In differential geometry , the term curvaturetensor may refer to the Riemann curvaturetensor of a Riemannian manifold &mdash see also Curvature of Riemannian manifolds the curvature of an affine connection or covariant derivative on tensors the curvature form of an Ehresmann connection see Ehresmann connection , connection principal bundle or connection vector bundle . See also Tensor disambiguation mathdab zh ... more details
General relativity In the mathematical field of differential geometry , the Riemann curvaturetensor , or Riemann Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel , is the most standard way to express curvature of Riemannian manifolds . It associates a tensor to each point of a Riemannian manifold i.e., it is a tensor field , that measures the extent to which the metric tensor is not locally isometric to a Euclidean space. The curvaturetensor can also be defined for any pseudo ... mathematical tool in the theory of general relativity , the modern theory of gravity , and the curvature of spacetime is in principle observable via the geodesic deviation equation . The curvaturetensor ... precise by the Jacobi field Jacobi equation . The curvaturetensor is given in terms of the Levi Civita ... and v , and so defines a tensor. Occasionally, the curvaturetensor is defined with the opposite ... v nabla u w . math The curvaturetensor measures noncommutativity of the covariant derivative , and as such is the integrability ... curvaturetensor. Coordinate expression In local coordinates math x mu math the Riemann curvature ... curvaturetensor has the following symmetries math R u,v R v,u math math langle R u,v w,z rangle ... tensor . These three identities form a complete list of symmetries of the curvaturetensor ... with such a curvaturetensor at some point. Simple calculations show that such a tensor has math n .... The Riemann tensor has only one functionally independent component. The Gaussian curvature coincides ... manifold, while the Ricci curvaturetensor of the surface is simply given by math operatorname Ric ... Decomposition of the Riemann curvaturetensor References citation first A.L. last Besse title ... DEFAULTSORT Riemann CurvatureTensor Category Tensors in general relativity Category Curvature ... R u,v w math is also called the curvature transformation or endomorphism . Geometrical meaning When ... original position. However, this property does not hold in the general case. The Riemann curvature ... more details
Riemann curvaturetensor . The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space. See the links below for further reading. Curvature of plane curves Cauchy defined the center of curvature C as the intersection point of two infinitesimal infinitely close normals to the curve, the radius of curvature as the distance from the point to C , and the curvature itself as the inverse of the radius of curvature. ref citation last1 Borovik first1 .... The Gauss curvature is thus the determinant of the shape tensor and the mean curvature is half its ...In mathematics , curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat ... on the context. There is a key distinction between extrinsic curvature , which is defined for objects embedded in another space usually a Euclidean space in a way that relates to the radius of curvature of circles that touch the object, and Curvature of Riemannian manifolds intrinsic curvature , which ... concept. The canonical example of extrinsic curvature is that of a circle , which everywhere has curvature ..., and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its ... or more dimensions it is described by a curvature vector that takes into account the direction of the bend as well as its sharpness. The curvature of more complex objects such as surface s or even ... C be a plane curve the precise technical assumptions are given below . The curvature of C at a point ... right 250px One way is geometrical. It is natural to define the curvature of a straight line to be identically zero. The curvature of a circle of radius R should be large if R is small and small if R is large. Thus the curvature of a circle is defined to be the reciprocal of the radius math kappa ... more details
