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Pseudo Riemannian manifold





Encyclopedia results for Pseudo Riemannian manifold

  1. Pseudo-Riemannian manifold

    In differential geometry , a pseudo Riemannian manifold ref citation last1 Benn first1 I.M. last2 Tucker ... A pseudo Riemannian manifold math , M,g math is a differentiable manifold math ,M math equipped ... can be generalized to the pseudo Riemannian case. In particular, the fundamental theorem of Riemannian geometry is true of pseudo Riemannian manifolds as well. This allows one to speak of the Levi Civita connection on a pseudo Riemannian manifold along with the associated curvature tensor . On the other ... case. For example, it is not true that every smooth manifold admits a pseudo Riemannian metric of a given ... not always inherit the structure of a pseudo Riemannian manifold for example, the metric tensor become ... year 1968 page 208 ref also called a semi Riemannian manifold is a generalization of a Riemannian manifold . It is one of many mathematical objects named after Bernhard Riemann . The key difference between a Riemannian manifold and a pseudo Riemannian manifold is that on a pseudo Riemannian manifold ... is called a pseudo Riemannian metric and its values can be positive, negative or zero. The signature of a pseudo Riemannian metric is var p var , var q var where both var p var and var q var are non negative. Lorentzian manifold A Lorentzian manifold is an important special case of a pseudo Riemannian manifold in which the signature of the metric is 1, var n var 1 or sometimes var n var 1 ... the most important subclass of pseudo Riemannian manifolds. They are important because of their physical ... Causal structure . Properties of pseudo Riemannian manifolds Just as Euclidean space math mathbb R n math can be thought of as the model Riemannian manifold , Minkowski space math mathbb R n 1,1 math with the flat Minkowski metric is the model Lorentzian manifold. Likewise, the model space for a pseudo Riemannian manifold of signature var p var , var q var is math mathbb R p,q math sup with the metric .... Vr nceanu & R. Ro ca 1976 Introduction to Relativity and Pseudo Riemannian Geometry , Bucarest Editura ...   more details



  1. Riemannian manifold

    Finsler manifold sub Riemannian manifold pseudo Riemannian manifold Metric tensor Hermitian manifold ... and the differential geometry of surfaces , a Riemannian manifold or Riemannian space M , g is a real ... geometric notions on a Riemannian manifold, such as angle s, lengths of curve s, area s or volume ... one could define Riemannian manifold as a metric space which is Isometry isometric to a smooth submanifold ... as metric spaces Usually a Riemannian manifold is defined as a smooth manifold with a smooth Section ... curve in the Riemannian manifold M , then we define its length L in analogy with the example ... space connected Riemannian manifold M becomes a metric space and even a intrinsic metric length ... . Riemannian metrics Let M be a differentiable manifold of dimension n . A Riemannian metric on M ... d x i otimes mathrm d x j. math Endowed with this metric, the differentiable manifold M , g is a Riemannian ... Euclidean metric . Let M , g be a Riemannian manifold and N M be a submanifold of M . Then the restriction .... Let M , g sup M sup be a Riemannian manifold, h M sup n k sup &rarr N sup k sup be a differentiable ... metric If f M &rarr N is a differentiable map and N , g sup N sup a Riemannian manifold, then the pullback ... submanifold inherits a metric from being embedded in a Riemannian manifold, and every covering space inherits a metric from covering a Riemannian manifold. Existence of a metric Every paracompact differentiable manifold admits a Riemannian metric. To prove this result, let M be a manifold and U sub .... Riemannian manifolds as metric spaces A connected space connected Riemannian manifold carries ... . Specifically, let M , g be a connected Riemannian manifold. Let c a,b M be a parametrized curve in M ... The diameter of a Riemannian manifold M is defined by math mathrm diam M sup p,q in M d p,q in mathbf ... completeness A Riemannian manifold M is geodesically complete if for all p M , the Exponential map Riemannian ... to an open proper submanifold of any other Riemannian manifold. The converse is not true ...   more details



