An antisymmetric tensor of second rank(a.k.a. 2-form).where is the wedge product (or outer product).
A geodesic mapping between two Riemannian manifolds is a diffeomorphism sending geodesics of into geodesics of , whose inverse also sends geodesics to geodesics (Ambartzumian 1982, p. 26).
When the Gaussian curvature is everywhere negative, a surface is called anticlastic and is saddle-shaped. A surface on which is everywhere positive is called synclastic. A point at which the Gaussian curvature is negative is called a hyperbolic point.
The angular acceleration is defined as the time derivative of the angular velocity ,
where is the unit normal vector and is one of the two principal curvatures.
Two points on a compact Riemann surface such that lies on every geodesic passing through , and conversely. An oriented surface where every point belongs to a Wiedersehen pair is called a Wiedersehen surface.The name is the German word for "seeing again" and was introduced by Blaschke.
A point on a regular surface is said to be hyperbolic if the Gaussian curvature or equivalently, the principal curvatures and , have opposite signs.
At each point on a given a two-dimensional surface, there are two "principal" radii of curvature. The larger is denoted , and the smaller . The "principal directions" corresponding to the principal radii of curvature are perpendicular to one another. In other words, the surface normal planes at the point and in the principal directions are perpendicular to one another, and both are perpendicular to the surface tangent plane at the point.
Let be a regular surface with points in the tangent space of . Then the third fundamental form is given bywhere is the shape operator.
For a unit speed curve on a surface, the length of the surface-tangential component of acceleration is the geodesic curvature . Curves with are called geodesics. For a curve parameterized as , the geodesic curvature is given bywhere , , and are coefficients of the first fundamental form and are Christoffel symbols of the second kind.
A ruled surface is a tangent developable of a curve if can be parameterized by . A tangent developable is a flat surface.
The maximum and minimum of the normal curvature and at a given point on a surface are called the principal curvatures. The principal curvatures measure the maximum and minimum bending of a regular surface at each point. The Gaussian curvature and mean curvature are related to and by(1)(2)This can be written as a quadratic equation(3)which has solutions(4)(5)
Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-space may be understood intrinsically to that surface. "Residents" of the surface may observe the Gaussian curvature of the surface without ever venturing into full three-dimensional space; they can observe the curvature of the surface they live in without even knowing about the three-dimensional space in which they are embedded.In particular, Gaussian curvature can be measured by checking how closely the arc lengths of circles of small radii correspond to what they should be in Euclidean space, . If the arc length of circles tends to be smaller than what is expected in Euclidean space, then the space is positively curved; if larger, negatively; if the same, 0 Gaussian curvature.Gauss (effectively) expressed the theorema egregium by saying that the Gaussian curvature at a point is given by where is the Riemann tensor, and and are an orthonormal..
A surface on which the Gaussian curvature is everywhere positive. When is everywhere negative, a surface is called anticlastic. A point at which the Gaussian curvature is positive is called an elliptic point.
A point on a regular surface is said to be planar if the Gaussian curvature and (where is the shape operator), or equivalently, both of the principal curvatures and are 0.
A patch (also called a local surface) is a differentiable mapping , where is an open subset of . More generally, if is any subset of , then a map is a patch provided that can be extended to a differentiable map from into , where is an open set containing . Here, (or more generally, ) is called the map trace of .
If is a regular patch on a regular surface in with normal , then(1)(2)(3)where , , and are coefficients of the second fundamental form and are Christoffel symbols of the second kind.
A point on a regular surface is said to be parabolic if the Gaussian curvature but (where is the shape operator), or equivalently, exactly one of the principal curvatures equals 0.
