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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] |
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] |
G2 Manifold
In differential geometry, a ''G''2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in ''G''2. The group G_2 is one of the five exceptional simple Lie groups. It can be described as the automorphism group of the octonions, or equivalently, as a proper subgroup of special orthogonal group SO(7) that preserves a spinor in the eight-dimensional spinor representation or lastly as the subgroup of the general linear group GL(7) which preserves the non-degenerate 3-form \phi, the associative form. The Hodge dual, \psi=*\phi is then a parallel 4-form, the coassociative form. These forms are calibrations in the sense of Reese Harvey and H. Blaine Lawson, and thus define special classes of 3- and 4-dimensional submanifolds. Properties All G_2-manifold are 7-dimensional, Ricci-flat, orientable spin manifolds. In addition, any compact manifold with holonomy equal to G_2 has finite fundamental group, non-zero first Pontryagin class, and non-zero third ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relativity Theorists
Relativity may refer to: Physics * Galilean relativity, Galileo's conception of relativity * Numerical relativity, a subfield of computational physics that aims to establish numerical solutions to Einstein's field equations in general relativity * Principle of relativity, used in Einstein's theories and derived from Galileo's principle * Theory of relativity, a general treatment that refers to both special relativity and general relativity ** General relativity, Albert Einstein's theory of gravitation ** Special relativity, a theory formulated by Albert Einstein, Henri Poincaré, and Hendrik Lorentz ** '' Relativity: The Special and the General Theory'', a 1920 book by Albert Einstein Social sciences * Linguistic relativity * Cultural relativity * Moral relativity Arts and entertainment Music * Relativity Music Group, a Universal subsidiary record label for releasing film soundtracks * Relativity Records, an American record label * Relativity (band), a Scots-Irish tradit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometers
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 structur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century French Mathematicians
The 1st century was the century spanning AD 1 ( I) through AD 100 ( C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or historical period. The 1st century also saw the appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius ( AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and instability, which was finally brought to an end by Vespasian, ninth Roman emper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra: *Infinite classes: ** prisms, **antiprisms. * Convex exceptional: ** 5 Platonic solids: regular convex polyhedra, ** 13 Archimedean solids: 2 quasiregular and 11 semiregular convex polyhedra. * Star (nonconvex) exceptional: ** 4 Kepler–Poinsot polyhedra: regular nonconvex polyhedra, ** 53 uniform star polyhedra: 14 quasiregular and 39 semiregular. Hence 5 + 13 + 4 + 53 = 75. There are also many degen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperkähler Manifold
In differential geometry, a hyperkähler manifold is a Riemannian manifold (M, g) endowed with three integrable almost complex structures I, J, K that are Kähler with respect to the Riemannian metric g and satisfy the quaternionic relations I^2=J^2=K^2=IJK=-1. In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio Calabi in 1979. Equivalent definition in terms of holonomy Equivalently, a hyperkähler manifold is a Riemannian manifold (M, g) of dimension 4n whose holonomy group is contained in the compact symplectic group . Indeed, if (M, g, I, J, K) is a hyperkähler manifold, then the tangent space is a quaternionic vector space for each point of , i.e. it is isomorphic to \mathbb^n for some integer n, where \mathbb is the algebra of quaternions. The compact symplectic group can be considered as the group of orthogonal transformations of \mathbb^n whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Calibrated Geometry
In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (''M'',''g'') of dimension ''n'' equipped with a differential ''p''-form ''φ'' (for some 0 ≤ ''p'' ≤ ''n'') which is a calibration, meaning that: * ''φ'' is closed: d''φ'' = 0, where d is the exterior derivative * for any ''x'' ∈ ''M'' and any oriented ''p''-dimensional subspace ''ξ'' of T''x''''M'', ''φ'', ''ξ'' = ''λ'' vol''ξ'' with ''λ'' ≤ 1. Here vol''ξ'' is the volume form of ''ξ'' with respect to ''g''. Set ''G''''x''(''φ'') = . (In order for the theory to be nontrivial, we need ''G''''x''(''φ'') to be nonempty.) Let ''G''(''φ'') be the union of ''G''''x''(''φ'') for ''x'' in ''M''. The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier (in 1966) Edmond Bonan introduced G2-manifolds and Spin(7)-manifolds, constructed all the parallel forms and showed that those manifolds w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion-Kähler Manifold
In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(''n'')·Sp(1) for some n\geq 2. Here Sp(''n'') is the sub-group of SO(4n) consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic n \times n matrix, while the group Sp(1) = S^3 of unit-length quaternions instead acts on quaternionic n-space ^n = ^ by right scalar multiplication. The Lie group Sp(n)\cdot Sp(1) \subset SO(4n) generated by combining these actions is then abstractly isomorphic to p(n) \times Sp(1) _2. Although the above loose version of the definition includes hyperkähler manifolds, the standard convention of excluding these will be followed by also requiring that the scalar curvature be non-zero— as is automatically true if the holonomy group equals the entire group Sp(''n'')·Sp(1). Early history Marcel Berger's 1955 paper on the classifica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