  1. Sub-Riemannian manifold

    In mathematics , a sub Riemannian manifold is a certain type of generalization of a Riemannian manifold . Roughly speaking, to measure distances in a sub Riemannian manifold, you are allowed to go only along curves tangent to so called horizontal subspaces . Sub Riemannian manifolds and so, a fortiori , Riemannian manifolds carry a natural intrinsic metric called the metric of Carnot Carath odory . The Hausdorff ... dimension unless it is actually a Riemannian manifold . Sub Riemannian manifolds often occur in the study ... math A,B,C,D, dots math are horizontal. A sub Riemannian manifold is a triple math M, H, g math , where math M math is a differentiable manifold , math H math is a completely non integrable horizontal ... . Any sub Riemannian manifold carries the natural intrinsic metric , called the metric of Carnot ... phase may be understood in the language of sub Riemannian geometry. The Heisenberg group , important to quantum mechanics , carries a natural sub Riemannian structure. Definitions By a distribution ... the orientation of the car. Therefore, the position of car can be described by a point in a manifold ... from one position to another? This defines a Carnot Carath odory metric on the manifold math mathbb R 2 times S 1. math A closely related example of a sub Riemannian metric can be constructed on a Heisenberg ... . Then choosing any smooth positive quadratic form on math H math gives a sub Riemannian metric on the group. Properties For every sub Riemannian manifold, there exists a Hamiltonian mechanics Hamiltonian , called the sub Riemannian Hamiltonian , constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub Riemannian manifold. The existence of geodesics of the corresponding Hamilton Jacobi equation s for the sub Riemannian Hamiltonian are given by the Chow ... last Risler editor2 first Jean Jacques title Sub Riemannian geometry url http books.google.com books ... first Andr editor2 last Risler. editor2 first Jean Jacques title Sub Riemannian geometry url http ...   more details



  1. Cut locus (Riemannian manifold)

    In Riemannian geometry , the cut locus of a point math p math in a manifold is roughly the set of all other points for which there are multiple minimizing geodesic geodesics connecting them from math p math , but it may contain additional points where the minimizing geodesic is unique, under certain circumstances. The distance function from p is a Smooth function smooth function except at the point p itself and the cut locus. Definition Fix a point math p math in a complete space complete Riemannian manifold math M,g math , and consider the tangent space math T pM math . It is a standard result that for sufficiently small math v math in math T p M math , the curve defined by the exponential map Riemannian geometry Riemannian exponential map , math gamma t exp p tv math for math t math belonging to the interval math 0,1 math is a geodesic minimizing geodesic , and is the unique minimizing geodesic connecting the two endpoints. Here math exp p math denotes the exponential map from math p math . The cut locus of math p math in the tangent space is defined to be the set of all vectors math v math in math T pM math such that math gamma t exp p tv math is a minimizing geodesic for math t in 0,1 math but fails to be minimizing for math t in 0,1 epsilon math for each math epsilon 0 math . The cut ... of math p math in math M math as the points in the manifold where the geodesics starting at math p ... to the manifold, and this is the largest such radius. The global injectivity radius is defined to be the infimum of the injectivity radius at p , over all points of the manifold. Characterization ... theorems in Riemannian geometry. Cut locus of a subset One can similarly define the cut locus of a submanifold of the Riemannian manifold, in terms of its normal exponential map. Notes reflist 2 See also Caustic mathematics References Petersen, Peter. Riemannian Geometry , 1st ed. Springer Verlag, 1998. Category Riemannian geometry de Schnittort ru ...   more details



  1. Riemannian

    Riemannian most often refers to Bernhard Riemann Riemannian geometry Riemannian manifold Pseudo Riemannian manifold Sub Riemannian manifold Riemannian submanifold Riemannian metric Riemannian circle Riemannian submersion Riemannian Penrose inequality Riemannian holonomy Riemann curvature tensor Riemannian connection Riemannian connection on a surface Riemannian symmetric space Riemannian volume form Riemannian bundle metric List of topics named after Bernhard Riemann but may also refer to Hugo Riemann Neo Riemannian theory music disambiguation ...   more details