Surface area is the area of a given surface. Roughly speaking, it is the "amount" of a surface (i.e., it is proportional to the amount of paint needed to cover it), and has units of distance squared. Surface area is commonly denoted for a surface in three dimensions, or for a region of the plane (in which case it is simply called "the" area).The following tables gives lateral surface areas for some common surfaces. Here, denotes the radius, the height, the ellipticity of a spheroid, the base perimeter, the slant height, the tube radius of a torus, and the radius from the rotation axis of the torus to the center of the tube (Beyer 1987). Note that many of these surfaces are surfaces of revolution, for which Pappus's centroid theorem can often be used to easily compute the surface area.surfaceconeconical frustumcubecylinderoblate spheroidprolate spheroidpyramidpyramidal frustumspherespherical lunetoruszoneEven simple..
There are three types of so-called fundamental forms. The most important are the first and second (since the third can be expressed in terms of these). The fundamental forms are extremely important and useful in determining the metric properties of a surface, such as line element, area element, normal curvature, Gaussian curvature, and mean curvature. Let be a regular surface with points in the tangent space of . Then the first fundamental form is the inner product of tangent vectors,(1)For , the second fundamental form is the symmetric bilinear form on the tangent space ,(2)where is the shape operator. The third fundamental form is given by(3)The first and secondfundamental forms satisfy(4)(5)where is a regular patch and and are the partial derivatives of with respect to parameters and , respectively. Their ratio is simply the normal curvature(6)for any nonzero tangent vector. The third fundamentalform is given in terms of the first and..
Let be an oriented regular surface in with normal . Then the support function of is the function defined by
A ruled surface is a normal developable of a curve if can be parameterized by , where is the normal vector (Gray 1993, pp. 352-354; first edition only).
Let be a regular surface with points in the tangent space of . Then the first fundamental form is the inner product of tangent vectors,(1)The first fundamental form satisfies(2)The first fundamental form (or line element) is givenexplicitly by the Riemannian metric(3)It determines the arc length of a curve on a surface.The coefficients are given by(4)(5)(6)The coefficients are also denoted , , and . In curvilinear coordinates (where ), the quantities(7)(8)are called scale factors.
A noncylindrical ruled surface alwayshas a parameterization of the form(1)where , , and is called the striction curve of . Furthermore, the striction curve does not depend on the choice of the base curve. The striction and director curves of the helicoid(2)are(3)(4)For the hyperbolic paraboloid(5)the striction and director curves are(6)(7)
Let be a unit tangent vector of a regular surface . Then the normal curvature of in the direction is(1)where is the shape operator. Let be a regular surface, , be an injective regular patch of with , and(2)where . Then the normal curvature in the direction is(3)where , , and are the coefficients of the first fundamental form and , , and are the coefficients of the second fundamental form.The maximum and minimum values of the normal curvature at a point on a regular surface are called the principal curvatures and .
The negative derivative(1)of the unit normal vector field of a surface is called the shape operator (or Weingarten map or second fundamental tensor). The shape operator is an extrinsic curvature, and the Gaussian curvature is given by the determinant of . If is a regular patch, then(2)(3)At each point on a regular surface , the shape operator is a linear map(4)The shape operator for a surface is given by the Weingartenequations.
A dual bivector is defined byand a self-dual bivector by
Let and be the principal curvatures, then their mean(1)is called the mean curvature. Let and be the radii corresponding to the principal curvatures, then the multiplicative inverse of the mean curvature is given by the multiplicative inverse of the harmonic mean,(2)In terms of the Gaussian curvature ,(3)The mean curvature of a regular surface in at a point is formally defined as(4)where is the shape operator and denotes the matrix trace. For a Monge patch with ,(5)(Gray 1997, p. 399).If is a regular patch, then the mean curvature is given by(6)where , , and are coefficients of the first fundamental form and , , and are coefficients of the second fundamental form (Gray 1997, p. 377). It can also be written(7)Gray (1997, p. 380).The Gaussian and mean curvature satisfy(8)with equality only at umbilic points, since(9)If is a point on a regular surface and and are tangent vectors to at , then the mean curvature of at is related to the..
The curvature of a surface satisfieswhere is the normal curvature in a direction making an angle with the first principal direction and and are the principal curvatures.
