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{{diffgeom}}

If you came to this wiki looking for a place to refer to definitions, facts, and proofs in Riemannian geometry, you need to know four things:

* Right now, we don't have much in these areas
* But we are rapidly expanding, and hope to soon be a destination of choice for Riemannian geometry
* By letting us know what you find missing, or what you find good, you can help us out
* By joining us, you can help us out even more!

Check out [[Diffgeom:Content scope]] to know about the content scope of Diffgeom for Riemannian geometry. As of now, we are trying to concentrate on the following themes and sub-themes of Riemannian geometry.

==Term definitions==

===Properties of Riemannian metric===

By a property of a Riemannian metric, we mean a property that, given any [[Riemannian metric]] on a [[differential manifold]], evalates to either true or false for that Riemannian metric. Further, the property should be invariant under isometries (in other words, it should depend ''only'' on the metric).

A full listing of these is available at:

[[:Category:Properties of Riemannian metrics]]

===Notions of curvature===

You may find the following preliminary articles and the following categories useful:

* [[:Category:Notions of curvature for Riemannian manifolds]]

If you want to know more about properties of manifolds that are defined in terms of curvature, check out:

* [[:Category:Curvature-based properties of Riemannian metrics]]

===Properties of local coordinates===

Local coordinates are a standard tool in the study of Riemannian manifolds. A list of the various kinds of local coordinates is available at:

* [[:Category: Properties of local coordinates]]

==Facts==

Ther eare many facts in Riemannian geometry, some of them being statements purely for a given differential manifold equipped with a Riemannian metric, while others are statements about the relation between (variation in) the Riemannian metric and the underlying differential/topological structure. Currently, we are aiming to put in a lot of facts -- the proofs will be filled in later, as the structure of the wiki becomes more and more emergent. Discussed below are various categorizations being used, that will help you locate precisely the result you want in Riemannian geometry.

===Dimension-based classification===

We have separate categories for results in different dimensions. Note that the category for results in a particular dimension mixes up Riemannian, differential, and topological results (because the dividing criterion is dimension). Important categories where you can get good listings are:

* [[:Category:Results in dimension 2]]
* [[:Category:Results in dimension 3]]
* [[:Category:Results in dimension 4]]
* [[:Category:Results in all dimensions]]

===Classification based on the kind of information yielded===

Here are some categories you might find useful:

* [[:Category:Results relating curvature to topology]]
* [[:Category:Results predicting the universal cover]]

Diffgeom:Guided tour of Riemannian geometry - Revision history

Vipul: 1 revision

Vipul at 07:58, 26 May 2007