  1. Riemannian geometry

    complicated structure of pseudo Riemannian manifold s, which in four dimensions are the main ... concepts Riemannian geometry is the branch of differential geometry that studies Riemannian manifold s, manifold smooth manifolds with a Riemannian metric , i.e. with an inner product on the tangent ... geometry , as well as Euclidean geometry itself. Any smooth manifold admits a Riemannian metric , which ... The following articles provide some useful introductory material Metric tensor Riemannian manifold ... curvature on a compact 2 dimensional Riemannian manifold is equal to math 2 pi chi M math where ... compact even dimensional Riemannian manifold, see generalized Gauss Bonnet theorem . Nash embedding ... geometry . They state that every Riemannian manifold can be isometrically embedding embedded in a Euclidean ... Sphere theorem . If M is a simply connected compact n dimensional Riemannian manifold with sectional ... epsilon n 0 math such that if an n dimensional Riemannian manifold has a metric with sectional curvature ... compact complete non negatively curved n dimensional Riemannian manifold, then M contains a compact ... Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional ... point. It implies that any two points of a simply connected complete Riemannian manifold with nonpositive ... manifold with negative sectional curvature is ergodic . If M is a complete Riemannian manifold ... × Z . Ricci curvature bounded below Myers theorem . If a compact Riemannian manifold has positive ... Riemannian manifold has nonnegative Ricci curvature and a straight line i.e. a geodesic which ... n 1 dimensional Riemannian manifold which has nonnegative Ricci curvature. Bishop Gromov inequality . The volume of a metric ball of radius r in a complete n dimensional Riemannian manifold ... isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete group discrete . Any smooth manifold of dimension math n geq 3 math admits a Riemannian metric with negative ...   more details



  1. Riemannian submanifold

    Notability Notability date January 2009 A Riemannian submanifold N of a Riemannian manifold M is a submanifold of M equipped with the Riemannian metric inherited from M . The image of an isometric immersion is a Riemannian submanifold. ref cite book last Chen first Bang Yen title Geometry of Submanifolds year 1973 publisher Mercel Dekker location New York isbn 0 8247 6075 1 pages 298 ref References Reflist Category Riemannian manifolds differential geometry stub ...   more details



  1. Geodesic manifold

    In mathematics , a complete manifold or geodesically complete manifold is a Pseudo Riemannian manifold pseudo Riemannian manifold for which every maximal inextendible geodesic is defined on math mathbb R math . Examples All compact space compact manifolds and all homogeneous space homogeneous manifolds are geodesically complete. Euclidean space math mathbb R n math , the sphere s math mathbb S n math and the torus tori math mathbb T n math with their usual Riemannian metric s are all complete manifolds. A simple example of a non complete manifold is given by the punctured plane math M mathbb R 2 setminus 0 math with its usual metric . Geodesics going to the origin cannot be defined on the entire real line. Path connectedness, completeness and geodesic completeness It can be shown that a finite dimensional Connected space Path connectedness path connected Riemannian manifold is a complete metric space if and only if it is geodesically complete. This is the Hopf Rinow theorem . This theorem does not hold for infinite dimensional manifolds. The example of a non complete manifold the punctured plane given above fails to be geodesically complete because, although it is path connected, it is not a complete metric space any sequence in the plane converging to the origin is a non converging Cauchy sequence in the punctured plane. References Citation last1 O Neill first1 Barrett title Semi Riemannian Geometry publisher Academic Press isbn 0 12 526740 1 year 1983 . See chapter 3, pp. 68 . DEFAULTSORT Complete Manifold Category Riemannian geometry Category Manifolds ...   more details



  1. Manifold

    two points, the arc length between the points hence it is a Riemannian manifold . History details History ... correspond to Riemannian manifold s with constant negative and positive curvature , respectively ... formalized as a manifold. Riemannian manifold s and Riemann surface s are named after Riemann ... Riemannian manifolds To measure distances and angles on manifolds, the manifold must be Riemannian. A Riemannian manifold is a differentiable manifold in which each tangent space is equipped with an Inner ... manifold. The Euclidean space itself carries a natural structure of Riemannian manifold the tangent ... to define an inner product. Any Riemannian manifold is a Finsler manifold. Lie groups Main Lie group ... of Riemannian manifolds curvature of a Riemannian manifold and the torsion differential ...Other uses Manifold disambiguation File Triangles spherical geometry .jpg thumb 300px The sphere surface of a ball mathematics ball is a two dimensional manifold since it can be represented by a collection ... manifold is a subset of Euclidean space which is locally the graph of a smooth perhaps vector valued function. A more general topological manifold can be described as a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold ... on into high dimensional space . More formally, every point of an n dimensional manifold has a neighborhood ... of a manifold may be more complicated. For example, any point on the usual two dimensional surface ... in the language of topology , they are not homeomorphic. The structure of a manifold is encoded ... in terms of the relatively well understood properties of simpler spaces. For example, a manifold is typically endowed with a differentiable structure that allows one to do calculus and a Riemannian metric that allows one to measure distance s and angle s. Symplectic manifold s serve as the phase ... dimensional Lorentzian manifold s model space time in general relativity . This seems out of place ...   more details



