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 Connections
Associated With
Includes
Source Paul Finsler (Wikipedia)
Source Paul Stäckel (Wikipedia)
Source Penrose-Hawking singularity theorems (Wikipedia)
Source Wikipedia: Pierre Ossian Bonnet
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 Robert Zimmer (Wikipedia)
Source Roland Weitzenböck (Wikipedia)
Source Rudolf Lipschitz (Wikipedia)
Source Sergei Bernstein, Biography (MacTutor History of Mathematics)
Source Shiing-Shen Chern (Wikipedia)
Source Shing-Tung Yau (Wikipedia)
Source Su Buqing (Wikipedia)
Source Wikipedia: Sumner Byron Myers
Source Wolfram MathWorld: Tensor
Source Tensor Field (Wikipedia)
Source Torsion Tensor (Wikipedia)
Source Tullio Levi-Civita (Wikipedia)
Source Weinstein Conjecture (Wikipedia)
Source Wilhelm Blaschke, Biography (MacTutor History of Mathematics)
Source Wikipedia: Willi Rinow
Source MacTutor History of Mathematics: William Browder
Showing 51 to 72 of 72 in Includes, page 2 of 2.
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.
In Group: Differential and Non-Euclidean Geometry, lead paragraph
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.
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.
View the Source 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.
View the Source Michael Atiyah (Wikipedia)
WikipediaIncludes: Michael Atiyah, Lead sectionQuote, Includes: Michael Atiyah, Lead section
a British-Lebanese mathematician specialising in geometry
View the Source 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.
View the Source Bochner's Formula (Wikipedia)
Includes: Bochner's Formula, LeadQuote, Includes: Bochner's Formula, Lead
Bochner's formula is a statement relating harmonic functions on a Riemannian manifold
View the Source Johann Christian Martin Bartels (MacTutor)
Includes: Johann Christian Martin Bartels, Biography, DorpatQuote, Includes: Johann Christian Martin Bartels, Biography, Dorpat
Bartels founded the Centre for Differential Geometry at Dorpat, and remained there until his death in 1836.
View the Source Dimitri Fedorovich Egorov (MacTutor)
Includes: Dmitri Egorov, SummaryQuote, Includes: Dmitri Egorov, Summary
worked in differential geometry and mathematical analysis
View the Source Edward Kasner (Wikipedia)
Includes: Edward Kasner, LeadQuote, Includes: Edward Kasner, Lead
specializing in differential geometry
View the Source Alan Weinstein (Wikipedia)
Includes: Alan Weinstein, LeadQuote, Includes: Alan Weinstein, Lead
working in the field of differential geometry, and especially in Poisson geometry
View the Source Edmond Bour (Wikipedia)
Includes: Edmond Bour, LeadQuote, Includes: Edmond Bour, Lead
Most of his work was on the deformation of surfaces
View the Source Shiing-Shen Chern (Wikipedia)
Includes: Shiing-Shen Chern, LeadQuote, Includes: Shiing-Shen Chern, Lead
He made fundamental contributions to differential geometry and topology.
View the Source Robert Zimmer (Wikipedia)
Includes: Robert Zimmer, Lead, research focus (differential geometry)Quote, Includes: Robert Zimmer, Lead, research focus (differential geometry)
differential geometry
View the Source Harry Rauch (Wikipedia)
Includes: Harry Rauch, LeadQuote, Includes: Harry Rauch, Lead
who worked on complex analysis and differential geometry
View the Source Paul Finsler (Wikipedia)
Includes: Paul Finsler, LeadQuote, Includes: Paul Finsler, Lead
Finsler's thesis work concerned differential geometry
View the Source Tullio Levi-Civita (Wikipedia)
Includes: Tullio Levi-Civita, Biography, 1917Quote, Includes: Tullio Levi-Civita, Biography, 1917
In 1917 he introduced the notion of parallel transport in Riemannian geometry
View the Source Donald C. Spencer (Wikipedia)
Includes: Donald C. Spencer, Lead, deformation theoryQuote, Includes: Donald C. Spencer, Lead, deformation theory
known for work on deformation theory of structures arising in differential geometry
View the Source Jean Frédéric Frenet (Wikipedia)
Includes: Jean Frédéric Frenet, LeadQuote, Includes: Jean Frédéric Frenet, Lead
These formulas are important in the theory of space curves (differential geometry)
View the Source Frenet-Serret formulas (Wikipedia)
Includes: Frenet-Serret formulas, LeadQuote, Includes: Frenet-Serret formulas, Lead
In differential geometry, the Frenet-Serret formulas describe the kinematic properties
View the Source MacTutor History of Mathematics: Joseph Alfred Serret (MacTutor)
MacTutor History of MathematicsIncludes: Joseph Alfred Serret, Biography, last sectionQuote, Includes: Joseph Alfred Serret, Biography, last section
Serret did important work in differential geometry.