Let a patch be given by the map , where is an open subset of , or more generally by , where is any subset of . Then (or more generally, ) is called the trace of .
A point on a regular surface is said to be elliptic if the Gaussian curvature or equivalently, the principal curvatures and have the same sign.
Let be a regular surface with points in the tangent space of . For , the second fundamental form is the symmetric bilinear form on the tangent space ,(1)where is the shape operator. The second fundamental form satisfies(2)for any nonzero tangent vector.The second fundamental form is given explicitly by(3)where(4)(5)(6)and are the direction cosines of the surface normal. The second fundamental form can also be written(7)(8)(9)(10)(11)(12)(13)(14)where is the normal vector, is a regular patch, and and are the partial derivatives of with respect to parameters and , respectively, or(15)(16)(17)
A curve on a surface whose tangents are always in the direction of principalcurvature. The equation of the lines of curvature can be writtenwhere and are the coefficients of the first and second fundamental forms.
The radius of curvature is given by(1)where is the curvature. At a given point on a curve, is the radius of the osculating circle. The symbol is sometimes used instead of to denote the radius of curvature (e.g., Lawrence 1972, p. 4).Let and be given parametrically by(2)(3)then(4)where and . Similarly, if the curve is written in the form , then the radius of curvature is given by(5)In polar coordinates , the radius of curvature is given by(6)where and (Gray 1997, p. 89).
(1)where is the radius vector and is the derivative with respect to time. Expressed in terms of the arc length,(2)where is the unit tangent vector, so the speed (which is the magnitude of the velocity) is(3)
A curve on a regular surface is a principal curve iff the velocity always points in a principal direction, i.e.,where is the shape operator and is a principal curvature. If a surface of revolution generated by a plane curve is a regular surface, then the meridians and parallels are principal curves.
The term "total curvature" is used in two different ways in differential geometry.The total curvature, also called the third curvature, of a space curve with line elements , , and along the normal, tangent, and binormal vectors respectively, is defined as the quantity(1)(2)where is the curvature and is the torsion (Kreyszig 1991, p. 47). The term is apparently also applied to the derivative directly , namely(3)(Kreyszig 1991, p. 47).The second use of "total curvature" is as a synonym for Gaussiancurvature (Kreyszig 1991, p. 131).
If (i.e., is an irrotational field) in a simply connected neighborhood of a point , then in this neighborhood, is the gradient of a scalar field ,for , where is the gradient operator. Consequently, the gradient theorem givesfor any path located completely within , starting at and ending at .This means that if , the line integral of is path-independent.
The torsion of a space curve, sometimes also called the "second curvature" (Kreyszig 1991, p. 47), is the rate of change of the curve's osculating plane. The torsion is positive for a right-handed curve, and negative for a left-handed curve. A curve with curvature is planar iff .The torsion can be defined by(1)where is the unit normal vector and is the unit binormal vector. Written explicitly in terms of a parameterized vector function ,(2)(3)(Gray 1997, p. 192), where denotes a scalar triple product and is the radius of curvature.The quantity is called the radius of torsion and is denoted or .
For a scalar function over a surface parameterized by and , the surface integral is given by(1)(2)where and are tangent vectors and is the cross product.For a vector function over a surface, the surfaceintegral is given by(3)(4)(5)where is a dot product and is a unit normal vector. If , then is given explicitly by(6)If the surface is surface parameterized using and , then(7)
A tensor-like object which reverses sign under inversion. Given a transformation matrix ,where det is the determinant. A pseudotensor is sometimesalso called a tensor density.
The indices of a contravariant tensor can be lowered, turning it into a covariant tensor , by multiplication by a so-called metric tensor , e.g.,
The Weyl tensor is the tensor defined by(1)where is the Riemann tensor, is the scalar curvature, is the metric tensor, and denotes the antisymmetric tensor part (Wald 1984, p. 40).The Weyl tensor is defined so that every tensorcontraction between indices gives 0. In particular,(2)(Weinberg 1972, p. 146). The number of independent components for a Weyl tensor in -D for is given by(3)(Weinberg 1972, p. 146). For , 4, ..., this gives 0, 10, 35, 84, 168, ... (OEIS A052472).
