Editing Differential geometry (section)

== Applications ==
{{Spacetime|cTopic=Introduction}}
Below are some examples of how differential geometry is applied to other fields of science and mathematics.
*In [[physics]], differential geometry has many applications, including:
**Differential geometry is the language in which [[Albert Einstein]]'s [[general theory of relativity]] is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature of [[spacetime]]. Understanding this curvature is essential for the positioning of [[satellites]] into orbit around the Earth. Differential geometry is also indispensable in the study of [[gravitational lensing]] and [[black holes]].
**[[Differential forms]] are used in the study of [[electromagnetism]].
**Differential geometry has applications to both [[Lagrangian mechanics]] and [[Hamiltonian mechanics]]. Symplectic manifolds in particular can be used to study [[Hamiltonian system]]s.
**Riemannian geometry and contact geometry have been used to construct the formalism of [[geometrothermodynamics]] which has found applications in classical equilibrium [[thermodynamics]].
*In [[chemistry]] and [[biophysics]] when modelling cell membrane structure under varying pressure.
*In [[economics]], differential geometry has applications to the field of [[econometrics]].<ref>{{cite book |editor-first=Paul |editor-last=Marriott |editor2-first=Mark |editor2-last=Salmon |title=Applications of Differential Geometry to Econometrics |publisher=Cambridge University Press |year=2000 |isbn=978-0-521-65116-5 }}</ref>
*[[Geometric modeling]] (including [[computer graphics]]) and [[computer-aided geometric design]] draw on ideas from differential geometry.
*In [[engineering]], differential geometry can be applied to solve problems in [[digital signal processing]].<ref>{{cite book |first=Jonathan H. |last=Manton |chapter=On the role of differential geometry in signal processing |year=2005 |doi=10.1109/ICASSP.2005.1416480|title=Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005 |volume=5 |pages=1021–1024 |isbn=978-0-7803-8874-1 |s2cid=12265584 }}</ref>
*In [[control theory]], differential geometry can be used to analyze nonlinear controllers, particularly [[geometric control]]<ref>{{cite book |first1=Francesco |last1=Bullo |first2=Andrew |last2=Lewis |title=Geometric Control of Mechanical Systems : Modeling, Analysis, and Design for Simple Mechanical Control Systems |publisher=Springer-Verlag |year=2010 |isbn=978-1-4419-1968-7 }}</ref>
* In [[probability]], [[statistics]], and [[information theory]], one can interpret various structures as Riemannian manifolds, which yields the field of [[information geometry]], particularly via the [[Fisher information metric]].
*In [[structural geology]], differential geometry is used to analyze and describe geologic structures.
*In [[computer vision]], differential geometry is used to analyze shapes.<ref>{{cite thesis |type=Ph.D. |date=May 2008 |first=Mario |last=Micheli |title=The Differential Geometry of Landmark Shape Manifolds: Metrics, Geodesics, and Curvature |url=https://www.math.ucla.edu/~micheli/PUBLICATIONS/micheli_phd.pdf |archive-date=June 4, 2011 |archive-url=https://web.archive.org/web/20110604092900/http://www.math.ucla.edu/~micheli/PUBLICATIONS/micheli_phd.pdf }}</ref>
*In [[image processing]], differential geometry is used to process and analyse data on non-flat surfaces.<ref>{{cite thesis |type=Ph.D. |first=Anand A. |last=Joshi |title=Geometric Methods for Image Processing and Signal Analysis |date=August 2008 |url=http://users.loni.ucla.edu/~ajoshi/final_thesis.pdf |archive-url=https://web.archive.org/web/20110720072929/http://users.loni.ucla.edu/~ajoshi/final_thesis.pdf |archive-date=2011-07-20 |url-status=live }}</ref>
*[[Grigori Perelman]]'s proof of the [[Poincaré conjecture]] using the techniques of [[Ricci flow]]s demonstrated the power of the differential-geometric approach to questions in [[topology]] and it highlighted the important role played by its analytic methods.
* In [[wireless|wireless communications]], [[Grassmannian| Grassmannian manifolds]] are used for [[beamforming]] techniques in [[MIMO|multiple antenna]] systems.<ref>{{cite journal |first1=David J. |last1=Love |first2=Robert W. Jr. |last2=Heath |title=Grassmannian Beamforming for Multiple-Input Multiple-Output Wireless Systems |journal=IEEE Transactions on Information Theory |volume=49 |issue=10 |date=October 2003 |pages=2735–2747 |doi=10.1109/TIT.2003.817466 |url=http://users.ece.utexas.edu/~rheath/papers/2002/grassbeam/paper.pdf |archive-url=https://wayback.archive-it.org/all/20081002134712/http://users.ece.utexas.edu/~rheath/papers/2002/grassbeam/paper.pdf |url-status=dead |archive-date=2008-10-02 |citeseerx=10.1.1.106.4187 }}</ref>
* In [[geodesy]], for calculating distances and angles on the mean sea level surface of the [[Earth]], modelled by an ellipsoid of revolution.
* In [[neuroimaging]] and [[brain-computer interface]], symmetric positive definite manifolds are used to model functional, structural, or electrophysiological connectivity matrices.<ref>{{cite arXiv
|last1  = Ju
|first1 = Ce
|last2  = Kobler 
|first2 = Reinmar
|last3  = Collas 
|first3 = Antoine
|last4  = Kawanabe 
|first4 = Motoaki
|last5  = Guan 
|first5 = Cuntai
|last6  = Thirion 
|first6 = Bertrand
|date  = 26 Apr 2025
|title = SPD Learning for Covariance-Based Neuroimaging Analysis: Perspectives, Methods, and Challenges
|arxiv = 2504.18882
|class =cs.LG}}</ref>