Difference between revisions of "Main Page"

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This is Diffgeom (the Differential Geometry wiki).
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Welcome to Diffgeom (the Differential Geometry wiki)!
  
Much of it is patterned on the [http://groupprops.wiki-site.com/ Group Properties wiki].
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Diffgeom is currently in a pre-formed state, and much of the information entered here may be incorrect, misleading or unreliable. Please do ''not'' rely on the information or content here.
  
It is currently in a very preliminary stage and may contain a number of factual errors. It is meant only for private use.
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==Using Diffgeom==
  
It may never become sturdy enough for public use!
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* '''How do I use Diffgeom?'''
  
=Contents=
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We are ourselves trying to figure out answers to that! Diffgeom has been patterned on the lines of Groupprops, the group theory wiki. We intend to put up the following pages soon: [[Diffgeom:Guided tour]] and [[Diffgeom:Exploration]].
 
 
==Basics==
 
 
 
*[[Topological manifold]]
 
*[[Differential manifold]]
 
*[[Riemannian manifold]]
 
*[[Connection]]
 
*[[Lie group]]
 
 
 
===Properties of connections===
 
 
 
*[[Linear connection]]
 
*[[Torsion of a linear connection]]
 
*[[Torsion-free linear connection]]
 
 
 
===Basic kinds of metrics===
 
 
 
*[[Riemannian metric]]
 
*[[Homogeneous metric]]
 
*[[Locally homogeneous metric]]
 
 
 
==Classification of differential manifolds according to the structure group==
 
 
 
*[[Pseudo-Riemannian manifold]]
 
*[[Almost Hermitian manifold]]
 
*[[Almost complex manifold]]
 
*[[Almost symplectic manifold]]
 
*[[Lorentzian manifold]]
 
*[[Oriented manifold]]
 
 
 
==Curvatures==
 
 
 
*[[Riemann curvature tensor]]
 
*[[Ricci curvature tensor]]
 
*[[Scalar curvature]]
 
*[[Sectional curvature]]
 
*[[Total scalar curvature]]
 
*[[Total absolute scalar curvature]]
 
*[[Average scalar curvature]]
 
*[[Mean curvature]]
 
*[[Average mean curvature]]
 
 
 
==Classification of metrics according to curvatures==
 
 
 
*[[Flat metric]]
 
*[[Ricci-flat metric]]
 
*[[Einstein metric]]
 
*[[Constant-curvature metric]]
 
*[[Constant-scalar curvature metric]]
 
*[[Zero-scalar curvature metric]]
 
 
 
==Special scalar functions on Riemannian manifolds==
 
 
 
*[[Injectivity radius of a manifold]]
 
*[[Isoperimetric constant]]
 
*[[Pinching constant]]
 
 
 
==Various theorems==
 
 
 
===Results on Ricci curvature===
 
 
 
*[[Bonnet-Myers theorem]]
 
*[[Schoen-Yau theorem]]
 
*[[Zhu's theorem]]
 
*[[Splitting theorem]]
 
 
 
===Results on sectional curvature===
 
 
 
*[[Cheeger-Gromoll conjecture]]
 
*[[Cohn-Vossen theorem]]
 
*[[Gromoll-Meyer theorem]]
 
*[[Cartan-Hadamard theorem]]
 
*[[Bieberbach theorem]]
 
*[[Finiteness theorem]]
 
*[[Schur's theorem]]
 
*[[Synge's theorem]]
 
 
 
===Results relating curvature to topology===
 
 
 
*[[Bieberbach theorem]]
 
*[[Bonnet-Myers theorem]]
 
*[[Cartan-Hadamard theorem]]
 
*[[Gauss-Bonnet theorem]]
 
*[[Synge's theorem]]
 
*[[Preissmann's theorem]]
 
 
 
 
 
 
 
*[[Splitting theorem]]
 
 
 
==Special topics==
 
 
 
*[[Minimal manifold]]
 
*[[Characteristic class]]
 
*[[Chern form]]
 
*[[Chern-Weil theorem]]
 

Revision as of 08:59, 7 September 2007

Welcome to Diffgeom (the Differential Geometry wiki)!

Diffgeom is currently in a pre-formed state, and much of the information entered here may be incorrect, misleading or unreliable. Please do not rely on the information or content here.

Using Diffgeom

  • How do I use Diffgeom?

We are ourselves trying to figure out answers to that! Diffgeom has been patterned on the lines of Groupprops, the group theory wiki. We intend to put up the following pages soon: Diffgeom:Guided tour and Diffgeom:Exploration.