A nonzero vector in -dimensional Lorentzian space is said to be positive timelike if it has imaginary (Lorentzian) norm and if its first component is positive. Symbolically, is positive timelike if bothandhold. Note that equation (6) above expresses the imaginary norm condition by saying, equivalently, that the vector has a negative squared norm.
The technique of extracting the content from geometric (tensor) equations by working in component notation and rearranging indices as required. Index gymnastics is a fundamental component of special and general relativity (Misner et al. 1973, pp. 84-89). Examples of index gymnastics include(1)(2)(3)(4)(5)(6)(7)(8)(Misner et al. 1973, p. 85), where is the metric tensor, is the Kronecker delta, is a comma derivative, is the antisymmetric tensor part, and is the symmetric tensor part.
A weak Riemannian metric on a smooth manifold is a tensor field which is both a weak pseudo-Riemannian metric and positive definite.In a very precise way, the condition of being a weak Riemannian metric is considerably less stringent than the condition of being a strong Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, any strong Riemannian metric provides an isomorphism between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injective linear maps from to (Marsden et al. 2002).
A nonzero vector in -dimensional Lorentzian space is said to be positive lightlike if it has zero (Lorentzian) norm and if its first component is positive. Symbolically, is positive lightlike if bothandhold. The collection of all positive lightlike vectors form the top half of the lightcone.
Harmonic coordinates satisfy the condition(1)or equivalently,(2)It is always possible to choose such a system. Using the d'Alembertian,(3)But since for harmonic coordinates, the result is a generalization of the harmonic equation(4)to(5)
"Poincaré transformation" is the name sometimes (e.g., Misner et al. 1973, p. 68) given to what other authors (e.g., Weinberg 1972, p. 26) term an inhomogeneous Lorentz transformationwhere is a Lorentz tensor.
The trace of a second-tensor rank tensor is a scalar given by the contracted mixed tensor equal to .The trace satisfies(1)For a small change in a given tensor , the trace can be computed approximately as(2)(3)(4)(5)(6)(7)
A tensor notation which considers the Riemann tensor as a matrix with indices and .
A differential k-form can be integrated on an -dimensional manifold. The basic example is an -form in the open unit ball in . Since is a top-dimensional form, it can be written and so(1)where the integral is the Lebesgue integral.On a manifold covered by coordinate charts , there is a partition of unity such that 1. has support in and 2. . Then(2)where the right-hand side is well-defined because each integration takes place in a coordinate chart. The integral of the -form is well-defined because, under a change of coordinates , the integral transforms according to the determinant of the Jacobian, while an -form pulls back by the determinant of the Jacobian. Hence,(3)is the same integral in either coordinate chart.For example, it is possible to integrate the 2-form(4)on the sphere . Because a point has measure zero, it is enough to integrate on , which can be covered by stereographic projection . Since(5)the pullback map of is(6)the integral of on..
The total number of contravariant and covariant indices of a tensor. The rank of a tensor is independent of the number of dimensions of the underlying space.An intuitive way to think of the rank of a tensor is as follows: First, consider intuitively that a tensor represents a physical entity which may be characterized by magnitude and multiple directions simultaneously (Fleisch 2012). Therefore, the number of simultaneous directions is denoted and is called the rank of the tensor in question. In -dimensional space, it follows that a rank-0 tensor (i.e., a scalar) can be represented by number since scalars represent quantities with magnitude and no direction; similarly, a rank-1 tensor (i.e., a vector) in -dimensional space can be represented by numbers and a general tensor by numbers. From this perspective, a rank-2 tensor (one that requires numbers to describe) is equivalent, mathematically, to an matrix.rankobject0scalar1vector2 matrixtensorThe..