  1. Riemannian circle

    Image Sphere halve.png thumb right A great circle divides the sphere in two equal Sphere hemisphere s In metric space theory and Riemannian geometry , the Riemannian circle named after Bernhard Riemann is a great circle equipped with its great circle distance . In more detail, the term refers to the circle equipped with its intrinsic Riemannian metric of a compact 1 dimensional manifold of total length 2 , as opposed to the extrinsic metric obtained by restriction of the Euclidean metric to the unit circle in the Plane geometry plane . Thus, the distance between a pair of points is defined to be the length of the shorter of the two arcs into which the circle is partitioned by the two points. Properties The diameter of the Riemannian circle is , in contrast with the usual value of 2 for the Euclidean diameter of the unit circle. The inclusion of the Riemannian circle as the equator or any great circle of the 2 sphere of constant Gaussian curvature 1, is an isometric imbedding in the sense of metric spaces there is no isometric imbedding of the Riemannian circle in Hilbert space in this sense . Gromov s filling conjecture A long standing open problem, posed by Mikhail Gromov mathematician Mikhail Gromov , concerns the calculation of the filling area conjecture filling area of the Riemannian circle. The filling area is conjectured to be 2 , a value attained by the hemisphere of constant Gaussian curvature 1. References Gromov, M. Filling Riemannian manifolds , Journal of Differential Geometry 18 1983 , 1&ndash 147. Category Riemannian geometry Category Circles Category Metric geometry ...   more details



  1. Riemannian submersion

    In differential geometry , a branch of mathematics , a Riemannian submersion is a Submersion mathematics submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Let M , g and N , h be two Riemannian manifolds and math f M to N math a submersion. Then f is a Riemannian submersion if and only if the isomorphism math df mathrm ker df perp rightarrow TN math is an isometry . Examples An example of a Riemannian submersion arises when a Lie group math G math acts isometrically, free action freely and proper action properly on a Riemannian manifold math M,g math . The projection math pi M rightarrow N math to the quotient space math N M G math equipped with the quotient metric is a Riemannian submersion. For example, component wise multiplication on math S 3 subset C 2 math by the group of unit complex numbers yields the Hopf fibration . Properties The sectional curvature of the target space of a Riemannian submersion can be calculated from the curvature of the total space by O Neill s formula math K N X,Y K M tilde X, tilde Y tfrac34 tilde X, tilde Y top 2 math where math X,Y math are orthonormal vector fields on math N math , math tilde X, tilde Y math their horizontal lifts to math M math , math , math is the Lie brackets and math Z top math is the projection of the vector field math Z math to the vertical distribution . In particular the lower bound for the sectional curvature of math N math is at least as big as the lower bound for the sectional curvature of math M math . Generalizations and variations Fiber bundle Submetry co Lipschitz map References citation title Spinors, Spectral Geometry, and Riemannian Submersions first1 Peter B. last1 Gilkey first2 John V. last2 Leahy first3 Jeonghyeong last3 Park url http www.emis.de monographs GLP year 1998 publisher Global Analysis Research Center, Seoul National University . Category Riemannian geometry Category Maps of manifolds ru ...   more details



  1. Pseudo-

    wiktionary pseudo For the novel with the original title Pseudo Hocus Bogus The prefix pseudo from Greek lying, false is used to mark something as falsity false , fraud ulent, or pretending to be something it is not. Biology prefix In biology and botany taxonomy the prefix pseudo or pseud can indicate a species with a visual similarity to another genus . An example is the Iris species Iris pseudacorus , by having leaves similar to those of Acorus calamus in the Acorus genus, having pseud acorus false acorus in its botanical name . See also lookfrom pseudo Falsehood Pseudorealism Deception Mimicry Pseudo.com Pseudo Blood of Our Own Category Prefixes Category Greek loanwords Category Biology prefixes and suffixes da Pseudo nl Pseudo sk Pseudo ...   more details



  1. Riemann manifold

    Riemann surface in complex analysis Riemannian manifold in Riemannian geometry Zariski Riemann space consisting of valuations mathdab ...   more details