in the form of the Riemann curvaturetensor . ref name Kline cite book title Mathematical thought ... tensor s, and the Riemann curvaturetensor . The exterior algebra of Hermann Grassmann ... scalar curvature Euclidean vector vector bivector , e.g. inverse metric tensor m 1 covector , linear ... tensor m 3 e.g. 3 form e.g. Riemann curvaturetensor ... m N e.g. N form i.e. determinant , volume ... Application of tensor theory in engineering Covariant derivative Curvature Diffusion MRI Mathematical ...other uses Dablink Note that in common usage, the term tensor is also used to refer to a tensor field . File Components stress tensor.svg right thumb 300px Stress, a second order tensor. The tensor s components ... and scalars themselves are also tensors. A tensor can be represented as a Array data structure ... of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number ... by a matrix, a 2 dimensional array, and therefore is a 2nd order tensor. A vector can be represented as a 1 dimensional array and is a 1st order tensor. Scalars are single numbers and are thus zeroth ... vectors . For example, the Stress mechanics Cauchy s stress theorem stress tensor stress tensor ... and applying the tensor to it results in an organized multidimensional array representing the tensor in that basis, or as it looks from that frame of reference. The coordinate independence of a tensor ... to be built in to the notion of a tensor in a geometrical or physical setting, and the precise form of the transformation law determines the type or valence of the tensor. Tensors are important ... University Press year 1972 ref History The concepts of later tensor analysis arose from the work ... 9783764328146 url http books.google.com books?id O6lixBzbc0gC ref The word tensor itself was introduced ... last Wilkins issue 7 9 ref to describe something different from what is now meant by a tensor ... in elementarer Darstellung publisher Von Veit year 1898 ref Tensor calculus was developed around ... more details
Unreferenced date December 2009 Seealso space form curvature of Riemannian manifolds sectional curvature In mathematics , constant curvature is a concept from differential geometry . Here, curvature refers to the sectional curvature of a space more precisely a manifold and is a single number determining its local geometry. The sectional curvature is said to be constant if it has the same value at all points. For example, a sphere is a surface of constant positive curvature. classification The geometries of constant curvature can be classified into the following three cases elliptic elliptic geometry constant positive sectional curvature Euclidean Euclidean geometry constant vanishing sectional curvature hyperbolic hyperbolic geometry constant negative sectional curvature properties every space of constant curvature is locally symmetric , i.e. its curvaturetensor is parallel math nabla mathrm R 0 math every space of constant curvature is locally maximally symmetric , i.e. it has math frac 1 2 n n 1 math number of Killing vector local isometries , where n is its dimension. conversely, there exists a similar but stronger statement every maximally symmetric space, i.e. a space which has math frac 1 2 n n 1 math global isometries , has constant curvature. the universal cover of a manifold of constant sectional curvature is one of the model spaces sphere sectional curvature positive flat manifold plane sectional curvature zero hyperbolic space hyperbolic manifold sectional curvature nagative a space of constant curvature which is geodesically complete is called space form and the study of space forms is intimately related to generalized crystallography see the article on space form for more details . two space forms are isomorphic if and only if they have the same dimension, their metrics ... Constant Curvature Category Differential geometry of surfaces Category Riemannian geometry Category Curvature mathematics nl Constante kromming zh ... more details
A curvature collineation often abbreviated to CC is vector field which preserves the Riemann tensor in the sense that, math mathcal L X R a bcd 0 math where math R a bcd math are the components of the Riemann tensor. The Set mathematics set of all smooth function smooth curvature collineations forms a Lie algebra under the Lie bracket operation if the smoothness condition is dropped, the set of all curvature collineations need not form a Lie algebra . The Lie algebra is denoted by math CC M math and may be infinity infinite dimension al. Every affine vector field is a curvature collineation. See also Conformal vector field Homothetic vector field Killing vector field Matter collineation Spacetime symmetries relativity stub Category Mathematical methods in general relativity ... more details
