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Kulkarni–Nomizu Product
In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two -tensors and gives as a result a -tensor. Definition If ''h'' and ''k'' are symmetric -tensors, then the product is defined via: :\begin (h k)(X_1, X_2, X_3, X_4) := &h(X_1, X_3)k(X_2, X_4) + h(X_2, X_4)k(X_1, X_3) \\ &- h(X_1, X_4)k(X_2, X_3) - h(X_2, X_3)k(X_1, X_4) \\ pt = &\begin h(X_1, X_3) &h (X_1, X_4)\\ k(X_2, X_3) &k (X_2, X_4) \end + \begin k(X_1, X_3) &k (X_1, X_4)\\ h(X_2, X_3) &h (X_2, X_4) \end \end where the ''X''''j'' are tangent vectors and , \cdot, is the matrix determinant. Note that h k = k h, as it is clear from the second expression. With respect to a basis \ of the tangent space, it takes the compact form :(h~\wedge\!\!\!\!\!\!\!\!\;\bigcirc~k)_ = (hk )(\partial_i, \partial_j, \partial_l,\partial_m) = 2h_k_ + 2 ... [...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 st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Decomposition
In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry. Definition of the decomposition Let (''M'',''g'') be a Riemannian or pseudo-Riemannian ''n''-manifold. Consider its Riemann curvature, as a (0,4)-tensor field. This article will follow the sign convention :R_=g_\Big(\partial_i\Gamma_^p-\partial_j\Gamma_^p+\Gamma_^p\Gamma_^q-\Gamma_^p\Gamma_^q\Big); written multilinearly, this is the convention :\operatorname(W,X,Y,Z)=g\Big(\nabla_W\nabla_XY-\nabla_X\nabla_WY-\nabla_Y,Z\Big). With this convention, the Ricci tensor is a (0,2)-tensor field defined by ''Rjk''=''gilRijkl'' and the scalar curvature is defined by ''R''=''gjkRjk.'' Define the traceless Ricci tensor :Z_=R_-\fracRg_, and then define three (0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g''''p'' on the tangent space ''T''''p''''M'' at each point ''p''. The family ''g''''p'' of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take ''g'' to be smooth, which means that for any smooth coordinate chart on ''M'', the ''n''2 functions :g\left(\frac,\frac\right):U\to\mathbb are smooth functions. These functions are commonly designated as g_. With further restrictions on the g_, one could also consider Lipschitz Riemannian metrics or measurable Riemannian metrics, among many other possibilities. A Riemannian metric (tensor) makes it possible to define several geometric notions on a Riemannian manifold, such as angle at an intersection, lengt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Of Riemannian Manifolds
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry. For a more elementary discussion see the article on curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as the differential geometry of surfaces. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications. Ways to express the curvature of a Riemannian manifold The Riemann curvature tensor The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection (or covariant differentiat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Tensor
In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how the volume of the body changes, but rather only how the shape of the body is distorted by the tidal force. The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor. This tensor has the same symmetries as the Riemann tensor, but satisfies the extra condition that it is trace-free: metric contraction on any pair of indices yields zero. It is obtained from the Riemann tensor by subtracting a tensor that is a linear express ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schouten Tensor
In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for by: :P=\frac \left(\mathrm -\frac g\right)\, \Leftrightarrow \mathrm=(n-2) P + J g \, , where Ric is the Ricci tensor (defined by contracting the first and third indices of the Riemann tensor), ''R'' is the scalar curvature, ''g'' is the Riemannian metric, J=\fracR is the trace of ''P'' and ''n'' is the dimension of the manifold. The Weyl tensor equals the Riemann curvature tensor minus the Kulkarni–Nomizu product of the Schouten tensor with the metric. In an index notation :R_=W_+g_ P_-g_ P_-g_ P_+g_ P_\, . The Schouten tensor often appears in 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 di ... because of its relatively simple conformal tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor. The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ravindra Shripad Kulkarni
Ravindra Shripad Kulkarni (born 1942) is an Indian mathematician, specializing in differential geometry. He is known for the Kulkarni–Nomizu product. Education and career Ravi S. Kulkarni received in 1968 his Ph.D. from Harvard University under Shlomo Sternberg with thesis ''Curvature and Metric''. For the academic year 1980–1981 he was a Guggenheim Fellow. He has served as the president of the Ramanujan Mathematical Society. Selected publications * * * * * * *with Allan L. Edmonds & John H. Ewing: *with Allan L. Edmonds & Robert E. Stong: *with Gregory Constantine: *with Hyman Bass Hyman Bass (; born October 5, 1932). The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the c ... References External links *Directory page at University of MichiganAuthor profilein the database zbMATH {{DEFAUL ...: * *with Krishnendu Gongopadhyay: as editor * with Ulrich Pinkall: References External linksConformal Geometry and Riemann Surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