  1. Simplicial manifold

    Expert subject Mathematics date February 2009 In mathematics , the term simplicial manifold commonly refers to either of two different types of objects, which combine attributes of a simplex with those of a manifold . Briefly a simplex is a generalization of the concept of a triangle into forms with more, or fewer, than two dimensions. Accordingly, a 3 simplex is the figure known as a tetrahedron . A manifold is simply a space which appears to be Euclidean space Euclidean following the laws of ordinary geometry, or more generally a flat Pseudo Riemannian manifold Pseudo Riemannian space in a given Neighborhood mathematics local neighborhood , though it can be greatly more complicated overall. The combination of these concepts gives us two useful definitions. A manifold made out of simplices A simplicial manifold is a simplicial complex for which the geometric realization is homeomorphic to a topological manifold . This can mean simply that a neighborhood mathematics neighborhood of each vertex i.e. the set of simplices that contain that point as a vertex is homeomorphic to a n dimensional ball mathematics ball . A manifold made from simplices can be locally flat, or can approximate a smooth curve, just as a large geodesic dome appears relatively flat over small areas, and approximates a Sphere hemisphere over its full extent. One can generalize this concept to more dimensions and other kinds of curved surfaces which makes it useful in various kinds of Computer simulation simulations . This notion of simplicial manifold is important in Regge calculus and Causal dynamical triangulation s as a way to discretize spacetime by triangulation triangulating it. A simplicial manifold ... manifold is also a simplicial object in the category mathematics category of manifold s. This is a special case of a simplicial space in which, for each n , the space of n simplices is a manifold. For example, if G is a Lie group , then the nerve category theory simplicial nerve of G has the manifold ...   more details



  1. Conformally flat manifold

    A pseudo Riemannian manifold pseudo Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation . More formally, let M , g be a pseudo Riemannian manifold. Then M , g is conformally flat if for each point x in M , there exists a neighborhood U of x and a smooth function f defined on U such that U , e sup 2 f sup g is flat manifold flat i.e. the Riemann curvature tensor curvature of e sup 2 f sup g vanishes on U . The function f need not be defined on all of M . Some authors use locally conformally flat to describe the above notion and reserve conformally flat for the case in which the function f is defined on all of M . Examples Every manifold with constant curvature constant sectional curvature is conformally flat. Every 2 dimensional pseudo Riemannian manifold is conformally flat. A 3 dimensional pseudo Riemannian manifold is conformally flat if and only the Cotton tensor vanishes. An n dimensional pseudo Riemannian manifold for n &ge 4 is conformally flat if and only if the Weyl tensor vanishes. Every compact space compact , simply connected , conformally flat Riemannian manifold is conformally equivalent to the n sphere round sphere . See also Weyl Schouten theorem conformal geometry Category Conformal geometry Category Riemannian geometry Category Manifolds differential geometry stub ...   more details



  1. Isometry (Riemannian geometry)

    In the study of Riemannian geometry in mathematics , a local isometry from one Pseudo Riemannian manifold pseudo Riemannian manifold to another is a map which pullback differential geometry pulls back the metric tensor on the second manifold to the metric tensor on the first. When such a map is also a diffeomorphism , such a map is called an isometry or isometric isomorphism , and provides a notion of isomorphism sameness in the category theory category Rm of Riemannian manifolds. Definition Let math M, g math and math M , g math be two Riemannian manifolds, and let math f M to M math be a diffeomorphism. Then math f math is called an isometry or isometric isomorphism if math g f g , , math where math f g math denotes the pullback differential geometry pullback of the rank 0, 2 metric tensor math g math by math f math . Equivalently, in terms of the push forward math f math , we have that for any two vector fields math v, w math on math M math i.e. sections of the tangent bundle math mathrm T M math , math g v, w g left f v, f w right . , math If math f math is a local diffeomorphism such that math g f g math , then math f math is called a local isometry . See also Myers Steenrod theorem References cite book author Lee, Jeffrey M. title Differential Geometry, Analysis and Physics year 2000 Category Riemannian geometry it Isometria geometria riemanniana ...   more details



  1. Riemannian theory

    orphan date February 2012 Image Minor as upside down major.png thumb right 350px Illustration of Riemann s dualist system minor as upside down major. Riemannian theory refers to the music theory musical theories of Hugo Riemann 1849 1919 . Riemann s dualist system for relating triad music triads was adapted from earlier 19th century harmony harmonic theorists. The term dualism refers to the emphasis on the inversional relationship between major and minor , with minor triad s being considered upside down versions of major triad s this dualism is what produces the change in direction described above. See also Utonality In the 1880 s, Riemann proposed a system of transformations that related triads directly to each other ref name Klumpenhouwer Klumpenhouwer, Henry, Some Remarks on the Use of Riemann Transformations, Music Theory Online 0.9 1994 ref See also Neo Riemannian theory Diatonic function Functional harmony Klang music Tonnetz Schenkerian analysis Sources reflist music theory stub Category Diatonic functions Category Music theory ...   more details



