Talk:Exponential map
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In the section on Gauss's lemma, I originally said "The differential of the exponential map at v evaluated on w (more compactly, d(expp)v(w)) is the parallel transport of w along the geodesic from p to expp(v). This is again just a reflection of the linearization of M: w in the double-tangent space can be "slid freely" to the origin of TpM along the straight line determined by v, by virtue of the linear structure of TpM, and so in the manifold, such a vector will be again "slid along" via parallel transport along the geodesic determined by v. (In fact this is used in the proof the Hopf-Rinow theorem). The crucial point is that the exponential map preserves the normality of vectors based at v. "
This is based on visualization of the situation in 2 dimensions where it is in fact true (it is also true for any vector parallel to v namely a scalar multiple of v, since the exponential map is linear). However I'm not sure if it is true in higher dimensions; the angle has to be preserved but it w could conceivably rotate around the geodesic. Hence I've removed this until I'm sure one way or the other. In the meantime perhaps someone else can confirm (or deny) this. [Hence it's a good demonstration that intuition is a great guide but also can mislead...]
Choni 10:43, 16 October 2005 (UTC)
Never mind. It's not even true in dimension 2. It's the whole starting point of the study of Jacobi fields.
Choni 05:03, 17 October 2005 (UTC)
