Any help will be thanked.

Fix at $$p$$ an orthonormal frame (written as a row matrix of vectors) $$E_0=\begin{bmatrix} e_1 & \cdots & e_n \end{bmatrix}$$ Extend it along each radial geodesic as a parallel frame $$E$$. Also, extend it as a frame of Jacobi fields $$J = \begin{bmatrix} J_1 & \dots & J_n \end{bmatrix},$$ where $$J(0)=0$$, $$J'(0)=E(0)$$. Let $$M$$ be the matrix-valued function along each radial geodesic such that $$J = EM.$$ Then you can show that the matrix-valued function $$A = M^{-1}M'$$ comprise the components of a symmetric $$2$$-tensor $$T$$, written with respect to the basis $$E$$. Moreover, the restriction of $$T$$ to tangent space of the geodesic sphere is equal to the second fundamental form of the sphere. $$T$$ is also the Hessian of the function $$r$$. Moreover, the Jacobi equation implies the matrix Riccati equation $$M'' + MK = 0,$$ which then implies that $$A' + A^2 + K = 0.$$ Taking the trace of this gives the identity for $$\Delta r$$.