are the Christoffel symbols of the metric. Unlike the Riemann curvaturetensor or the Ricci tensor , which ... tensor has only one independent component and it can be easily expressed in terms of the scalar curvature ... tensor of an n dimensional Euclidean space vanishes identically, so the scalar curvature ... R to represent three different things the Riemann curvaturetensor math R ijk l math or math R abcd math the Ricci tensor math R ij math the scalar curvature R These three are then distinguished from ... for the full Riemann curvaturetensor. See also Basic introduction to the mathematics of curved spacetime ...Unreferenced date December 2009 In Riemannian geometry , the scalar curvature or Ricci scalar is the simplest curvature invariant of a Riemannian manifold . To each point on a Riemannian manifold, it assigns ..., the scalar curvature represents the amount by which the volume of a geodesic ball in a curved ..., the scalar curvature is twice the Gaussian curvature , and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity. In general relativity , the scalar curvature is the Lagrangian ... metrics are known as Einstein manifold Einstein metrics . The scalar curvature is defined as the trace of the Ricci tensor , and it can be characterized as a multiple of the average of the sectional curvature s at a point. Unlike the Ricci tensor and sectional curvature, however, global results involving only the scalar curvature are extremely subtle and difficult. One of the few is the positive ... , which seeks extremal metrics in a given conformal class for which the scalar curvature is constant. Definition The scalar curvature is usually denoted by S other notations are Sc , R . It is defined as the Trace linear algebra trace of the Ricci curvaturetensor with respect to the metric tensor metric math S mbox tr g , operatorname Ric . math The trace depends on the metric since the Ricci tensor ... more details
In Riemannian geometry , the Schouten tensor is a second order tensor which is introduced by Jan Arnoldus Schouten . It is defined by, for n 3 dimensions, math P frac 1 n 2 left Ric frac R 2 n 1 g right , math where Ric is the Ricci tensor , R is the scalar curvature , g is the Riemannian metric and n is the dimension of the manifold. The Weyl tensor equals the Riemann curvaturetensor minus the Kulkarni&ndash Nomizu product of the Schouten tensor with the metric. See also Weyl Schouten theorem Cotton tensor Category Riemannian geometry Category Tensors differential geometry stub ru ... more details
of gravitation such as general relativity , curvature scalars play an important role in telling distinct spacetimes apart. Two of the most basic curvature invariants in general relativity are the Curvature ... familiar quadratic invariants of the electromagnetic field tensor in classical electromagnetism. An important ... mathematics syzygies for the zero th order invariants of the Riemann tensor. They have limitations ... from Minkowski spacetime using any number of curvature invariants of any order . See also Cartan Karlhede algorithm Carminati McLenaghan invariants Curvature invariant general relativity ..., 2nd Edition, Cambridge Univ. Press 2003 Curvature invariants are studied in Chapter 9 ref references DEFAULTSORT Curvature Invariant Category Riemannian geometry relativity stub differential geometry ... more details
In differential geometry , the Ricci curvaturetensor , named after Gregorio Ricci Curbastro , represents ... . The Riemannian curvaturetensor of math M math is the math 1,3 math tensor defined by math R X,Y ... curvaturetensor and the Christoffel symbols , one has math R alpha beta R rho alpha rho beta ... curvature , since knowing it is equivalent to knowing the Ricci curvaturetensor. The Ricci curvature ... in dimensions 2 and 3 does the Ricci tensor determine the full curvaturetensor. A notable exception ... tensor. The tensor was introduced by Ricci for this reason. If the Ricci curvature function ... constant Ricci curvature, or to be an Einstein manifold . This happens if and only if the Ricci tensor Ric is a constant multiple of the metric tensor g . The Ricci curvature is usefully thought of as a multiple ... Ricci tensor harvnb Galloway 2000 . These results show that positive Ricci curvature has strong ... &fnof sup g does not change the Ricci curvature. Trace free Ricci tensor In Riemannian geometry and general ... tensor, math S math is the scalar curvature , math g math is the metric tensor , and math n math ... . If math nabla math denotes an affine connection, then the curvaturetensor math R math is the math ... geometry Category Tensors in general relativity Category Curvature mathematics ca Tensor de ... from that of ordinary Euclidean n space. The Ricci tensor is defined on any pseudo Riemannian manifold , as a trace mathematics trace of the Riemann curvaturetensor . Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold harv Besse 1987 p 43 . ref ... connection , the Ricci tensor need not be symmetric. ref In relativity theory , the Ricci tensor is the part of the curvature of space time that determines the degree to which matter will tend to converge ... tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison cf. comparison theorem with the geometry of a constant curvature space form . If the Ricci ... more details