  1. Einstein manifold

    In differential geometry and mathematical physics , an Einstein manifold is a Riemannian manifold Riemannian or pseudo Riemannian manifold whose Ricci tensor is proportional to the metric tensor metric . They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein equations with cosmological constant , although the dimension, as well as the signature, of the metric can be arbitrary, unlike the four dimensional Lorentzian manifold s usually studied in general relativity . If M is the underlying n dimensional manifold and g is its metric tensor the Einstein condition means that math mathrm Ric k ,g, math for some constant k , where Ric denotes the Ricci tensor of g . Einstein manifolds with k 0 are called Ricci flat manifold s. The Einstein condition and Einstein s equation In local coordinates the condition that M ,  g be an Einstein manifold is simply math R ab k ,g ab . math Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by math R nk , math where n is the dimension of M . In general relativity , Einstein s equation with a cosmological constant &Lambda is math R ab frac 1 2 g ab R g ab Lambda 8 pi T ab , math written in geometrized units with G c 1. The stress energy tensor T sub ab sub gives the matter and energy ... Riemannian manifolds Category Albert Einstein Manifold Category Mathematical physics de Einsteinsche ... include Any manifold with constant sectional curvature is an Einstein manifold&mdash in particular ... Study metric . Calabi Yau manifold s admit a unique Einstein metric that is also K hler metric K hler . Among closed manifold closed , oriented , 4 manifold s, only those that satisfy the Hitchin&ndash Thorpe inequality can be Einstein manifolds. Applications Four dimensional Riemannian ... as 4 dimensional hyperk hler manifold s in the Ricci flat case, and quaternion K hler manifold s otherwise ...   more details



  1. Fundamental theorem of Riemannian geometry

    of the Fundamental theorem of Riemannian geometry proceeds by showing that a torsion free metric connection on a Riemannian manifold is necessarily given by the Koszul formula math begin matrix 2 g nabla ...In Riemannian geometry , the fundamental theorem of Riemannian geometry states that on any Riemannian manifold or pseudo Riemannian manifold there is a unique torsion differential geometry torsion free metric affine connection connection , called the Levi Civita connection of the given metric. Here a metric or Riemannian connection is a connection which preserves the metric tensor . More precisely blockquote Let math M,g math be a Riemannian manifold or pseudo Riemannian manifold then there is a unique connection math nabla math which satisfies the following conditions for any vector fields math X,Y,Z math we have math partial X langle Y,Z rangle langle nabla X Y,Z rangle langle Y, nabla X Z rangle math , where math partial X langle Y,Z rangle math denotes the derivative of the function math langle Y,Z rangle math along vector field math X math . for any vector fields math X,Y math , math nabla XY nabla YX X,Y math , br where math X,Y math denotes the Lie bracket s for vector field s math X,Y math . blockquote The first condition means that the metric tensor is preserved by parallel transport , while the second condition expresses the fact that the torsion differential geometry torsion of math nabla math is zero. An extension of the fundamental theorem states that given a pseudo Riemannian manifold there is a unique connection preserving the metric tensor with any given vector valued 2 form as its torsion. The following technical proof presents a formula for Covariant derivative ... hand, compatibility with the Riemannian metric implies that math partial k g ij langle nabla partial ... DEFAULTSORT Fundamental Theorem Of Riemannian Geometry Category Connection mathematics Category Theorems in Riemannian geometry Category Articles containing proofs Category Fundamental theorems Riemannian ...   more details



  1. Hadamard manifold

    In mathematics , a Hadamard manifold , named after Jacques Hadamard &mdash sometimes called a Cartan Hadamard manifold , after lie Cartan &mdash is a Riemannian manifold M ,  g that is complete space complete and simply connected space simply connected , and has everywhere non positive sectional curvature . Examples The real line R with its usual metric is a Hadamard manifold with constant sectional curvature equal to 0. Standard n dimensional hyperbolic space H sup n sup is a Hadamard manifold with constant sectional curvature equal to &minus 1. See also Cartan Hadamard theorem Hadamard space References cite arxiv last Mourougane first Christophe title Interpolation in non positively curved K hler manifolds date 7 Mar 2001 eprint math 0103045 class math.CV Category Riemannian manifolds topology stub fr Vari t de Hadamard ...   more details



