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List Of Things Named After Élie Cartan
These are things named after Élie Cartan (9 April 1869 – 6 May 1951), a French mathematician. Mathematics and physics * Cartan calculus * Cartan connection, Cartan connection applications * Cartan's criterion * Cartan decomposition * Cartan's equivalence method * Cartan formalism (physics) * Cartan involution * Cartan's magic formula * Cartan relations ** Cartan map * Cartan matrix * Cartan pair * Cartan subalgebra * Cartan subgroup * Cartan's method of moving frames * Cartan's theorem, a name for the closed-subgroup theorem * Cartan's theorem, a name for the theorem on highest weights * Cartan's theorem, a name for Lie's third theorem * Einstein–Cartan theory ** Einstein–Cartan–Evans theory * Cartan–Ambrose–Hicks theorem * Cartan–Brauer–Hua theorem * Cartan–Dieudonné theorem * Cartan–Hadamard manifold * Cartan–Hadamard theorem * Cartan–Iwahori decomposition * Cartan-Iwasawa-Malcev theorem * Cartan–Kähler theorem * Cartan–Karlhede algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He also made significant contributions to general relativity and indirectly to quantum mechanics. He is widely regarded as one of the greatest mathematicians of the twentieth century. His son Henri Cartan was an influential mathematician working in algebraic topology. Life Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan (1837–1917) and Anne Cottaz (1841–1927). Joseph Cartan was the village blacksmith; Élie Cartan recalled that his childhood had passed under "blows of the anvil, which started every morning from dawn", and that "his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning-wheel". Élie had an elder sister J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed-subgroup Theorem
In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if is a closed subgroup of a Lie group , then is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding. One of several results known as Cartan's theorem, it was first published in 1930 by Élie Cartan, who was inspired by John von Neumann's 1929 proof of a special case for groups of linear transformations.; . Overview Let G be a Lie group with Lie algebra \mathfrak. Now let H be an arbitrary closed subgroup of G. It is necessary to show that H is a smooth embedded submanifold of G. The first step is to identify something that could be the Lie algebra of H, that is, the tangent space of H at the identity. The challenge is that H is not assumed to have any smoothness and therefore it is not clear how one may define its tangent space. To proceed, define the "Lie algebra" \mathfrak of H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Compact Subgroup
In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the classification of Lie groups and especially semi-simple Lie groups. Maximal compact subgroups of Lie groups are ''not'' in general unique, but are unique up to conjugation – they are essentially unique. Example An example would be the subgroup O(2), the orthogonal group, inside the general linear group GL(2, R). A related example is the circle group SO(2) inside SL(2, R). Evidently SO(2) inside GL(2, R) is compact and not maximal. The non-uniqueness of these examples can be seen as any inner product has an associated orthogonal group, and the essential uniqueness corresponds to the essential uniqueness of the inner product. Definition A maximal compact subgroup is a maximal subgroup amongst compact subgroups – a ''maximal (compact subgrou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan–Hadamard Theorem
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898. Élie Cartan generalized the theorem to Riemannian manifolds in 1928 (; ; ). The theorem was further generalized to a wide class of metric spaces by Mikhail Gromov in 1987; detailed proofs were published by for metric spaces of non-positive curvature and by for general locally convex metric spaces. Riemannian geometry The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected complete Riemannian manifold of non-positive sectional curvature is diffeomorphic to R''n''. In fact, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Manifold
In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold (M, g) that is complete and simply connected and has everywhere non-positive sectional curvature. By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space \R^n. Furthermore it follows from the Hopf–Rinow theorem that every pairs of points in a Cartan–Hadamard manifold may be connected by a unique geodesic segment. Thus Cartan–Hadamard manifolds are some of the closest relatives of \R^n. Examples The Euclidean space \R^n with its usual metric is a Cartan-Hadamard manifold with constant sectional curvature equal to 0. Standard n-dimensional hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan–Dieudonné Theorem