The permutation tensor, also called the Levi-Civita tensor or isotropic tensor of rank 3 (Goldstein 1980, p. 172), is a pseudotensor which is antisymmetric under the interchange of any two slots. Recalling the definition of the permutation symbol in terms of a scalar triple product of the Cartesian unit vectors,(1)the pseudotensor is a generalization to an arbitrary basisdefined by(2)(3)where(4)and , where is the metric tensor. is nonzero iff the vectors are linearly independent.When viewed as a tensor, the permutation symbol is sometimes known as the Levi-Civita tensor. The permutation tensor of rank four is important in general relativity, and has components defined as(5)(Weinberg 1972, p. 38). The rank four permutation tensor satisfies the identity(6)
The vector Laplacian can be generalized to yieldthe tensor Laplacian(1)(2)(3)(4)(5)where is a covariant derivative, is the metric tensor, , is the comma derivative (Arfken 1985, p. 165), and(6)is a Christoffel symbol of the secondkind.
Orthogonal contravariant and covariantsatisfywhere is the Kronecker delta.
where is the Ricci curvature tensor, is the scalar curvature, and is the metric tensor. (Wald 1984, pp. 40-41). It satisfies(Misner et al. 1973, p. 222).
A dyadic, also known as a vector direct product, is a linear polynomial of dyads consisting of nine components which transform as(1)(2)(3)Dyadics are often represented by Gothic capital letters. The use of dyadics is nearly archaic since tensors perform the same function but are notationally simpler.A unit dyadic is also called the idemfactor and is defined such that(4)In Cartesian coordinates,(5)and in spherical coordinates(6)
A nonzero vector in -dimensional Lorentzian space is said to be negative timelike if it has imaginary (Lorentzian) norm and if its first component is negative. Symbolically, is negative timelike if bothandhold. Note that equation (6) above expresses the imaginary norm condition by saying, equivalently, that the vector has a negative squared norm.
Dyads extend vectors to provide an alternative description to second tensor rank tensors. A dyad of a pair of vectors and is defined by . The dot product is defined by(1)(2)and the colon product by(3)
A nonzero vector in -dimensional Lorentzian space is said to be negative lightlike if it has zero (Lorentzian) norm and if its first component is negative. Symbolically, is negative lightlike if bothandhold. The collection of all negative lightlike vectors form the bottom half of the lightcone.
where is the Dirac operator , is the covariant derivative on spinors, is the scalar curvature, and is the self-dual part of the curvature of .
A catastrophe which can occur for two control factors and one behavior axis. The cusp catastrophe is the universal unfolding of the singularity and has the equation . The equation also has a cusp catastrophe.
It is conjectured that any convex body in -dimensional Euclidean space has an interior point lying on normals through distinct boundary points (Croft et al. 1991). This has been proved for and 3 by Heil (1979ab, 1985). It is known that higher dimensions always contain at least a 6-normal point, but the general conjecture remains open.
where is the torsion. The symbol is also sometimes used instead of .
In a rectangular room (a cuboid) with dimensions , a spider is located in the middle of one wall one foot away from the ceiling. A fly is in the middle of the opposite wall one foot away from the floor. If the fly remains stationary, what is the shortest total distance (i.e., the geodesic) the spider must crawl along the walls, ceiling, and floor in order to capture the fly? The answer, , can be obtained by "flattening" the walls as illustrated above. Note that his distance is shorter than the the spider would have to travel if first crawling along the wall to the floor, then across the floor, then up one foot to get to the fly. The puzzle was originally posed in an English newspaper by Dudeney in 1903 (Gardner 1958).A twist to the problem can be obtained by a spider that suspends himself from strand of cobweb and thus takes a shortcut by not being forced to remain glued to a surface of the room. If the spider attaches a strand of cobweb to the wall at his starting..
Given an antisymmetric second tensor rank tensor , a dual pseudotensor is defined by(1)where(2)(3)
Let , , and be three polar vectors, and define(1)(2)where det is the determinant. The is a third-tensor rank tensor, then the dual scalar is defined as(3)where is the permutation tensor.
