Branches of Mathematic
Differential Geometry
Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. Beyond its classical roots in the study of curves and surfaces, the field supplies the mathematical language of modern physics: differential geometry is the language in which Albert Einstein's general theory of relativity is expressed. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhat notion of size, distance, shape or curvature can be defined intrinsically on a smooth space, without reference to any surrounding space it might sit inside. 1 Key DebateWhether Carl Friedrich Gauss deserves priority for discovering non-Euclidean geometry ahead of the 1829 to 1832 published treatises of Janos Bolyai and Nikolai Lobachevsky. Gauss told Bolyai's father he had developed such a geometry years earlier but never published it, leaving his private claim impossible to verify against the two mathematicians' public work. 2 Classification
Pure or AppliedBoth / Interdisciplinary 1 Differential Geometry
Connections
Associated With
Includes
Source Differential Geometry (Wikipedia)
Source Myers's theorem (Wikipedia)
Source Carathéodory conjecture (Wikipedia)
Source Differential Geometry (Wikipedia)
Source Cartan-Hadamard theorem, Wikipedia
Source Élie Cartan (Wikipedia)
Source Filling Area Conjecture (Wikipedia)
Source Laplace-Beltrami Operator (Wikipedia)
Source Manifold (Wikipedia)
Source Michael Atiyah (Wikipedia)
Source Rauch comparison theorem, Wikipedia
Introduced the Ricci flow, the defining tool of geometric analysis in differential geometry, and developed it into the program that proved the Poincare and geometrization conjectures.
Source MacTutor History of Mathematics Archive
Source Torsion Tensor (Wikipedia)
Sources
1. Differential Geometry (Wikipedia)
WikipediaIntroduction
A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure.
Applications section, physics subsection, general relativity sentence
Differential geometry is the language in which Albert Einstein's general theory of relativity is expressed.
Lead section, pure or applied classification
It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky.
Includes: Carl Friedrich Gauss, Intrinsic geometry section
Gauss introduced the Gauss map, Gaussian curvature, first and second fundamental forms, proved the Theorema Egregium.
Includes: Bernhard Riemann, Intrinsic geometry section
Riemann introduced the notion of a Riemannian metric and the Riemannian curvature tensor for the first time, and began the systematic study of differential geometry in higher dimensions.
View the Source 2. History of Non-Euclidean Geometry (Wikipedia)
WikipediaDevelopment of non-Euclidean geometry sectionQuote, Development of non-Euclidean geometry section
Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, though he did not publish.
Manifold (Wikipedia)
Wikimedia FoundationIncludes: Manifold, lead paragraphQuote, Includes: Manifold, lead paragraph
Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done.
Michael Atiyah (Wikipedia)
WikipediaIncludes: Michael Atiyah, Lead sectionQuote, Includes: Michael Atiyah, Lead section
a British-Lebanese mathematician specialising in geometry
MacTutor History of Mathematics Archive
University of St Andrews, School of Mathematics and StatisticsIncludes: Richard Hamilton, https://mathshistory.st-andrews.ac.uk/Biographies/Hamilton_Richard/Quote, Includes: Richard Hamilton, https://mathshistory.st-andrews.ac.uk/Biographies/Hamilton_Richard/
Richard Hamilton was an American mathematician famed for his important contributions to proving the Poincare Conjecture.
Torsion Tensor (Wikipedia)
Includes: Torsion Tensor, Lead sentenceQuote, Includes: Torsion Tensor, Lead sentence
In differential geometry, the torsion tensor is a tensor that is associated to any affine connection.
Laplace-Beltrami Operator (Wikipedia)
Filling Area Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Filling Area Conjecture, Lead sentenceQuote, Includes: Filling Area Conjecture, Lead sentence
In differential geometry, Mikhail Gromov's filling area conjecture asserts that the hemisphere has minimum area among the orientab
Carathéodory conjecture (Wikipedia)
Myers's theorem (Wikipedia)
Cartan-Hadamard theorem, Wikipedia
Rauch comparison theorem, Wikipedia
Élie Cartan (Wikipedia)
Includes: Elie Cartan, Lead paragraph [in-branch 2]Quote, Includes: Elie Cartan, Lead paragraph [in-branch 2]
differential geometry
Dissenting Readings (1 dissenting reading)
Key Debate
Gauss's biographers point to correspondence, not only his 1832 remark to Bolyai's father, as evidence of priority: letters to Christian Ludwig Gerling and Friedrich Wilhelm Bessel already describe a self-consistent non-Euclidean geometry years before Bolyai's 1832 appendix to his father's Tentamen or Lobachevsky's 1829 paper in the Kazan Messenger. On this view Gauss avoided publishing to sidestep the reaction of philosophers committed to Euclidean space as a Kantian necessity, not from doubt in the result. The prevailing view among historians remains that unpublished private correspondence, however early, does not establish discovery in the sense that matters for credit: Bolyai and Lobachevsky each published complete developed systems that others could read, check and build on, and it is that act of publication, not private possession, that the field credits.
A dissenting reading, from Carl Friedrich Gauss and his later biographersHistory of Non-Euclidean Geometry (Wikipedia), Wikipedia
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