View the Source Eugenio Beltrami (MacTutor)
Includes: Eugenio Beltrami, SummaryQuote, Includes: Eugenio Beltrami, Summary
Eugenio Beltrami contributed to work in differential geometry on curves and surfaces
View the Source Elwin Bruno Christoffel (MacTutor)
Includes: Elwin Bruno Christoffel, Biography, geometry and tensor analysisQuote, Includes: Elwin Bruno Christoffel, Biography, geometry and tensor analysis
Christoffel published papers on function theory including conformal mappings, geometry and tensor analysis
View the Source Alfred Enneper (Wikipedia)
Includes: Alfred Enneper, Lead, minimal surfacesQuote, Includes: Alfred Enneper, Lead, minimal surfaces
He studied minimal surfaces and parametrized Enneper's minimal surfaces in 1864.
View the Source Albert Ribaucour (Wikipedia)
Includes: Albert Ribaucour, Lead, mathematicsQuote, Includes: Albert Ribaucour, Lead, mathematics
particularly in differential geometry and minimal surfaces.
View the Source Eugenio Calabi (Wikipedia)
Includes: Eugenio Calabi, LeadQuote, Includes: Eugenio Calabi, Lead
specializing in differential geometry, partial differential equations and their applications
View the Source Rudolf Lipschitz (Wikipedia)
Includes: Rudolf Lipschitz, Lead, differential geometryQuote, Includes: Rudolf Lipschitz, Lead, differential geometry
and differential geometry
View the Source Gregorio Ricci-Curbastro (Wikipedia)
Includes: Gregorio Ricci-Curbastro, LeadQuote, Includes: Gregorio Ricci-Curbastro, Lead
He is most famous as the discoverer of tensor calculus.
View the Source Shing-Tung Yau (Wikipedia)
Includes: Shing-Tung Yau, LeadQuote, Includes: Shing-Tung Yau, Lead
Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis
View the Source Ashutosh Mukherjee (Wikipedia)
Includes: Ashutosh Mukherjee, The young mathematicianQuote, Includes: Ashutosh Mukherjee, The young mathematician
He also made lasting contributions in differential geometry
View the Source MacTutor History of Mathematics: William Browder
MacTutor History of MathematicsIncludes: William Browder, SummaryQuote, Includes: William Browder, Summary
specialising in algebraic topology, differential topology and differential geometry
View the Source Roland Weitzenböck (Wikipedia)
Includes: Roland Weitzenböck, LeadQuote, Includes: Roland Weitzenböck, Lead
was an Austrian mathematician working on differential geometry who introduced the Weitzenböck connection
View the Source Harold Rosenberg (mathematician) (Wikipedia)
Includes: Harold Rosenberg, LeadQuote, Includes: Harold Rosenberg, Lead
is an American mathematician who works on differential geometry
View the Source Christiaan Huygens (Wikipedia)
Includes: Christiaan Huygens, Lead, field of workQuote, Includes: Christiaan Huygens, Lead, field of work
As a mathematician, Huygens developed his theory of evolutes and wrote on games of chance and the problem of points in Van Rekeningh in Spelen van Gluck, which Frans van Schooten translated and published as De Ratiociniis in Ludo Aleae (1657).