  1. Finsler manifold

    are smooth outside the origin. Riemannian manifold s but not pseudo Riemannian manifold s are special cases of Finsler manifolds. Randers manifolds Let M , a be a Riemannian manifold and b a differential ... 0 to T TM setminus 0 quad quad v tfrac 1 2 big I mathcal L H J big . math In analogy with the Riemannian manifold Riemannian case, there is a version math D dot gamma D dot gamma X t R dot gamma ... , as in the Riemannian manifold Riemannian case. Notes reflist References Citation editor1 last Antonelli ...In mathematics , particularly differential geometry , a Finsler manifold is a differentiable manifold together with the structure of an intrinsic equation intrinsic quasimetric space Quasimetrics quasimetric space in which the length of any rectifiable curve nowrap a , b M is given by the functional length functional math L gamma int a b F gamma t , dot gamma t ,dt, math where F x ,    is a Minkowski norm or at least an asymmetric norm on each tangent space T sub x sub M . Finsler manifolds non trivially generalize Riemannian manifold s in the sense that they are not necessarily infinitesimally Euclidean space Euclidean . This means that the asymmetric norm on each tangent space is not necessarily induced by an inner product metric tensor . harvs txt authorlink lie Cartan last Cartan ... dissertation harv Finsler 1918 . Definition A Finsler manifold is a differentiable manifold M together ... , F is a Randers manifold , a special case of a non reversible Finsler manifold. Kropina manifolds F 3 M , M , F is a Kropina manifold after V. K. Kropina 1959 . Smooth quasimetric spaces Let M , d be a quasimetric so that M is also a differentiable manifold and d is compatible with the differential ... . A constant speed curve is a geodesic of a Finsler manifold if its short enough segments ... &rarr M with fixed endpoints a x and b y . Canonical spray structure on a Finsler manifold The Euler ... pages 582 586 http www.ams.org notices 199609 chern.pdf S. Chern Finsler geometry is just Riemannian ...   more details



  1. Kähler manifold

    manifold. Every Riemannian metric on a Riemann surface is K hler, since the condition for to be closed ... l tude des vari t s k hl riennes 1958 DEFAULTSORT Kahler manifold Category Riemannian ...In mathematics , a K hler manifold is a manifold with unitary group unitary structure a G structure U n structure satisfying an integrability condition . In particular, it is a Riemannian manifold , a complex manifold , and a symplectic manifold , with these three structures all mutually compatible. This threefold ... Hermitian manifold . If the Sp structure is integrable but the complex structure need not be , the notion is an almost K hler manifold if the complex structure is integrable but the Sp structure need not be , the notion is a Hermitian manifold . K hler manifolds are named after the mathematician ... geometric generalization of complex Algebraic variety algebraic varieties . Definition A manifold with a Hermitian metric is an almost Hermitian manifold a K hler manifold is a manifold with a Hermitian ... defined as a complex manifold with an additional structure or a symplectic manifold with an additional structure, or a Riemannian manifold with an additional structure . One can summarize the connection between the three structures via math h g i omega math , where h is the Hermitian form, g is the Riemannian metric , i is the almost complex manifold almost complex structure , and math omega math is the almost symplectic manifold almost symplectic structure . A K hler metric on a complex manifold ... manifold. The metric on a K hler manifold locally satisfies math g i bar j frac partial 2 K partial z i partial bar z j math for some function K , called the K hler potential. A K hler manifold, the associated ... manifold s for more details. Examples Complex Euclidean space C sup n sup with the standard Hermitian metric is a K hler manifold. A torus C sup n sup a full lattice group lattice inherits ... submanifold of a K hler manifold is K hler. In particular, any Stein manifold embedded in C sup ...   more details