The contraction of a tensor is obtained by setting unlike indices equal and summing according to the Einstein summation convention. Contraction reduces the tensor rank by 2. For example, for a second-rank tensor,The contraction operation is invariant under coordinate changes sinceand must therefore be a scalar.When is interpreted as a matrix, the contraction is the same as the trace.Sometimes, two tensors are contracted using an upper index of one tensor and a lower of the other tensor. In this context, contraction occurs after tensor multiplication.
A tensor is said to satisfy the double contraction relation when(1)This equation is satisfied by(2)(3)(4)where the hat denotes zero trace, symmetric unit tensors. These tensors are used to define the spherical harmonic tensor.
Written in the notation of partial derivatives, the d'Alembertian in a flat spacetime is defined bywhere is the speed of light.The operator usually called the d'Alembertian is also the Laplacian on a flat manifold of Lorentzian signature.
An th-rank tensor in -dimensional space is a mathematical object that has indices and components and obeys certain transformation rules. Each index of a tensor ranges over the number of dimensions of space. However, the dimension of the space is largely irrelevant in most tensor equations (with the notable exception of the contracted Kronecker delta). Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly two indices) to an arbitrary number of indices.Tensors provide a natural and concise mathematical framework for formulating and solving problems in areas of physics such as elasticity, fluid mechanics, and general relativity.The notation for a tensor is similar to that of a matrix (i.e., ), except that a tensor , , , etc., may have an arbitrary number of indices. In addition, a tensor with rank may be of mixed type , consisting of so-called "contravariant"..
The index associated to a metric tensor on a smooth manifold is a nonnegative integer for whichfor all . Here, the notation denotes the quadratic form index associated with .The index of a metric tensor provides an alternative tool by which to define a number of various notions typically associated to the signature of . For example, a Lorentzian manifold can be defined as a pair for which and for which , a definition equivalent to its more typical definition as a manifold of dimension no less than two equipped with a tensor of metric signature (or, equivalently, ).
A second-tensor rank symmetric tensor is defined as a tensor for which(1)Any tensor can be written as a sum of symmetric and antisymmetric parts(2)(3)The symmetric part of a tensoris denoted using parentheses as(4)(5)Symbols for the symmetric and antisymmetricparts of tensors can be combined, for example(6)(Wald 1984, p. 26).The product of a symmetric and an antisymmetric tensor is 0. This can be seen as follows. Let be antisymmetric, so(7)(8)Let be symmetric, so(9)Then(10)(11)(12)A symmetric second-tensor rank tensor has scalar invariants(13)(14)
A covariant tensor, denoted with a lowered index (e.g., ) is a tensor having specific transformation properties. In general, these transformation properties differ from those of a contravariant tensor.To examine the transformation properties of a covariant tensor, first consider thegradient(1)for which(2)where . Now let(3)then any set of quantities which transform according to(4)or, defining(5)according to(6)is a covariant tensor.Contravariant tensors are a type of tensor with differing transformation properties, denoted . To turn a contravariant tensor into a covariant tensor (index lowering), use the metric tensor to write(7)Covariant and contravariant indices can be used simultaneously in a mixedtensor.In Euclidean spaces, and more generally in flat Riemannian manifolds, a coordinate system can be found where the metric tensor is constant, equal to Kronecker delta(8)Therefore, raising and lowering indices is trivial,..
The covariant derivative of a contravariant tensor (also called the "semicolon derivative" since its symbol is a semicolon) is given by(1)(2)(Weinberg 1972, p. 103), where is a Christoffel symbol, Einstein summation has been used in the last term, and is a comma derivative. The notation , which is a generalization of the symbol commonly used to denote the divergence of a vector function in three dimensions, is sometimes also used.The covariant derivative of a covariant tensor is(3)(Weinberg 1972, p. 104).Schmutzer (1968, p. 72) uses the older notation or .