. In this case the form is an alternative description of the Riemann curvaturetensorcurvaturetensor , i.e. math ,R X,Y Omega X,Y , math using the standard notation for the Riemannian curvaturetensor, Bianchi identities If math theta math is the canonical vector valued 1 form on the frame bundle ...In differential geometry , the curvature form describes curvature of a connection form connection on a principal bundle . It can be considered as an alternative to or generalization of curvaturetensor in Riemannian geometry . Definition Let G be a Lie group with Lie algebra math mathfrak g math , and P B be a principal bundle principal G bundle . Let be an Ehresmann connection on P which is a math mathfrak g math valued Differential form one form on P . Then the curvature form is the math mathfrak g math valued 2 form on P defined by math Omega d omega 1 over 2 omega, omega D omega. math Here math d math stands for exterior derivative , math cdot, cdot math is defined by math alpha otimes X, beta otimes Y alpha wedge beta otimes X, Y mathfrak g math and D denotes the exterior covariant derivative . In other terms, math , Omega X,Y d omega X,Y omega X , omega Y . math Curvature form in a vector bundle If E B is a vector bundle. then one can also think of as a matrix of 1 forms and the above formula becomes the structure equation math , Omega d omega omega wedge omega, math where math wedge math is the Exterior power wedge product . More precisely, if math omega i j math and math Omega i j math denote components of and correspondingly, so each math omega i j math is a usual 1 form and each math Omega i j math is a usual 2 form then math Omega i j d omega i j sum k omega i k wedge omega k j . math For example, for the tangent bundle of a Riemannian manifold , the structure ... principal bundle Basic introduction to the mathematics of curved spacetime Chern Simons form Curvature of Riemannian manifolds Gauge theory curvature Category Differential geometry Category Curvature ... more details
In differential geometry , the Weyl curvaturetensor , named after Hermann Weyl , is a measure of the curvature of spacetime or, more generally, a Pseudo Riemannian manifold . Like the Riemann curvaturetensor , the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic . The Weyl tensor differs from the Riemann curvaturetensor in that it does not convey information on how ... force. The Ricci curvature , or trace linear algebra trace component of the Riemann tensor contains ... through regions of space devoid of matter. More generally, the Weyl curvature is the only component of curvature for Ricci flat manifold s and always governs the method of characteristics characteristics of the field equations of an Einstein manifold . In dimensions 2 and 3 the Weyl curvaturetensor vanishes identically. In dimensions 4, the Weyl curvature is generally nonzero. If the Weyl tensor ... . Definition The Weyl tensor can be obtained from the full curvaturetensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a 0,4 valence tensor by contracting ... tensor , s is the scalar curvature , and h small O small k denotes the Kulkarni&ndash Nomizu product ... decomposition , expresses the Riemann curvaturetensor into its irreducible representation irreducible ... scalar the scalar curvature and brackets around indices refers to the Antisymmetric tensor antisymmetric ... tensor. Petrov classification Weyl curvature hypothesis Weyl scalar Cotton tensor References Citation ... tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace free Tensor contraction Metric contraction metric contraction on any pair of indices yields zero. In general relativity , the Weyl curvature is the only part of the curvature that exists in free space&mdash a solution of the Einstein ... system in which the metric tensor is proportional to a constant tensor. This fact was a key component ... more details
Expert subject Physics date December 2008 Unreferenced date December 2008 In general relativity , the magnetogravitic tensor is one of the three pieces appearing in the Bel decomposition of the Riemann tensor . The magnetogravitic tensor can be interpreted physically as a specifying possible spin spin force s on spinning bits of matter, such as spinning test particle s. See also Papapetrou Dixon equations Curvature invariant s References reflist Category Tensors in general relativity relativity stub ... more details
