Talk:Riemannian geometry

I have altered the statement that Euclidean geometry is a subset of Riemannian geometry. The set of theorems of Riemannian geometry could be said to be a subset of the set of theorems of Euclidean geometry, if one were to construe the former to mean propositions true in all Riemannian manifolds. On the other hand, the class of spaces that satisfy the axioms of Riemannian geometry is a subclass of those that satisfy the axioms of Euclidean geometry. Not a set, but rather a proper class. Michael Hardy 19:57 Mar 12, 2003 (UTC)

Doesn't the fact that the set of theorems of Riemannian geometry could be said to be a subset of the set of theorems of Euclidean geometry imply that Euclidean geometry is a subset of Riemannian geometry?

The article mentions Euclidean spaces, but doesn't currently say anything about the relation between Riemannian geometry and Euclidean geometry; I think this relation should be mentioned. —Kri (talk) 13:40, 1 November 2018 (UTC)[reply]

This page has problems, in relation to the Riemannian manifold coverage elsewhere. The initial posting seems to have been about the Riemannian geometry of constant negative curvature. I'm not quite sure now what the thrust is.

Charles Matthews 19:01 29 Jun 2003 (UTC)

Riemannian geometry is the original name for geometry which deals with non-euclidean spaces. Historically, it is concrete. It is important to preserve the timeline for epistemological reasons. Also to give credit where credit is due, such that the things that the inventor had to say about their invention don't go unheard. They are important and the inventor has earned the right to be heard by inventing.

Riemannian geometry is prior to the Riemannian manifold.

Kevin Baas -2003.12.07

--- The page does have problems: it is a little bit of a hwole lot and nothing substantial of anything. Where there are headlines, those should be separate pages all together. Is an orthonormal frame riemannian geometry? No, it is a topic based off of riemannion geometry. It should be a page of it's own, at most linked to. same with the other topics. The point of this page is to give people an idea of what riemannian geometry is, not to throw a bunch of esoteric and advanced topics at them with no explanation or introduction. -Kevin Baas -2003.12.07

There was a question on an edit summary: isn't a line just a geodesic? a line is a geodesic if and only if it is the shortest path between two points.

Here are some rough definitions:

Line - a continuous one-dimensional extension, usually residing in a space. usually thought to be of infinite lenght, though sometimes used as shorthand for a line segment.

line segment - a continuous, 1-dimension extension from one point to another, of finite length.

geodesic - the shortest path between points, see calculus of variations.

curve - a continuous function defined on a space, often thought of as one-dimensional, but not thus restricted.

trajectory - a continuous function defined on a space, parametrized by a variable such as "t" (for time), often thought of as one-dimensional, but not thus restricted.

a given line is not neccessarily a geodesic. it is concievable to have a geodesic plane between two lines, this making a geodesic not neccessarily a line, but i don't know if the strict definition of the term includes such a generalization. Kevin Baas | talk 20:09, 2004 Aug 3 (UTC)

Intro has some issues

The intro uses "one" to avoid using "you." I'm not sure which WP:TC to use. Advice?-- Thinboy00 ^talk/_contribs 02:08, 9 October 2007 (UTC)[reply]

Just realized, first of all i'm referring to the section called "introduction," secondly are sections allowed to be called "introduction?" That's the purpose of the area before the sections.-- Thinboy00 ^talk/_contribs 02:11, 9 October 2007 (UTC)[reply]

History

It would be nice to get some history of the topic as well as a description of what it consists of. skeptical scientist (talk) 20:20, 30 March 2008 (UTC)[reply]

This article is very badly written

I am worried about the article and very angry about the statement: "There is no easy introduction to Riemannian geometry[citation needed]. It is generally recommended[who?] that one should work in the subject for quite a while to build some geometric intuition, usually by doing enormous amounts of calculations." What is this trying to convey? That Riemannian geometry is about calculating stuff? I feel that this is not the case. Calculations are done to build up an understanding and it is certainly not true that all Riemannian geometers do is calculate. --PS T 09:55, 21 January 2009 (UTC)[reply]

I agree that this article is very badly written but, to me, the biggest problem is the overwhelming amount of jargon and failure to explain things in plain English rather than any factual errors.Newzild (talk) 02:19, 13 September 2020 (UTC)[reply]

Passing by to agree with Newzild and say that a full rewriting is needed. I have engineering degrees and I can't understand the introductory jargon. Since this is an encyclopedia, I expected something along the lines of "Riemann is one of the geometries that apply to space, not only to the plane as the traditional, Euclidean geometry". After saying the basics, I guess it's ok to mention that it is a "branch of differential geometry", "inner product on the tangent space", but most people have no idea of what those are -- and this is not a Geometry treatise. Wikipedia has several areas filled with jargon, useless for nonspecialized people (Statistics is another example). Encyclopedias should be useful to smart but nonspecialized people. Vmkern (talk) 18:27, 1 July 2022 (UTC)[reply]

From the article:

This list is oriented to those who already know the basic definitions and want to know what these definitions are about.

