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)

[edit] Split in two?

What do people think about splitting this article into articles two pieces:

  • Exponential map (Lie theory)
  • Exponential map (Riemannian geometry)

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)

But why would you want to split them? The article is rather short, and the two concepts coincide when a manifold has both a Lie group and a Riemannian structure and the two structures are compatible (at least that's what the article says). To me it looks that the reader will get a better idea about the two concepts and their relationship if they are in the same article. Oleg Alexandrov (talk) 03:59, 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)

Note that the two concepts do NOT coincide when the Lie group has a left-invariant structure, which is the usual meaning of the term "compatible" in this context. They coincide in the much more special case of a bi-invariant metric, which is a far less interesting special case. A book that details their differences and interrelationships is "Gromov's almost flat manifolds" by Buser and Karcher. Katzmik (talk) 15:49, 16 April 2008 (UTC)
If a split is done, the links to this page need to be disambiguated.
My own take is that since the concepts are very related, things should be like now: one section to one concept, another section to the other one, and a section describing when the two notions coincide.
But I am not a differential geometer, and don't feel strongly about it. As long as somebody is willing to do a good job at splitting the page, and fixing the links to point to the right article, I would be happy. Oleg Alexandrov (talk) 04:59, 17 April 2008 (UTC)