Diffgeom:Guided tour of Riemannian geometry

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This article is about the diffgeom wiki itself

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:

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

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:

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:

Classification based on the kind of information yielded

Here are some categories you might find useful: