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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)
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
- 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)
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. 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. --PST 09:55, 21 January 2009 (UTC)
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. 220.127.116.11 (talk) 17:23, 22 June 2010 (UTC)
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."
How do you pronounce Riemannian
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)