the exponential map at p . The sectional curvature is a smooth real valued function on the 2 Grassmannian fiber bundle bundle over the manifold. The sectional curvature determines the Riemann curvaturetensorcurvaturetensor completely. Definition Given a Riemannian manifold and two linearly independent ... over langle u,u rangle langle v,v rangle langle u,v rangle 2 math Here R is the Riemann curvaturetensor ... tensorcurvature of Riemannian manifolds curvaturecurvature Category Riemannian geometry Category ...In Riemannian geometry , the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds . The sectional curvature K &sigma sub p sub depends on a two dimensional plane &sigma sub p sub in the tangent space at p . It is the Gaussian curvature of that section &mdash the surface ... curvature in fact depends only on the 2 plane sub p sub in the tangent space at p spanned by u and v . It is called the sectional curvature of the 2 plane &sigma sub p sub , and is denoted K &sigma sub p sub . Manifolds with constant sectional curvature Riemannian manifold s with constant sectional curvature are the most simple. These are called space form s. By rescaling the metric there are three possible cases negative curvature &minus 1, hyperbolic geometry zero curvature, Euclidean geometry positive curvature 1, elliptic geometry The model manifolds for the three geometries ... complete , simply connected Riemannian manifolds of given sectional curvature. All other complete constant curvature manifolds are quotients of those by some group of isometry isometries . If for each point in a connected Riemannian manifold of dimension three or greater the sectional curvature is independent of the tangent 2 plane, then the sectional curvature is in fact constant on the whole manifold. Toponogov s theorem Toponogov s theorem affords a characterization of sectional curvature in terms ... has non negative curvature, then for all sufficiently small triangles math d z,m 2 ge tfrac12d z,x ... more details
Image Gaussian curvature.PNG thumb From left to right a surface of negative Gaussian curvature hyperboloid , a surface of zero Gaussian curvature cylinder geometry cylinder , and a surface of positive Gaussian curvature sphere . In differential geometry , the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvature s, sub 1 sub and sub 2 sub , of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances ... curvature is defined as math Kappa kappa 1 kappa 2 , math . where math kappa 1 math and math kappa 2 math are the Principal curvature principal curvatures . Alternative definitions It is also given ... where math nabla i nabla mathbf e i math is the covariant derivative and g is the metric tensor . At a point p on a regular surface in R sup 3 sup , the Gaussian curvature is also given by math K mathbf p det S mathbf p , math where S is the shape operator . A useful formula for the Gaussian curvature ... of f vanishes this can always be attained by a suitable rigid motion . Then the Gaussian curvature ... curvature Image Hyperbolic triangle.svg thumb The sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle. The surface integral of the Gaussian curvature over some region of a surface is called the total curvature . The total curvature of a geodesic ... on a surface of positive curvature will exceed math pi math , while the sum of the angles of a triangle on a surface of negative curvature will be less than math pi math . On a surface of zero curvature, such as the Euclidean plane , the angles will sum to precisely math pi math . math ... states that Gaussian curvature of a surface can be determined from the measurements of length on the surface ... of the Gaussian curvature of a surface S in R sup 3 sup certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the inner ... more details
