Talk:Exponential map (Riemannian geometry)

message: 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)
thread_title: Gauss lemma

message: 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)
thread_title:  Moved unrelated material 

message: I had to place the  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 . 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)
replies: text: Well, I'm not an expert neither, but the part about Lie algebras and groups seems correct to me and there are examples too. Concerning the Riemannian geometry and the case of a general manifold with affine connection, I agree that some examples would be nice.. Franp9am (talk) 22:06, 27 August 2011 (UTC)
replies: text: I removed the expert tag, and then looked here, at this talk page.  I agree that this article is thin, lacking in details.  But the same could be said for the vast majority of all articles in math. When we even have an article for the topic ... Conversely, I saw nothing factually incorrect in this article. Normally, when I see the "expert" tag, I recoil, and think "uh oh, there is something wrong with this article".  There is nothing I could see wrong here.  And one does not need to be an "expert" to flesh out this article: any reasonably competent student studying this topic should be able to expand this article in several ways.linas (talk) 16:27, 13 July 2012 (UTC)
thread_title:  Expert Needed