A strong Riemannian metric on a smooth manifold is a tensor field which is both a strong pseudo-Riemannian metric and positive definite.In a very precise way, the condition of being a strong Riemannian metric is considerably more stringent than the condition of being a weak Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, strong Riemannian metrics provide an isomorphism between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injective linear maps from to .
A semi-Riemannian manifold is said to be Lorentzian if and if the index associated with the metric tensor satisfies .Alternatively, a smooth manifold of dimension is Lorentzian if it comes equipped with a tensor of metric signature (or, equivalently, ).
A Lorentz transformation is a four-dimensional transformation(1)satisfied by all four-vectors , where is a so-called Lorentz tensor. Lorentz tensors are restricted by the conditions(2)with the Minkowski metric (Weinberg 1972, p. 26; Misner et al. 1973, p. 68).Here, the tensor indices run over 0, 1, 2, 3, with being the time coordinate and being space coordinates, and Einstein summation is used to sum over repeated indices. There are a number of conventions, but a common one used by Weinberg (1972) is to take the speed of light to simplify computations and allow to be written simply as for . The group of Lorentz transformations in Minkowski space is known as the Lorentz group.An element in four-space which is invariant under a Lorentz transformation is said to be a Lorentz invariant; examples include scalars, elements of the form , and the interval between two events (Thorn 2012).Note that while some authors (e.g., Weinberg 1972,..
A contravariant tensor is a tensor having specific transformation properties (cf., a covariant tensor). To examine the transformation properties of a contravariant tensor, first consider a tensor of rank 1 (a vector)(1)for which(2)Now let , then any set of quantities which transform according to(3)or, defining(4)according to(5)is a contravariant tensor. Contravariant tensors are indicated with raised indices, i.e., .Covariant tensors are a type of tensor with differing transformation properties, denoted . However, in three-dimensional Euclidean space,(6)for , 2, 3, meaning that contravariant and covariant tensors are equivalent. Such tensors are known as Cartesian tensor. The two types of tensors do differ in higher dimensions, however.Contravariant four-vectors satisfy(7)where is a Lorentz tensor.To turn a covariant tensor into a contravariant tensor (index raising), use the metric tensor to write(8)Covariant and..
The Lie derivative of a spinor is defined bywhere is the image of by a one-parameter group of isometries with its generator. For a vector field and a covariant derivative , the Lie derivative of is given explicitly bywhere and are Dirac matrices (Choquet-Bruhat and DeWitt-Morette 2000).
Contracting tensors with in the Bianchi identities(1)gives(2)Contracting again,(3)or(4)or(5)
A four-vector is said to be lightlike if its four-vector norm satisfies .One should note that the four-vector norm is nothing more than a special case of the more general Lorentzian inner product on Lorentzian -space with metric signature : In this more general environment, the inner product of two vectors and has the formwhereby one defines a vector to be lightlike precisely when .Lightlike vectors are sometimes called null vectors. The collection of all lightlike vectors in a Lorentzian space (e.g., in the Minkowski space of special relativity) is known as the light cone. One often draws distinction between lightlike vectors which are positive and those which are negative.
The components of the gradient of the one-form are denoted , or sometimes , and are given by(Misner et al. 1973, p. 62). Note that Schmutzer (1968, p. 70) uses the older notation .
The Lie derivative of tensor with respect to the vector field is defined by(1)Explicitly, it is given by(2)where is a comma derivative. The Lie derivative of a metric tensor with respect to the vector field is given by(3)where denotes the symmetric tensor part and is a covariant derivative.
