This is a contributed entry in progress on Differential Geometry.

Index of Differential Geometry

• 1.0 Euclidean and analytical geometry
• 1.1 Riemannian geometry
• 1.2 Pseudo-Riemannian geometry
• 1.3 Cauchy-Riemannian geometry
• 1.4 Finsler geometry
• 1.5 Symplectic geometry
• 1.6 Contact geometry
• 1.7 Complex and Kähler geometry
• 1.8 Affine differential geometry
• 1.9 Projective differential geometry
• 2.0 noncommutative geometry
• 2.1 Synthetic differential geometry
• 2.2 Abstract differential geometry
• 2.3 Discrete differential geometry

Fundamental concepts of differential geometry

• Bundles
• Connections (connexions)
• manifolds
• Submanifolds
• Differentiable manifolds
• Cross manifolds
• Hypersurfaces
• tensors and Forms
• operators on forms and Integration
• Rigidity
• Involutive distributions
• Jacobi fields and conjugate points
• First and second variations
• geodesics
• Lie groups in differential geometry
• Lie derivatives
• xyz
• xwz
• tangent spaces

Differential geometry (DG) applications

• In physics both Electromagnetism and general relativity (GR) employ extensively DG concepts and tools; our Universe was represented in Einstein's GR as a smooth manifold equipped with a pseudo-Riemannian metric, that describes the curvature of space-time; however, refined GR models of space-times in the inflationary universe regard the space-times as a directed sequence of space-times or a limit.
• Symplectic manifolds are especially useful to the study of Hamiltonian systems.
• In engineering there are numerous applications of differential geometry in digital signal processing, architectural design, image enhancement, and so on.
• The Fisher information metric is used in information theory and advanced statistics.
• computer graphics and CAD computer-aided design are based on differential geometry.
• In Geophysics, differential geometry is routinely used to analyze and describe geologic structures, as well as several other applications

Often differential geometry is considered to be one of the more applied areas of mathematics.

Bibliography

1

Ethan D. Bloch. (1996). A First Course in Geometric Topology and Differential Geometry.

2

William L. Burke. (1985). Applied Differential Geometry.

3

do Carmo, Manfredo Perdigao (1994). Riemannian Geometry.

4

Theodore Frankel (2004). The geometry of physics: an introduction.

5

Alfred Gray. (1998). Modern Differential Geometry of Curves and Surfaces with Mathematica.

6

Michael Spivak. (1999). A Comprehensive Introduction to Differential Geometry (5 Volumes).

7

John McCleary.(1994). Geometry from a Differentiable Viewpoint.

8

Noel. J. Hicks. 1965. Notes on Differential Geometry. Van Nostrand Reinhold: New York; Free download