In mathematics , the mean curvature math H math of a surface math S math is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedding ... curvature math K i math given at math p math . Of those curvatures math K i math , at least one ... curvature s of math S math . The mean curvature at math p in S math is then the average of the principal ... the mean curvature is given as math H frac 1 n sum i 1 n kappa i . math More abstractly, the mean curvature is the trace of the second fundamental form divided by n or equivalently, the shape operator . Additionally, the mean curvature math H math may be written in terms of the covariant derivative ... vector, and math g ij math the metric tensor . A surface is a minimal surface if and only if the mean curvature is zero. Furthermore, a surface which evolves under the mean curvature of the surface math S math , is said to obey a heat equation heat type equation called the mean curvature flow equation. The sphere is the only embedded surface of constant positive mean curvature without boundary or singularities ... euclid.pjm 1102702809 ref Surfaces in 3D space For a surface defined in 3D space, the mean curvature ... the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal the curvature is positive if the surface curves away from the normal. The formula above ... z S x, y math , and using downward pointing normal the doubled mean curvature expression is math ... 1 r math comes from the derivative of math z S r S left scriptstyle sqrt x 2 y 2 right math . Mean curvature ... surface. main Minimal surface A minimal surface is a surface which has zero mean curvature at all ... of constant mean curvature . See also Gaussian curvature Mean curvature flow Inverse mean curvature ... . curvature Category Differential geometry Category Differential geometry of surfaces Category Surfaces Category Curvature mathematics de Mittlere Kr mmung es Curvatura media fr Courbure moyenne ko it Curvatura ... more details
In differential geometry and general relativity , the Bach tensor is a tensor of rank 2 which is conformally invariant in dimension n 4. It is the only known conformally invariant tensor that is algebraically independent of the Weyl tensor . ref P. Szekeres, Conformal Tensors. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 304, No. 1476 Apr. 2, 1968 , pp. http www.jstor.org pss 2416002 113 122 ref In abstract index notation abstract indices the Bach tensor is given by math B ab P cd W a c b d nabla c nabla aP bc nabla c nabla cP ab math where math W math is the Weyl tensor , and math P math the Schouten tensor given in terms of the Ricci tensor math R ab math and scalar curvature math R math by math P ab frac 1 n 2 left R ab frac R 2 n 1 g ab right math . References Reflist Category Tensors Category Tensors in general relativity geometry stub relativity stub ru ... more details
Image Minimal surface curvature planes en.svg thumb 300px right Saddle surface with normal planes in directions ... have different curvature s for different normal planes at p . The principal curvatures at p , denoted k sub 1 sub and k sub 2 sub , are the maximum and minimum values of this curvature. Here the curvature of a curve is by definition the multiplicative inverse reciprocal of the radius of the osculating circle . The curvature is taken to be positive if the curve turns in the same direction ... the curvature takes its maximum and minimum values are always perpendicular, a result of Leonhard Euler ... from the spectral theorem because they can be given as the principal axes of a symmetric tensor ... 2 sub of the two principal curvatures is the Gaussian curvature , K , and the average k sub 1 sub k sub 2 sub 2 is the mean curvature , H . If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface . For a minimal surface , the mean curvature is zero at every point. Formal definition Let M be a surface in Euclidean ... eigenvectors principal directions , then the sectional curvature of M at p is given by math K .... The Monkey saddle is one surface with an isolated flat umbilic. Lines of curvature The lines of curvature or curvature lines are curves which are always tangent to a principal direction they are integral curve s for the principal direction fields . There will be two lines of curvature through each ... of curvature form one of three configurations star , lemon and monstar derived from lemon star ref ... of curvature near umbilics widths 150px Image TensorLemon.png Lemon Image TensorMonstar.png Monstar Image TensorStar.png Star gallery In these figures, the red curves are the lines of curvature for one family of principal directions, and the blue curves for the other. When a line of curvature has a local extremum of the same principal curvature then the curve has a ridge differential geometry ... more details
Wikt Radius of curvature may refer to Radius of curvature mathematics Radius of curvature optics Radius of curvature applications , in geodesy and materials science The reciprocal of the curvature , in differential geometry Radius , for a sphere lingo The radius of the osculating circle in differential geometry of curves Minimum railway curve radius disambiguation ... more details
