Talk:Riemannian geometry
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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)
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)