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Tractor Bundle
In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an Group action (mathematics), effective group representation, representation of the conformal group (see associated bundle). The term ''tractor'' is a portmanteau of "Tracy Thomas" and "twistor", the bundle having been introduced first by T. Y. Thomas as an alternative formulation of the Cartan connection, Cartan conformal connection, and later rediscovered within the formalism of local twistors and generalized to projective connections by Michael Eastwood ''et al.'' in Bailey, T. N.; Eastwood, M. G.; Gover, A. R., "Thomas's structure bundle for conformal, projective and related structures", ''Rocky Mountain J.'' 24 (1994), 1191–1217. References [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Geometry
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry. Conformal manifolds A conformal manifold is a pseudo-Riemannian manifold equipped with an equivalence class of metric tensors, in which two metrics ''g'' and ''h'' are equivalent if and only if :h = \lambda^2 g , where ''λ'' is a real-valued smooth function defined on the manifold and is called the conformal factor. An equivalence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Manifold
In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry. Conformal manifolds A conformal manifold is a pseudo-Riemannian manifold equipped with an equivalence class of metric tensors, in which two metrics ''g'' and ''h'' are equivalent if and only if :h = \lambda^2 g , where ''λ'' is a real-valued smooth function defined on the manifold and is called the conformal factor. An equivalence class o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system. The term ''representation of a group'' is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Group
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space. Several specific conformal groups are particularly important: * The conformal orthogonal group. If ''V'' is a vector space with a quadratic form ''Q'', then the conformal orthogonal group is the group of linear transformations ''T'' of ''V'' for which there exists a scalar ''λ'' such that for all ''x'' in ''V'' *:Q(Tx) = \lambda^2 Q(x) :For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations. * The conformal group of the sphere is generated by the inversions in circles. This group is also known as the Möbius group. * In Euclidean space E''n'', , the conformal group is generated by inversions in hyperspheres. * In a pseudo-Euclidean space E''p'',''q'', the conformal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Bundle
In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G. For a fiber bundle ''F'' with structure group ''G'', the transition functions of the fiber (i.e., the cocycle) in an overlap of two coordinate systems ''U''α and ''U''β are given as a ''G''-valued function ''g''αβ on ''U''α∩''U''β. One may then construct a fiber bundle ''F''′ as a new fiber bundle having the same transition functions, but possibly a different fiber. An example A simple case comes with the Möbius strip, for which G is the cyclic group of order 2, \mathbb_2. We can take as F any of: the real number line \mathbb, the interval 1,\ 1/math>, the real number line less the point 0, or the two-point set \. The action of G on these (the non-identity element acting as x\ \rightarrow\ -x in each cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Portmanteau
A portmanteau word, or portmanteau (, ) is a blend of wordsGarner's Modern American Usage , p. 644. in which parts of multiple words are combined into a new word, as in ''smog'', coined by blending ''smoke'' and ''fog'', or ''motel'', from ''motor'' and ''hotel''. In , a portmanteau is a single morph that is analyzed as representing two (or more) underlying s. When portmanteaus shorte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (''repère mobile''). The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. In relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Connection
In conformal differential geometry, a conformal connection is a Cartan connection on an ''n''-dimensional manifold ''M'' arising as a deformation of the Klein geometry given by the celestial ''n''-sphere, viewed as the homogeneous space :O+(n+1,1)/''P'' where ''P'' is the stabilizer of a fixed null line through the origin in R''n''+2, in the orthochronous Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch phy ... O+(n+1,1) in ''n''+2 dimensions. Normal Cartan connection Any manifold equipped with a conformal structure has a canonical conformal connection called the normal Cartan connection. Formal definition A conformal connection on an ''n''-manifold ''M'' is a Cartan geometry modelled on the conformal sphere, where the latter is viewed as a homogeneous space for O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Twistor
Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administration * Local news, coverage of events in a local context which would not normally be of interest to those of other localities * Local union, a locally based trade union organization which forms part of a larger union Arts, entertainment, and media * Local (comics), ''Local'' (comics), a limited series comic book by Brian Wood and Ryan Kelly * Local (novel), ''Local'' (novel), a 2001 novel by Jaideep Varma * Local TV LLC, an American television broadcasting company * Locast, a non-profit streaming service offering local, over-the-air television * The Local (film), ''The Local'' (film), a 2008 action-drama film * ''The Local'', English-language news websites in several European countries Computing * .local, a network address component * L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Connection
In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold. The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection. Much like affine connections, projective connections also define geodesics. However, these geodesics are not affinely parametrized. Rather they are projectively parametrized, meaning that their preferred class of parameterizations is acted upon by the group of fractional linear transformations. Like an affine connection, projective connections have associated torsion and curvature. Projective space as the model geometry The first step in defining any Cartan connection is to consider the flat case: in which the connection corresponds to the Maurer-Cartan form on a homogeneous space. In the projective setting, the underlying manifold M of the homogeneous space is the projective space RPn which we shall represent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