In differential geometry , the Einstein tensor also trace reversed Ricci tensor , named after Albert Einstein , is used to express the curvature of a Riemannian manifold . In general relativity , the Einstein tensor occurs in the Einstein field equations for gravitation describing spacetime curvature in a manner consistent with energy considerations. Definition The Einstein tensor math mathbf G math is a rank 2 tensor defined over Riemannian manifold s. In index free notation it is defined as math mathbf G mathbf R frac 1 2 mathbf g R, math where math mathbf R math is the Ricci tensor , math mathbf g math is the metric tensor and math R math is the scalar curvature . In component form, the previous equation reads as math G mu nu R mu nu 1 over2 g mu nu R. math The Einstein tensor is symmetric math G mu nu G nu mu , math and, like the stress energy tensor , divergenceless math G mu nu nu 0 ,. math Explicit form The Ricci tensor depends only on the metric tensor , so the Einstein tensor can be defined directly with just the metric tensor. However, this expression is complex and rarely quoted in textbooks. The complexity of this expression can be shown using the formula for the Ricci tensor ... align math where math delta alpha beta math is the Kronecker tensor and the Christoffel symbol math ... derivatives of the metric tensor vanish and the component form of the Einstein tensor is considerably ... denote Antisymmetric tensor antisymmetrization over bracketed indices, i.e. math g alpha beta, gamma ... linear algebra trace of the Einstein tensor can be computed by Tensor contraction contract ing the equation in the Definition definition with the metric tensor math g mu nu math . In math n math dimensions ... 3 space, 1 time gives math G , math , the trace of the Einstein tensor, as the negative of math R , math , the Ricci tensor s trace. Thus another name for the Einstein tensor is the trace reversed Ricci tensor . Use in general relativity The Einstein tensor allows the Einstein field equations without ... more details
tensor fields , that is, fields defined over a manifold which define a tensor at every point of the manifold. An example is the Riemann curvaturetensor . Geometric introduction Intuitively, a vector ... tensor torsion and Affine connection curvature tensors built from them are. Notation The notation .... Applications The curvaturetensor is discussed in differential geometry and the stress ...Unreferenced date August 2008 In mathematics , physics and engineering , a tensor field assigns a tensor to each point of a mathematical space typically a Euclidean space or manifold . Tensor fields are used ... physics stress and strain tensor strain in materials, and in numerous applications in the physical sciences and engineering. As a tensor is a generalization of a scalar physics scalar a pure number representing a value, like length and a Euclidean vector vector a geometrical arrow in space , a tensor ... velocity at each point of the Earth s surface. The general idea of tensor field combines the requirement ... tensor &mdash with the idea that we don t want our notion to depend on the particular method ... explanation The contemporary mathematical expression of the idea of tensor field breaks it down into a two ... the tensor product concept is independent of any choice of basis, taking the tensor product of two ... of tensor field , namely as a section fiber bundle section of some tensor bundle . There are vector bundles which are not tensor bundles the M bius band for instance. This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space math V otimes cdots otimes V otimes ... . See also tangent bundle and cotangent bundle . Given two tensor bundles E M and F M , a map A E F from the space of sections of E to sections of F can be considered itself as a tensor section ..., where f is a smooth function on M . Thus a tensor is not only a linear map on the vector space ... more details
