在Bishop-Gromov体积比较定理证明中, Laplace比较定理是非常关键的工具. 我们将对于Laplace比较定理进行详细研究, 并给出在黎曼流形一点处的测地圆盘上Laplace比较定理的刚性结论.
            
首先, 我们介绍Jacobi场, 第二变分公式以及Morse指标形式, 并指出Morse指标形式的本质是从代数的视角看待第二变分公式. 其次, 我们介绍Laplace算子和距离函数, 并对Laplace比较定理进行了详细研究, 整理出如下结论: 对于黎曼流形$M$上一条无共轭点的测地线$\gamma$,  如果沿$\gamma$的切空间中含切向$\dot{\gamma}$的平面之截面曲率为常值, 那么沿$\gamma$的初值为0的法Jacobi场单位化后是平行向量场, 并且我们给出了该Jacobi场的表达式. 最后, 基于Laplace比较定理的取等条件和初值为0的法Jacobi场的表达式, 我们给出了在黎曼流形一点处的测地圆盘上Laplace比较定理的刚性结论.

In the proof of the Bishop-Gromov volume comparison theorem, Laplacian comparison theorem is a very critical tool. We will conduct a detailed study of Laplacian comparison theorem and give the rigid conclusion of Laplacian comparison theorem on the geodesic disk at a point on the Riemannian manifold.  
            
Firstly, we introduce the Jacobi field, the second variation formula and the Morse index form, and point out that the essence of the Morse index form is to view the second variation formula from an algebraic perspective. Secondly, we introduce the Laplace operator and distance function, conduct a detailed study on the Laplacian comparison theorem, and draw the following conclusion: For a geodesic $\gamma$ without conjugate points on the Riemannian manifold $M$, if the sectional curvature of the plane containing the tangent direction $\dot{\gamma}$ in the tangent space along $\gamma$ is a constant value, then the normal Jacobi field along $\gamma$ with an initial value of 0 is normalized to be a parallel vector field, and we give the expression of this Jacobi field. Finally, based on the equality conditions of Laplacian comparison theorem and the expression of the normal Jacobi field with an initial value of 0, we give the rigid conclusion of the Laplacian comparison theorem on the geodesic disk at a point on the Riemannian manifold.