Differential geometry

From Mathematics wiki

Contents 1 Preliminaries 2 Metric structure 3 Connections and curvature 4 Isometries 5 Differential forms 6 Integration 7 Principal bundles 8 Curves 9 Surfaces 10 Uniformization and Liouville theory 11 Ricci flow 12 Supermanifolds 13 Resources 14 Appendix

Preliminaries

• Differential topology

Metric structure

• Metric tensor
• Parallel transport
• Geodesic

Connections and curvature

• Connection
• Affine connection
• Levi-Civita connection
• Covariant derivative
• Christoffel symbol
• Divergence
• Curvature
• Torsion
• Riemann tensor
• Ricci tensor
• Ricci scalar
• Orthonormal frame
• Laplace-Beltrami operator
• Spin connection
• Riemann normal coordinates
• Isothermal coordinates

Isometries

• Isometries
• Killing vector fields
• Weyl transformation
• Conformal symmetry

Differential forms

• Volume form
• Laplace-deRham operator
• Hodge dual
• Cartan formalism
• Covariant exterior derivative
• Curvature form
• Torsion form

Integration

• Stokes' theorem
• Gauss–Bonnet theorem

Principal bundles

• Principal bundle
• Connection form
• Curvature form

See also: Characteristic class (Algebraic topology)

Curves

• Frenet-Serret equations

Surfaces

Uniformization and Liouville theory

• Uniformization theorem
• Klein-Poincaré uniformization theorem
• Liouville theory

Ricci flow

• Metric on the space of metrics

Supermanifolds

Resources

• Books

Appendix

• Useful formulae