math is the tensor math S alpha beta gamma delta lambda mu partial beta omega delta lambda beta partial mu omega g beta mu partial lambda omega math The Riemann curvaturetensor transforms as math widetilde ...In differential geometry , the Cotton tensor on a pseudo Riemannian manifold of dimension n is a third order tensor field tensor concomitant of the metric tensor metric , like the Weyl curvature Weyl tensor . The concept is named after mile Cotton . Just as the vanishing of the Weyl tensor for n ... flat , the same is true for the Cotton tensor for n 3, while for n 3 it is identically zero. In coordinates, and denoting the Ricci tensor by R sub ij sub and the scalar curvature by R , the components of the Cotton tensor are math C ijk nabla k R ij nabla j R ik frac 1 2 n 1 left nabla j Rg ik nabla k Rg ij right . math The Cotton tensor can be regarded as a vector valued ... order trace free tensor density math C i j nabla k left R li frac 1 4 Rg li right epsilon klj , math sometimes called the Cotton York tensor . The proof of the classical result that for n 3 the vanishing of the Cotton tensor is equivalent the metric being conformally flat is given .... This tensor density is uniquely characterized by its conformal properties coupled with the demand ... manifolds, we obtain the Ricci tensor by contracting the transformed Riemann tensor to see ... math Combining all these facts together permits us to conclude the Cotton York tensor transforms as math ... W, math where the gradient is plugged into the symmetric part of the Weyl tensor W . Symmetries The Cotton tensor has the following symmetries math C ijk C ikj , math and therefore math C ijk 0. , math In addition the Bianchi formula for the Weyl tensor for can be rewritten as math delta W ... reflist Cite journal first S. J. last Aldersley title Comments on certain divergence free tensor ... origyear 1925 year 1977 isbn 0691080267 A. Garcia, F.W. Hehl, C. Heinicke, A. Macias, The Cotton tensor ... more details
In Riemannian geometry , the geodesic curvature math k g math of a curve lying on a submanifold of the ambient space measures how far the curve is from being a geodesic . For instance it applies to Curvature Curves on surfaces curves on surfaces . The notion of geodesic curvature allows to distinguish the part of the curvature in ambient space that is due to the submanifold the normal curvature math k n math and the one that comes from the curve itself. The curvature math k math of the curve is related to these two by math k sqrt k g 2 k n 2 math . In particular geodesics have zero geodesic curvature they are straight , and that is their definition, so that math k k n math , which explains why they appear to be curved in ambient space whenever the submanifold is. Definition Consider a curve ... math s math , with unit tangent vector math T math . The geodesic curvature is the norm of the projection ... curvature is the norm of the projection of math dT ds math on the normal bundle to the submanifold ... Euclidean space. The normal curvature of math S 2 math is identically 1. Great circles have curvature math k 1 math , which implies zero geodesic curvature, thus they are geodesics. Smaller circles of radius math r math will have curvature math 1 r math and geodesic curvature math k g sqrt 1 r 2 r math . Some results involving geodesic curvature The geodesic curvature is no other than the usual curvature of the curve when computed intrinsically in the submanifold math M math . It does not depend on the way the submanifold math M math sits in math bar M math . On the contrary the normal curvature ... Codazzi equations . The Gauss Bonnet theorem . See also Curvature Darboux frame Gauss Codazzi equations ... Surfaces isbn 0 486 63433 7 . springer id G g044070 title Geodesic curvature first Yu.S. last Slobodyan year 2001 . External links Mathworld urlname GeodesicCurvature title Geodesic curvaturecurvature Category Curvature mathematics Category Differential geometry of surfaces Category Geodesic mathematics ... more details
In geometry , center of curvature of a curve is found at a point that is at a distance equal to the Radius of curvature mathematics radius of curvature lying on the normal vector . It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the center of curvature. Cauchy defined the center of curvature C as the intersection point of two infinitesimal infinitely close normals to the curve. ref citation last1 Borovik first1 Alexandre author1 link Alexandre Borovik last2 Katz first2 Mikhail G. author2 link Mikhail Katz arxiv 1108.2885 doi 10.1007 s10699 011 9235 x issue journal Foundations of Science pages title Who gave you the Cauchy Weierstrass tale? The dual history of rigorous calculus volume year 2011 ref See also Curvature Differential geometry of curves Ref notes Reflist References Citation last1 Hilbert first1 David author1 link David Hilbert last2 Cohn Vossen first2 Stephan author2 link Stephan Cohn Vossen title Geometry and the Imagination publisher Chelsea location New York edition 2nd isbn 978 0 8284 1097 8 year 1952 Category Curves Category Differential geometry Category Geometric centers Curvature differential geometry stub am ca Centre de curvatura es Centro de curvatura ... more details
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