Huh? If I knew the basic definitions, why would I want to know what they were about? The article could at least be kind enough to state what these basic definitions actually are. I suggest that this list be re-written so that it is oriented towards people with a less bizarre state of prior knowledge --- for instance, people who have the appropriate mathematical prerequisites to understand differential geometry, but who don't already know it. As it is, the article isn't much good as an introduction to the topic. 82.22.96.82 (talk) 17:23, 22 June 2010 (UTC)[reply]

Soul theorem error

"In particular, if M has strictly positive curvature everywhere, then it is diffeomorphic to Rn. G. Perelman in 1994 gave an astonishingly elegant/short proof of the Soul Conjecture: M is diffeomorphic to Rn if it has positive curvature at only one point."

Shouldn't it say "positive curvature at any one point"? ᛭ LokiClock (talk) 12:24, 25 April 2012 (UTC)[reply]

How do you pronounce Riemannian

Can someone help explain how Riemannian is pronounced? Is it Ri-Men-ean or REM-ANIAN — Preceding unsigned comment added by 2607:FB90:425:6E86:AC5A:86C2:B125:898C (talk) 15:16, 12 September 2016 (UTC)[reply]

It's ree-MONN-ee-in.

Projecting a sphere to a plane

The illustration, "Projecting a sphere to a plane." doesn't do that at all. At best, it projects a circular disk on one plane to a parallel surface. Drawing in the sphere does nothing.WithGLEE (talk) 12:53, 12 June 2018 (UTC)[reply]

moving GR infoboxes on Riemannian geometry topics to pseudo-Riemannian ones

the relativity infoboxes on Riemannian geometry should be moved to pseudo-Riemannian geometry.

while Riemannian geometry may have inspired pseudo-Riemannian geometry, which this is one way people can understand its impact on science, the latter is not the former.

i don't think it's appropriate to talk about Riemannian geometry in the terms of physics.

i would like to move any GR/SR infoboxes on Riemannian concepts to the pseudo-equivalent if they exist. if the equivalent does not exist, i would like to remove it.

enough is enough. we're taking our math back. 198.53.108.48 (talk) 18:32, 14 June 2021 (UTC)[reply]

I'm not sure a RfC was really all that necessary for this, and as far as RfCs go, this one is a bit malformed. Regarding the substance here, there's actually three related links in {{General relativity sidebar}}, which is a bit weird: Riemannian geometry, Riemannian manifold, and Lorentzian manifold (which links to a section in the pseudo-Riemannian manifold article you want included in the template).

I've moved Riemannian manifold to Pseudo-Riemannian manifold in the template (and added a sidebar there), and removed Lorentzian manifold as it would otherwise be linked to twice. I don't think this article (Riemannian geometry) should be removed from the actual template as it's kind of the more basic maths that was generalized/built upon in the math in the development of general relativity. But, I can see a reason not to include physics sidebars in more general mathematical articles when the connection isn't completely direct, so I can see the argument to remove the actual sidebar from this article. I'll leave that for someone else's judgement though. ‑‑Volteer1 (talk) 11:29, 15 June 2021 (UTC)[reply]

hi @Volteer1:, thank you for this intermediate solution. with regards to the malformed RfC: it seems i am incapable of making a proper one. could you please fix up my original request so that it works as-expected? thank you — Preceding unsigned comment added by 198.53.108.48 (talk • contribs)

The problem isn't that this isn't an actual RfC, it's that it's not really phrased like one, and it is probably unnecessary when a normal discussion would suffice. I've thought about it some more and I agree with you about removing the sidebar on this article: it's an article on a very general mathematical concept, and not actually the one directly used in GR anyway, so it is a bit out of place here; I've removed it. As this seems to not be that contentious and we've come to an agreement we're happy with, I'll just take off the RfC template and call it a day. ‑‑Volteer1 (talk) 04:48, 16 June 2021 (UTC)[reply]

"Pinched" requires a definition

It is hard for me to understand the thinking of someone who uses the word "pinched" in a Wikipedia mathematics article without defining it or even linking to a definition of it.

If sectional curvatures are "strictly pinched" between 1/4 and 1, does that mean they all lie in the open interval (1/4, 1) ?

If so, could someome knowledgeable about this subject please state this in the article?

If not, could someome knowledgeable about this subject please state the correct definition in the article?