A Note on the Differential of the Exponential Map and Jacobi Fields in a Symmetric Space
A Note on the Differential of the Exponential Map and Jacobi Fields in a Symmetric Space
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对称空间中指数映射与雅可比场微分的一个注解
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发表时间：
1984
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通讯作者：
H. Taniguchi
中科院分区：
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作者：
H. Taniguchi
Rauch[4]指出，对称空间中雅可比场的微分方程式相对于适当的沿测地线运动的标架具有常系数。使用一种稍有不同的方法，我们在这里给出了坐标自由形式下的雅可比场的显式形式。在S 1中的引理中，我们还给出了指数映射的微分与雅可比场之间的关系。这为众所周知的公式(Helason[1]，第二章)提供了另一种证明。IV，定理4.1)关于对称空间中指数映射的微分。S 1.设$M$是具有无挠仿射联络的流形
ABLA$，并考虑EXP$=EXP：T.M
右行M，$$xem$，并让$dexp_{x}：t_{x}M
右行T_{exp X}M$，
It was remarked by Rauch [4] that the differential equation for Jacobi fields in a symmetric space has constant coefficients with respect to suitable moving frames along a geodesic. Using a slightly different method, we give here an explicit form of a Jacobi field in a coordinate free form. We also give, in the Lemma in S 1, a relationship between the differential of the exponential map and Jacobi fields. This provides an alternate proof of the well known formula (Helgason [1], Chap. IV, Theorem 4.1) on the differential of the exponential map in a symmetric space. S 1. Let $M$ be a manifold with a torsion-free affine connection $
abla$ , and consider exp $=exp.:T.M
ightarrow M,$ $xeM$, and let $dexp_{X}:T_{x}M
ightarrow T_{exp X}M$,