View the Source Charles Dupin (Wikipedia)
Includes: Charles Dupin, Article textQuote, Includes: Charles Dupin, Article text
descriptive and differential geometry
View the Source Nicolas Fuss (Wikipedia)
Includes: Nicolas Fuss, Lead, field of workQuote, Includes: Nicolas Fuss, Lead, field of work
He contributed to spherical trigonometry, differential equations, the optics of microscopes and telescopes, differential geometry, and actuarial science.
View the Source Heinrich Scherk (Wikipedia)
Includes: Heinrich Scherk, Lead, minimal surfacesQuote, Includes: Heinrich Scherk, Lead, minimal surfaces
notable for his work on minimal surfaces and the distribution of prime numbers
View the Source Su Buqing (Wikipedia)
Includes: Su Buqing, Lead, fieldQuote, Includes: Su Buqing, Lead, field
He was the founder of differential geometry in China
View the Source Michael Spivak (Wikipedia)
Includes: Michael Spivak, LeadQuote, Includes: Michael Spivak, Lead
was an American mathematician specializing in differential geometry
View the Source MacTutor History of Mathematics, University of St Andrews: Louis Puissant (biography)
MacTutor History of Mathematics, University of St AndrewsIncludes: Louis Puissant, Biography, map projectionQuote, Includes: Louis Puissant, Biography, map projection
The map was produced with considerable detail, the projection used spherical trigonometry, truncated power series and differential geometry.
View the Source Paul Stäckel (Wikipedia)
Includes: Paul Stäckel, LeadQuote, Includes: Paul Stäckel, Lead
active in the areas of differential geometry, number theory, and non-Euclidean geometry
View the Source Herbert Busemann (Wikipedia)
Includes: Herbert Busemann, Lead, fieldQuote, Includes: Herbert Busemann, Lead, field
was a German-American mathematician specializing in convex and differential geometry
View the Source Hidehiko Yamabe (MacTutor)
Includes: Hidehiko Yamabe, Biography, Minnesota yearsQuote, Includes: Hidehiko Yamabe, Biography, Minnesota years
differential equations and differential geometry
View the Source George Henri Halphen (MacTutor)
Includes: Georges Henri Halphen, Summary, differential geometryQuote, Includes: Georges Henri Halphen, Summary, differential geometry
projective differential geometry
View the Source Georges Reeb (MacTutor)
Includes: Georges Reeb, SummaryQuote, Includes: Georges Reeb, Summary
Georges Reeb was a French mathematician who worked in differential topology, differential geometry, topological dynamics and non-standard analysis.
View the Source Albert Wangerin (MacTutor)
Includes: Friedrich Heinrich Albert Wangerin, Summary, differential geometryQuote, Includes: Friedrich Heinrich Albert Wangerin, Summary, differential geometry
Albert Wangerin was a German mathematician who worked on potential theory, spherical functions and differential geometry.
View the Source 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.
View the Source 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
View the Source Carathéodory conjecture (Wikipedia)
Myers's theorem (Wikipedia)
Cartan-Hadamard theorem, Wikipedia
Rauch comparison theorem, Wikipedia
Élie Cartan (Wikipedia)
Includes: Élie Cartan, Lead paragraph [in-branch 2]Quote, Includes: Élie Cartan, Lead paragraph [in-branch 2]
differential geometry
View the Source Wikipedia: Sumner Byron Myers
Includes: Sumner Byron Myers, LeadQuote, Includes: Sumner Byron Myers, Lead
specializing in topology and differential geometry
View the Source Wikipedia: Pierre Ossian Bonnet
Includes: Pierre Ossian Bonnet, LeadQuote, Includes: Pierre Ossian Bonnet, Lead
important contributions to the differential geometry of surfaces
View the Source Wikipedia: Nicolaas Kuiper
Includes: Nicolaas Kuiper, LeadQuote, Includes: Nicolaas Kuiper, Lead
He completed his Ph.D. in differential geometry from the University of Leiden in 1946 under the supervision of Willem van der Woude.
