Talk:Gauss's lemma (Riemannian geometry)

{{helpme}} ^{request for help can be viewed at User talk:Zadigus. Pumpmeup 11:10, 4 February 2008 (UTC)}[reply]

Retitled[edit]

I have moved this page from "Lemme de Gauss" to the correct English title. JohnCD (talk) 16:48, 4 February 2008 (UTC)[reply]

Forcing PNG display[edit]

Some of the <math> formulae were displayed much smaller than others. It appears that by default the system displays simpler formulae in HTML and only uses PNG where it has to, giving rise to a very uneven appearance. However by adding a backslash followed by a space character at the end of the simpler expressions, just before </math>, one can force it to display PNG. I have made this change throughout: the display is not pretty, but it is at least consistent. I have checked the result both in Firefox and in IE6. JohnCD (talk) 21:21, 5 February 2008 (UTC)[reply]

Balls[edit]

My grasp of differential geometry is on a par with my grasp of French, but... I think $T_{p}$ must be the tangent space at $p$ (the set of all tangents/vectors/velocities at point $p$ on $M$ ) - which should be stated. But then isn't the ball $B_{\epsilon }(0)$ the set of all small velocities/tangent-vectors (at $p$ ), in which case it's a subset of $T_{p}$ not $T_{p}M$ , so isn't $T_{p}M\supset B_{\epsilon }(0)$ a bit wrong??? --catslash (talk) 22:31, 5 February 2008 (UTC)[reply]

T_{p}M

is the tangent space to M at p.

B_{\epsilon }(0)

is the ball in

T_{p}M

of tangent vectors of length less than epsilon. What's the problem? Algebraist 18:07, 7 February 2008 (UTC)[reply]

OK fair enough, it's just that I was expecting something more ball-shaped - or am I confusing the tangent space and the tangent bundle? - yes I am. I thought

T_{p}

was the tangent space at

p

and so

T_{p}M

was the tangent bundle.

--catslash (talk) 19:46, 7 February 2008 (UTC)[reply]

Draft of translation[edit]

User:JohnCD made a draft translatio to be found at User:JohnCD/Gauss. Andreas ^(T) 02:32, 15 February 2008 (UTC)[reply]

Translation moved into main article; maths could do with a more concise rewrite?[edit]

I have replaced the French version with my translation into English. The original French can be found in the early versions of 4 - 6 Feb. Comments so far have said little about the translation but have been discouraging about the maths:

No offense, but it reads like a cliche differential geometry textbook, too many formulas. WP:NOT#TEXTBOOK may be relevant. Arcfrk (talk) 03:49, 13 February 2008 (UTC)[reply]

I'm just a bit sceptical of the exclamation points. Those French can be so passionate :-) Pete St.John (talk) 22:54, 12 February 2008 (UTC)[reply]

I think it's probably worth rewriting rather than translating. The current article fails to state Gauss' lemma in plain terms. Rather it relies on far too many equations, and fancy gadgets like the double tangent bundle TTM. Furthermore, it reinvents the wheel with respect to the exponential map (which Wikipedia already has an article on). The proof is not at all transparent; it would be better to give a few lines summarizing the main ideas. Finally, the actual statement of the theorem is misleading at best (and completely wrong at worst): the exponential map is not an isometry, though the statement here attempts to present it as though it were. Personally, I think a stub with a clear statement of the theorem, perhaps a few lines summarizing the main points of the proof, and a description of the applications of the theorem (to geodesic convexity, for instance) is preferable to the current mess. Silly rabbit (talk) 02:26, 14 February 2008 (UTC)[reply]

What is T_0 exp_p (v) in the equation line in "Introduction" section prefaced by "over v, we obtain"?

I've done a modest amount of reading in differential geometry and most of the terms are familiar, but not the one involving T_0. By the context it seems to be referring to a vector in T_p M, but I have never seen T_0 used that way. John M Lee's textbook "Riemannian Manifolds - Introduction to Curvature" uses T_0 M to refer to the tangent space to the vector space T_p M considered as a manifold, located at the point of the zero vector. But that interpretation doesn't fit here. Some explanation is needed as to what is meant by T_0 in this context.

It seems worth having this article (it was a redlink from the "Gauss's Lemma" DAB page), but there is clearly a more concise version to be written, and I am leaving this one here mainly in the hope that someone able to write that version (as I am not) will be inspired to do so. JohnCD (talk) 20:10, 19 February 2008 (UTC)[reply]

correction[edit]

corrected the formula $\langle T_{v}\exp _{p}(v),T_{v}\exp _{p}(w)\rangle _{q}=\langle v,w\rangle _{p}.$ , so that the metric is now evaluated in the right tangent spaces.