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Hypercomplex Manifold
In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a way that the quaternions I, J, K define integrable almost complex structures. If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex. Examples Every hyperkähler manifold is also hypercomplex. The converse is not true. The Hopf surface :\bigg(\backslash 0\bigg)/ (with acting as a multiplication by a quaternion q, , q, >1) is hypercomplex, but not Kähler, hence not hyperkähler either. To see that the Hopf surface is not Kähler, notice that it is diffeomorphic to a product S^1\times S^3, hence its odd cohomology group is odd-dimensional. By Hodge decomposition, odd cohomology of a compact Kähler manifold are always even-dimensional. In fact Hidekiyo Wakakuwa proved that on a compact hyperkähler manifold \ b_\equiv 0 \ mod \ 4. Misha Verbi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominic Joyce
Dominic David Joyce Fellow of the Royal Society, FRS (born 8 April 1968) is a British mathematician, currently a professor at the University of Oxford and a fellow of Lincoln College, Oxford, Lincoln College since 1995. His undergraduate and doctoral studies were at Merton College, Oxford. He undertook a DPhil in geometry under the supervision of Simon Donaldson, completed in 1992. After this he held short-term research posts at Christ Church, Oxford, as well as Princeton University and the University of California, Berkeley in the United States. Joyce is known for his construction of the first known explicit examples of compact Joyce manifolds (i.e., manifolds with G2 (mathematics), G2 holonomy). He has received the London Mathematical Society Junior Whitehead Prize and the European Mathematical Society Young Mathematicians Prize. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. Selected publications * * * with Yinan Song:arxiv.or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nuclear Physics (journal)
''Nuclear Physics A'', ''Nuclear Physics B'', ''Nuclear Physics B: Proceedings Supplements'' and discontinued ''Nuclear Physics'' are peer-reviewed scientific journals published by Elsevier. The scope of ''Nuclear Physics A'' is nuclear and hadronic physics, and that of ''Nuclear Physics B'' is high energy physics, quantum field theory, statistical systems, and mathematical physics. ''Nuclear Physics'' was established in 1956, and then split into ''Nuclear Physics A'' and ''Nuclear Physics B'' in 1967. A supplement series to ''Nuclear Physics B'', called ''Nuclear Physics B: Proceedings Supplements'' has been published from 1987 onwards. ''Nuclear Physics B'' is part of the SCOAP3 initiative. Abstracting and indexing ''Nuclear Physics A'' * Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate Analytics, formerly the Institute for Scientific Information and Thomson Reuters. It is published online and in several different printed sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called ''Surveys in Differential Geometry''. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University. History The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009. In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Harv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
Quaternionic Manifold
In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language of ''G''-structures on a manifold. Specifically, a quaternionic ''n-''manifold can be defined as a smooth manifold of real dimension 4''n'' equipped with a torsion-free \operatorname(n, \mathbb)\cdot\mathbb^\times-structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds. Definitions The enhanced quaternionic general linear group If we regard the quaternionic vector space \mathbb^n\cong\R^ as a right \mathbb-module, we can identify the algebra of right \mathbb-linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twistor Space
In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation \nabla_^\Omega_^=0 . It was described in the 1960s by Roger Penrose and Malcolm MacCallum. According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force. Informal motivation In the (translated) words of Jacques Hadamard: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying four-dimensional space \mathbb^4 it might be valuable to identify it with \mathbb^2. However, since there is no canonical way of doing so, instead all isomorphisms respecting orientation and metric between the two are considered. It turns out that complex projective 3-space \mathbb^3 parametrizes such isomorphisms together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dmitry Kaledin
Dmitri (russian: Дми́трий); Church Slavic form: Dimitry or Dimitri (); ancient Russian forms: D'mitriy or Dmitr ( or ) is a male given name common in Orthodox Christian culture, the Russian version of Greek Demetrios (Δημήτριος ''Dēmētrios'' ). The meaning of the name is "devoted to, dedicated to, or follower of Demeter" (Δημήτηρ, ''Dēmētēr''), "mother-earth", the Greek goddess of agriculture. Short forms of the name from the 13th–14th centuries are Mit, Mitya, Mityay, Mit'ka or Miten'ka (, or ); from the 20th century (originated from the Church Slavic form) are Dima, Dimka, Dimochka, Dimulya, Dimusha etc. (, etc.) St. Dimitri's Day The feast of the martyr Saint Demetrius of Thessalonica is celebrated on Saturday before November 8 ld Style October 26 The name day (именины): October 26 (November 8 on the Julian Calendar) See also: Eastern Orthodox liturgical calendar. The Saturday before October 26/November 8 is called Demetrius Saturd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edmond Bonan
Edmond Bonan (born 27 January 1937 in Haifa, Mandatory Palestine) is a French mathematician, known particularly for his work on special holonomy. Biography After completing his undergraduate studies at the École polytechnique, Bonan went on to write his 1967 University of Paris doctoral dissertation in Differential geometry under the supervision of André Lichnerowicz. From 1968 to 1997, he held the post of lecturer and then professor at the University of Picardie Jules Verne in Amiens, where he currently holds the title of professor emeritus. Early in his career, from 1969 to 1981, he also lectured at the École Polytechnique. See also * G2 manifold * G2 structure * Spin(7) manifold *Holonomy * Quaternion-Kähler manifold * Calibrated geometry * Hypercomplex manifold *Hyperkähler manifold *Uniform polyhedron In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, and is known for his work on the differential geometry of smooth fiber bundles, notably the introduction of the concepts of Ehresmann connection and of jet bundles, and for his seminar on category theory. Life Ehresmann was born in Strasbourg (at the time part of the German Empire) to an Alsatian-speaking family; his father was a gardener. After World War I, Alsace returned part of France and Ehresmann was taught in French at Lycée Kléber. Between 1924 and 1927 he studied at the École Normale Supérieure (ENS) in Paris and obtained agrégation in mathematics. After one year of military service, in 1928-29 he taught at a French school in Rabat, Morocco. He studied further at the University of Göttingen during the years 1930–31, and at Princeton University in 1932–34. He co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space. On any manifold of positive dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K3 Surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface :x^4+y^4+z^4+w^4=0 in complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