  1. Differentiable manifold

    Barden. Structures on manifolds Pseudo Riemannian manifolds A Riemannian manifold is a differentiable ... manifold can be given a pseudo Riemannian structure there are topological restrictions on doing so. A Finsler manifold is a generalization of a Riemannian manifold, in which the inner product is replaced ... a rank 4 Riemann curvature tensor . On a Riemannian manifold one has notions of length, volume, and angle. Any differentiable manifold can be given a Riemannian structure. A pseudo Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an metric ... of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable. A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus . Any manifold can be described ... differentiable chart. Note that a differentiable manifold as it stands does not have any metric structure or any notion of orthogonality. The addition of metric or pseudo metric structure corresponds to the linear space mentioned above actually being Euclidean space or pseudo Euclidean space . In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure . Any topological manifold can be given a differential structure locally by using the homeomorphism .... He motivated the idea of a manifold by an intuitive process of varying a given object in a new ... principle . A modern definition of a 2 dimensional manifold was given by Hermann Weyl in his 1913 book on Riemann surface s. ref See H. Weyl 1955 . ref The widely accepted general definition of a manifold ... Definition A presentation of a topological manifold is a second countable Hausdorff space Hausdorff ... of patching together pieces of a space to make a manifold the manifold produced also contains the data of how it has been patched together. However, different atlases patchings may produce the same manifold ...   more details



  1. Hermitian manifold

    In mathematics , a Hermitian manifold is the complex analog of a Riemannian manifold . Specifically, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian form Hermitian inner product on each holomorphic tangent space . One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure . Complex structure is essentially an almost ... manifold M can therefore be specified by either a Hermitian metric h as above, a Riemannian metric ... manifold admits a Hermitian metric. This follows directly from the analogous statement for Riemannian metric. Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct .... Every almost Hermitian manifold M has a canonical volume form which is just the Riemannian volume ... G structure U n structure on the manifold. By dropping this condition we get an almost Hermitian manifold . On any almost Hermitian manifold we can introduce a fundamental 2 form , or cosymplectic ... bundle E over a smooth manifold M is a smoothly varying definite bilinear form positive definite ... p sub and math h p zeta, bar zeta 0 math for all nonzero in E sub p sub . A Hermitian manifold is a complex manifold with a Hermitian metric on its holomorphic tangent space . Likewise, an almost Hermitian manifold is an almost complex manifold with a Hermitian metric on its holomorphic tangent space. On a Hermitian manifold the metric can be written in local holomorphic coordinates z sup sub ... of a positive definite Hermitian matrix . Riemannian metric and associated form A Hermitian metric h on an almost complex manifold M defines a Riemannian metric g on the underlying smooth manifold. The metric g is defined to be the real part of h math g 1 over 2 h bar h . math The form ... uniquely determine the other two. The Riemannian metric g and associated 1,1 form are related by the almost ... u,v 1 over 2 left g u,v g Ju,Jv right . math Choosing a Hermitian metric on an almost complex manifold ...   more details



  1. Sasakian manifold

    index.php Sasakian manifold EoM page, Sasakian manifold Category Riemannian geometry ...In differential geometry , a Sasakian manifold named after Shigeo Sasaki is a contact manifold math M, theta math equipped with a special kind of Riemannian metric math g math , called a Sasakian metric. Definition A Sasakian metric is defined using the construction of the Riemannian cone . Given a Riemannian manifold math M,g math , its Riemannian cone is a product math M times Bbb R 0 , math of math M math with a half line math Bbb R 0 math , equipped with the cone metric math t 2 g dt 2, , math where math t math is the parameter in math Bbb R 0 math . A manifold math M math equipped with a 1 form math theta math is contact if and only if the 2 form math t 2 ,d theta 2t , dt cdot theta , math on its cone is symplectic this is one of the possible definitions of a contact structure . A contact Riemannian manifold is Sasakian, if its Riemannian cone with the cone metric is a K hler manifold with K hler form math t 2 ,d theta 2t ,dt cdot theta. , math Examples As an example consider math S 2n 1 hookrightarrow Bbb R 2n, Bbb C n, math where the right hand side is a natural K hler manifold ... endowed with contact form math theta frac12 dz sum i y i ,dx i math and the Riemannian metric math .... The Reeb vector field The homothetic vector field on the cone over a Sasakian manifold is defined ... structure J . The Reeb vector field on the Sasaskian manifold is defined to be math xi J t partial partial ... and in particular with all isometry isometries of the Sasakian manifold. If the orbits of the vector ... manifold at unit radius is a unit vector field and tangential to the embedding. Sasaki Einstein manifolds A Sasakian manifold math M math is one with the Riemannian cone K hler. If the cone is, in addition, Ricci flat , math M math is called Sasaki Einstein if it is hyperk hler manifold hyperk hler , math M math is called 3 Sasakian . Any 3 Sasakian manifold is an Einstein manifold and a spin manifold ...   more details




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