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.


1 Answer 1


The best places, I think, to learn about Jacobi fields and comparison theorem are in the writings of Eschenburg, Karcher, and their co-authors. Heintze-Karcher is a classic.

The key trick, when working with Jacobi fields, is to write them with respect to an orthonormal frame that is parallel along an appropriately chosen family of geodesics. Then you are working simply with matrix-valued functions of two variables (the parameter of each constant speed geodesic and the parameterization of the family).

You can get what you want, I believe, via the following path (the subscripts label the columns and supersecripts the rows of a matrix).

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 $\Delta r$.


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