Branches of Mathematics
Differential Geometry
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Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.
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 Cross-Tradition Connections
Associated With
Includes
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.
Sources
1. Differential Geometry (Wikipedia)
WikipediaIntroductionQuote, Introduction
A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure.
1. Differential Geometry (Wikipedia)
WikipediaIncludes: Carl Friedrich Gauss, Intrinsic geometry sectionQuote, Includes: Carl Friedrich Gauss, Intrinsic geometry section
Gauss introduced the Gauss map, Gaussian curvature, first and second fundamental forms, proved the Theorema Egregium.
1. Differential Geometry (Wikipedia)
WikipediaIncludes: Bernhard Riemann, Intrinsic geometry sectionQuote, 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.
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.
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