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Spinh Structure
In spin geometry, a spinʰ structure (or quaternionic spin structure) is a special classifying map that can exist for orientable manifolds. Such manifolds are called spinʰ manifolds. H stands for the quaternions, which are denoted \mathbb and appear in the definition of the underlying spinʰ group. Definition Let M be a n-dimensional orientable manifold. Its tangent bundle TM is described by a classifying map M\rightarrow\operatorname(n) into the classifying space \operatorname(n) of the special orthogonal group \operatorname(n). It can factor over the map \operatorname^\mathrm(n)\rightarrow\operatorname(n) induced by the canonical projection \operatorname^\mathrm(n)\twoheadrightarrow\operatorname(n) on classifying spaces. In this case, the classifying map lifts to a continuous map M\rightarrow\operatorname^\mathrm(n) into the classifying space \operatorname^\mathrm(n) of the spinʰ group \operatorname^\mathrm(n), which is called ''spinʰ structure''. Let \operatorname^\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Geometry
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics. An important generalisation is the theory of symplectic Dirac operators in symplectic spin geometry and symplectic topology, which have become important fields of mathematical research. See also * Contact geometry * Symplectic topology * Spinor * Spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\co ... * Spin manifold Books * * Differential topology Differential geometry {{physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Projective Plane
In mathematics, the complex projective plane, usually denoted or is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2,Z_3)\neq (0,0,0) where, however, the triples differing by an overall rescaling are identified: :(Z_1,Z_2,Z_3) \equiv (\lambda Z_1,\lambda Z_2, \lambda Z_3); \quad \lambda \in \C, \qquad \lambda \neq 0. That is, these are homogeneous coordinates in the traditional sense of projective geometry. Topology The Betti numbers of the complex projective plane are :1, 0, 1, 0, 1, 0, 0, ..... The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane. The nontrivial homotopy groups of the complex projective plane are \pi_2=\pi_5=\mathbb. The fundamental group is trivial and all other higher homotopy groups are those of the 5-sphere, i.e. torsion. Algebraic geometry In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Geometry And Physics
The ''Journal of Geometry and Physics'' is a scientific journal in mathematical physics. Its scope is to stimulate the interaction between geometry and physics by publishing primary research and review articles which are of common interest to practitioners in both fields. The journal is published by Elsevier since 1984. The Journal covers the following areas of research: ''Methods of:'' * Algebraic and Differential Topology * Algebraic Geometry * Real and Complex Differential Geometry * Riemannian and Finsler Manifolds * Symplectic Geometry * Global Analysis, Analysis on Manifolds * Geometric Theory of Differential Equations * Geometric Control Theory * Lie Groups and Lie Algebras * Supermanifolds and Supergroups * Discrete Geometry * Spinors and Twistors ''Applications to:'' * Strings and Superstrings * Noncommutative Topology and Geometry * Quantum Groups * Geometric Methods in Statistics and Probability * Geometry Approaches to Thermodynamics * Classical and Quantum Dynamical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christian Bär
Christian Bär (born 17 September 1962 in Kaiserslautern) is a German mathematician, whose research concerns differential geometry and mathematical physics. Bär enrolled on Ph.D. studies at the University of Bonn as a student of Hans Werner Ballmann, and obtained his Ph.D. in 1990. He was elected president of the German Mathematical Society, having assumed the post in 2011. Selected papers * * References External linksAuthor profilein the database zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastru ... * * {{DEFAULTSORT:Bar, Christian 20th-century German mathematicians University of Bonn alumni 1962 births Living people Differential geometers German mathematical physicists People from Kaiserslautern 21st-century German mathematicians Presidents of the Germ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots. Connected sum at a point A connected sum of two ''m''-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 Genus
Â, â ( a- circumflex) is a letter of the Inari Sami, Skolt Sami, Romanian, Vietnamese and Mizo alphabets. This letter also appears in French, Friulian, Frisian, Portuguese, Turkish, Walloon, and Welsh languages as a variant of the letter " a". It is included in some romanization systems for Khmer, Persian, Balinese, Sasak, Russian, and Ukrainian. Berber languages "â" can be used in Berber Latin alphabet to represent . Emilian-Romagnol  is used to represent ːin Emilian dialects, as in Bolognese ''câna'' aːna"cane". Faroese , who translated the Gospel of Matthew into Faroese in 1823, used â to denote a non-syllabic a, as in the following example:  is not used in modern Faroese, however. French , in the French language, is used as the letter with a circumflex accent. It is a remnant of Old French, where the vowel was followed, with some exceptions, by the consonant . For example, the modern form ''bâton'' () comes from the Old French ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Class
In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundle E over M, its k-th Pontryagin class p_k(E) is defined as :p_k(E) = p_k(E, \Z) = (-1)^k c_(E\otimes \Complex) \in H^(M, \Z), where: *c_(E\otimes \Complex) denotes the 2k-th Chern class of the complexification E\otimes \Complex = E\oplus iE of E, *H^(M, \Z) is the 4k-cohomology group of M with integer coefficients. The rational Pontryagin class p_k(E, \Q) is defined to be the image of p_k(E) in H^(M, \Q), the 4k-cohomology group of M with rational coefficients. Properties The total Pontryagin class :p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\Z), is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e., :2p(E\oplus F)=2p(E)\smile p(F) for two vector bundles E and F over M. In terms of the individual Pon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betti Number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The ''n''th Betti number represents the rank of the ''n''th homology group, denoted ''H''''n'', which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if H_n(X) \cong 0 then b_n(X) = 0, if H_n(X) \cong \mathbb then b_n(X) = 1, if H_n(X) \cong \mathbb \oplus \mathbb then b_n(X) = 2, if H_n(X) \cong \mathbb \oplus \mathbb\oplus \mathbb then b_n(X) = 3, etc. Note that only the ranks of infinite groups are considered, so for example if H_n(X) \cong \mathbb^k \oplus \mathbb/(2), where \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected Space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Manifold
In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only Connected space, connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP''n'' is a closed ''n''-dimensional manifold. The complex projective space CP''n'' is a closed 2''n''-dimensional manifold. A Real line, line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Properties Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. If M is a closed connected n-manifold, the n-th homology group H_(M;\mathbb) is \mathbb or 0 depending on whether M is Orientability, orientable or not. Moreover, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
