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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)
What do people think about splitting this article into articles two pieces:
This page will then become a disambiguation page (with additional links to exponential function and matrix exponential). I think the material on the relationship between the two concepts would fit better in the Riemannian geometry fork, although we can leave a summary of it in the Lie theory fork as well. -- Fropuff ( talk) 03:11, 7 February 2008 (UTC)
Primarily because they are different (although related) concepts. It will force editors to link to the correct concept, rather than making readers figure out which one is meant. Secondly, in some cases, the two concepts can differ. If G is a Lie group with a left-invariant metric (but not a bi-invariant one) there will be two distinct exponential maps: the Lie-theoretic one and the Riemannian one. This happens, for example, in the special linear group and many other noncompact Lie groups. The two concepts coincide only when the metric is bi-invariant. Finally, the term "exponential map" can also just mean the exponential function so this page should properly be a disambiguation page. I'm not altogether opposed to not splitting; it just struck me that it might be a good idea. -- Fropuff ( talk) 04:29, 7 February 2008 (UTC)
The consensus seems clear (the article to split) and accordingly I carried out the splitting. -- Taku ( talk) 00:28, 6 December 2014 (UTC)
I moved the material on discrete dynamical systems to a new article exponential map (discrete dynamical systems) because it is unrelated to the topic of the current article. Sławomir Biały ( talk) 15:36, 2 July 2009 (UTC)
Shouldn't the Chaos Theory template also be moved to the exponential map (discrete dynamical systems) article? 93.223.155.227 ( talk) 12:07, 12 March 2014 (UTC)
I had to place the {{ expert}} template on this page, because it's in such bad shape. It doesn't seem at all coherent. I came to the article hoping to learn how to explicitly calculate the flow of a vector field using the exponential map. There's only passing mention to his, and no examples given at all. The problem seems to be that the exponential map turns up in so many different guises that each editor's contribution is made from a different theoretical point of view. It all needs to be brought together. This article should be a key priority; the exponential is a well known and well used object. I don't know enough about it myself; that's why I put {{ expert}}. I do, however, know enough about Wikipedia and editing to be able to help anyone that might need any help. Let's hope we can get this article up to scratch. — Fly by Night ( talk) 19:38, 22 November 2010 (UTC)