Talk:PlanetPhysics/Index of Differential Geometry

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This is a contributed entry in progress on \htmladdnormallink{Differential Geometry}{http://planetphysics.us/encyclopedia/DifferentialGeometry.html}.

\section{Index of Differential Geometry} \begin{itemize} \item 1.0 Euclidean and analytical geometry \item 1.1 \htmladdnormallink{Riemannian geometry}{http://planetphysics.us/encyclopedia/NonabelianAlgebraicTopology3.html} \item 1.2 Pseudo-Riemannian geometry \item 1.3 Cauchy-Riemannian geometry \item 1.4 Finsler geometry \item 1.5 Symplectic geometry \item 1.6 Contact geometry \item 1.7 Complex and K\"ahler geometry \item 1.8 Affine differential geometry \item 1.9 Projective differential geometry \item 2.0 \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} \item 2.1 Synthetic differential geometry \item 2.2 Abstract differential geometry \item 2.3 Discrete differential geometry \end{itemize}

\subsection{Fundamental concepts of differential geometry} \begin{itemize} \item Bundles \item Connections (connexions) \item \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} \item Submanifolds \item Differentiable manifolds \item Cross manifolds \item Hypersurfaces \item \htmladdnormallink{tensors}{http://planetphysics.us/encyclopedia/Tensor.html} and Forms \item \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} on forms and Integration \item Rigidity \item Involutive distributions \item Jacobi \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} and conjugate points \item First and second variations \item \htmladdnormallink{geodesics}{http://planetphysics.us/encyclopedia/GeodesicEquation.html} \item \htmladdnormallink{Lie groups}{http://planetphysics.us/encyclopedia/BilinearMap.html} in differential geometry \item Lie derivatives \item xyz \item xwz \item \htmladdnormallink{tangent spaces}{http://planetphysics.us/encyclopedia/BilinearMap.html} \end{itemize}

\subsection{Differential geometry (DG) applications}

\begin{itemize} \item In physics both \htmladdnormallink{Electromagnetism}{http://planetphysics.us/encyclopedia/Electromagnetism.html} and \htmladdnormallink{general relativity}{http://planetphysics.us/encyclopedia/SR.html} (\htmladdnormallink{GR}{http://planetphysics.us/encyclopedia/SR.html}) employ extensively DG \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} and tools; our \htmladdnormallink{Universe}{http://planetphysics.us/encyclopedia/MultiVerses.html} was represented in \htmladdnormallink{Einstein's}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} GR as a smooth manifold equipped with a pseudo-Riemannian \htmladdnormallink{metric}{http://planetphysics.us/encyclopedia/MetricTensor.html}, 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 {\em limit}. \item Symplectic manifolds are especially useful to the study of \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html} \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}. \item In engineering there are numerous applications of differential geometry in digital signal processing, architectural design, image enhancement, and so on. \item The Fisher information metric is used in information theory and advanced statistics. \item \htmladdnormallink{computer}{http://planetphysics.us/encyclopedia/Program3.html} graphics and CAD {\em computer-aided design} are based on differential geometry. \item In Geophysics, differential geometry is routinely used to analyze and describe geologic structures, as well as several other applications \end{itemize}

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

\begin{thebibliography}{99}

\bibitem{Bloch96} Ethan D. Bloch. (1996). A First Course in Geometric Topology and Differential Geometry.

\bibitem{Burke} William L. Burke. (1985). Applied Differential Geometry.

\bibitem{CP94} do Carmo, Manfredo Perdigao (1994). Riemannian Geometry.

\bibitem{Frankel2k4} Theodore Frankel (2004). The geometry of physics: an introduction.

\bibitem{Gray98} Alfred Gray. (1998). Modern Differential Geometry of Curves and Surfaces with Mathematica.

\bibitem{Spivak99} Michael Spivak. (1999). A Comprehensive Introduction to Differential Geometry (5 Volumes).

\bibitem{McCleary94} John McCleary.(1994). Geometry from a Differentiable Viewpoint.

\bibitem{Hicks65} Noel. J. Hicks. 1965. Notes on Differential Geometry. Van Nostrand Reinhold: New York; \htmladdnormallink{Free download}{http://www.wisdom.weizmann.ac.il/~yakov/scanlib/hicks.pdf}

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