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an ''n''- dimensional symmetric bilinear space can be described as the composition of at most ''n'' reflections. The notion of a symmetric bilinear space is a generalization of Euclidean space whose structure is defined by a symmetric bilinear form (which need not be positive definite, so is not necessarily an inner product – for instance, a pseudo-Euclidean space is also a symmetric bilinear space). The orthogonal transformations in the space are those automorphisms which preserve the value of the bilinear form between every pair of vectors; in Euclidean space, this corresponds to preserving distances and angles. These orthogonal transformations form a group under composition, called the orthogonal group. For example, in the two-dimensional Euclidean plane, every orthogonal transformation is either a reflection across a line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan–Brauer–Hua Theorem
In abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings. It says that given two division rings such that ''xKx''−1 is contained in ''K'' for every ''x'' not equal to 0 in ''D'', either ''K'' is contained in the center of ''D'', or . In other words, if the unit group of ''K'' is a normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ... of the unit group of ''D'', then either or ''K'' is central . References * * Theorems in ring theory {{Abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan–Ambrose–Hicks Theorem
In mathematics, the Cartan–Ambrose–Hicks theorem is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words, behavior of the curvature tensor under parallel translation determines the metric. The theorem is named after Élie Cartan, Warren Ambrose, and his PhD student Noel Hicks.0, the exponential maps : \exp_x:B_r(x)\subset T_xM\rightarrow M, \exp_y:B_r(y)\subset T_yN\rightarrow N are local diffeomorphisms. Here, B_r(x) is the ball centered on x of radius r. One then defines a diffeomorphism f:B_r(x)\rightarrow B_r(y) by : f=\exp_y\circ I\circ \exp_x^. When is f an isometry? Intuitively, it should be an isometry if it satisfies the two conditions: * It is a linear isometry at the tangent space of every point on B_r(x), that is, it is an isometry on the infinitesimal patches. * It preserves the curvature tensor at the tangent space of every point on B_r(x), that is, it preserves how ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein–Cartan–Evans Theory
Einstein–Cartan–Evans theory or ECE theory was an attempted unified theory of physics proposed by the Welsh chemist and physicist: "ECE Theory was discovered by chemist, physicist, and mathematician, Myron Wyn Evans...". Myron Wyn Evans (May 26, 1950 – May 2, 2019), which claimed to unify general relativity, quantum mechanics and electromagnetism. The hypothesis was largely published in the journal ''Foundations of Physics Letters'' between 2003 and 2005. Several of Evans's central claims were later shown to be mathematically incorrect and, in 2008, the new editor of ''Foundations of Physics'', Nobel laureate Gerard 't Hooft, published an editorial note effectively retracting the journal's support for the hypothesis. Scope Earlier versions of the theory were called " O(3) electrodynamics". Evans claims that he is able to derive a generally covariant field equation for electromagnetism and gravity, similar to that derived by Mendel Sachs. Evans argues that Einstein's theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein–Cartan Theory
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstein–Cartan theory is the simplest Poincaré gauge theory. Overview Einstein–Cartan theory differs from general relativity in two ways: (1) it is formulated within the framework of Riemann–Cartan geometry, which possesses a locally gauged Lorentz symmetry, while general relativity is formulated within the framework of Riemannian geometry, which does not; (2) an additional set of equations are posed that relate torsion to spin. This difference can be factored into by first reformulating general relativity onto a Riemann–Cartan geometry, replacing the Einstein–Hilbert action over Riemannian geometry by the Palatini action over Riemann–Cartan geometry; and second, removing the zero torsion constraint from the Palatini action, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie's Third Theorem
In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra \mathfrak over the real numbers is associated to a Lie group ''G''. The theorem is part of the Lie group–Lie algebra correspondence. Historically, the third theorem referred to a different but related result. The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a group action on a smooth manifold. The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a ''local'' Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem. Historical notes The equivalence between the category of simply connected real Lie groups and finite-dimensional real Lie algebras is usually called (in the literature of the second half ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