View the Source Wikipedia: Alexis Clairaut
Includes: Alexis Claude Clairaut, LeadQuote, Includes: Alexis Claude Clairaut, Lead
In mathematics he is also credited with Clairaut's theorem on mixed partial derivatives, Clairaut's equation, and Clairaut's relation in differential geometry. == Biography == === Childhood and early life === Clairaut was born in Paris, France, to Jean-Baptiste and Catherine Petit Clairaut.
View the Source Wikipedia: Willi Rinow
Includes: Willi Rinow, LeadQuote, Includes: Willi Rinow, Lead
Willi Ludwig August Rinow (February 28th, 1907 in Berlin, March 29th, 1979 in Greifswald) was a German mathematician who specialized in differential geometry and topology.
View the Source Cartan-Hadamard conjecture (Wikipedia)
Includes: Cartan-Hadamard Conjecture, Lead paragraphQuote, Includes: Cartan-Hadamard Conjecture, Lead paragraph
In mathematics, the Cartan-Hadamard conjecture is a fundamental problem in Riemannian geometry and geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan-Hadamard manifolds.
View the Source Tensor Field (Wikipedia)
Includes: Tensor field, Lead paragraphQuote, Includes: Tensor field, Lead paragraph
Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences.
View the Source Penrose-Hawking singularity theorems (Wikipedia)
Includes: Penrose-Hawking Singularity Theorems, Lead paragraphQuote, Includes: Penrose-Hawking Singularity Theorems, Lead paragraph
The Penrose singularity theorem is a theorem in semi-Riemannian geometry and its general relativistic interpretation predicts a gravitational singularity in black hole formation.
View the Source Heinrich Maschke, Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsIncludes: Heinrich Maschke, BiographyQuote, Includes: Heinrich Maschke, Biography
who worked on group theory and on differential geometry
View the Source Wolfram MathWorld: Hopf Conjecture
Wolfram MathWorldIncludes: Hopf Conjecture, Subject classificationsQuote, Includes: Hopf Conjecture, Subject classifications
Calculus and Analysis Differential Geometry
View the Source Wolfram MathWorld: Tensor
Wolfram MathWorldIncludes: Tensor, Subject classificationsQuote, Includes: Tensor, Subject classifications
Calculus and Analysis Differential Geometry
View the Source Wolfram MathWorld: Kronecker Delta
Wolfram MathWorldIncludes: Kronecker Delta, Subject classificationsQuote, Includes: Kronecker Delta, Subject classifications
Calculus and Analysis Differential Geometry
View the Source Weinstein Conjecture (Wikipedia)
Wikimedia FoundationIncludes: Weinstein Conjecture, Lead sectionQuote, Includes: Weinstein Conjecture, Lead section
the conjecture claims that on a compact contact manifold, its Reeb vector field should carry at least
View the Source Double bubble theorem - Wikipedia
Includes: Double Bubble Theorem, Lead sectionQuote, Includes: Double Bubble Theorem, Lead section
In the mathematical theory of minimal surfaces, the double bubble theorem states that the shape
View the Source Sergei Bernstein, Biography (MacTutor History of Mathematics)
Includes: Sergei Bernstein, SummaryQuote, Includes: Sergei Bernstein, Summary
Sergei Bernstein was a Ukrainian mathematician who made important contributions to partial differential equations, differential geometry, probability theory and approximation theory.
View the Source Wilhelm Blaschke, Biography (MacTutor History of Mathematics)
Includes: Wilhelm Blaschke, SummaryQuote, Includes: Wilhelm Blaschke, Summary
Wilhelm Blaschke was an Austrian mathematician whose research was on various aspects of geometry. He also initiated the study of topological differential geometry.
View the Source Gaspard Monge, Biography (MacTutor History of Mathematics)
Includes: Gaspard Monge, SummaryQuote, Includes: Gaspard Monge, Summary
Gaspard Monge is considered the father of differential geometry because of his work Application de l'analyse à la géométrie where he introduced the concept of lines of curvature of a surface in 3 -space.
View the Source Joseph Plateau, Biography (MacTutor History of Mathematics)
Includes: Joseph Plateau, SummaryQuote, Includes: Joseph Plateau, Summary
Joseph Plateau was a Belgian mathematician best known for Plateau's problem on surfaces of minimal area.
View the Source 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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