Christoffel symbols of the second kind are the second type of tensor-like object derived from a Riemannian metric which is used to study the geometry of the metric. Christoffel symbols of the second kind are variously denoted as (Walton 1967) or (Misner et al. 1973, Arfken 1985). They are also known as affine connections (Weinberg 1972, p. 71) or connection coefficients (Misner et al. 1973, p. 210).Unfortunately, there are two different definitions of the Christoffel symbolof the second kind.Arfken (1985, p. 161) defines(1)(2)(3)where is a partial derivative, is the metric tensor,(4)where is the radius vector, and(5)Therefore, for an orthogonal curvilinear coordinate system, by this definition,(6)The symmetry of definition (6) means that(7)(Walton 1967).This Christoffel symbol of the second kind is related to the Christoffel symbol of the first kind by(8)Walton (1967) lists Christoffel symbols of the second..
The first type of tensor-like object derived from a Riemannian metric which is used to study the geometry of the metric. Christoffel symbols of the first kind are variously denoted , , , or . They are also known as connections coefficients (Misner et al. 1973, p. 210).The Christoffel symbol of the first kind is defined by(1)(2)(3)where is the metric tensor, is a Christoffel symbol of the second kind, and(4)But(5)(6)(7)so(8)
The scalar curvature, also called the "curvature scalar" (e.g., Weinberg 1972, p. 135; Misner et al. 1973, p. 222) or "Ricci scalar," is given bywhere is the metric tensor and is the Ricci curvature tensor.
The Riemann tensor (Schutz 1985) , also known the Riemann-Christoffel curvature tensor (Weinberg 1972, p. 133; Arfken 1985, p. 123) or Riemann curvature tensor (Misner et al. 1973, p. 218), is a four-index tensor that is useful in general relativity. Other important general relativistic tensors such that the Ricci curvature tensor and scalar curvature can be defined in terms of .The Riemann tensor is in some sense the only tensor that can be constructed fromthe metric tensor and its first and second derivatives,(1)where are Christoffel symbols of the first kind and is a comma derivative (Schmutzer 1968, p. 108; Weinberg 1972). In one dimension, . In four dimensions, there are 256 components. Making use of the symmetry relations,(2)the number of independent components is reduced to 36. Using the condition(3)the number of coordinates reduces to 21. Finally, using(4)20 independent components are left (Misner..
For a smooth harmonic map ,where is the gradient, Ric is the Ricci curvature tensor, and Riem is the Riemann tensor.
If any set of points is displaced by where all distance relationships are unchanged (i.e., there is an isometry), then the vector field is called a Killing vector.(1)so let(2)(3)(4)(5)(6)(7)(8)where is the Lie derivative.An ordinary derivative can be replaced with a covariant derivative in a Lie derivative, so we can take as the definition(9)(10)which gives Killing's equation(11)where denotes the symmetric tensor part and is a covariant derivative.A Killing vector satisfies(12)(13)(14)where is the Ricci curvature tensor and is the Riemann tensor.In Minkowski space, there are 10 Killing vectors(15)(16)(17)(18)The first group is translation, the second rotation,and the final corresponds to a "boost."
The equation defining Killing vectors.where is the Lie derivative and is a covariant derivative.
The covariant derivative of the Riemanntensor is given by(1)Permuting , , and (Weinberg 1972, pp. 146-147) gives the Bianchi identities(2)which can be written concisely as(3)(Misner et al. 1973, p. 221), where denoted the antisymmetric tensor part. Wald (1984, p. 39) calls(4)the Bianchi identity, where is the covariant derivative, and is the Riemann tensor.
An antisymmetric (also called alternating) tensor is a tensor which changes sign when two indices are switched. For example, a tensor such that(1)is antisymmetric.The simplest nontrivial antisymmetric tensor is therefore an antisymmetric rank-2 tensor, which satisfies(2)Furthermore, any rank-2 tensor can be written as a sumof symmetric and antisymmetric parts as(3)The antisymmetric part of a tensor is sometimes denoted using the special notation(4)For a general rank- tensor,(5)where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example(6)(Wald 1984, p. 26).
The Ricci curvature tensor, also simply known as the Ricci tensor (Parker and Christensen 1994), is defined bywhere is the Riemann tensor.Geometrically, the Ricci curvature is the mathematical object that controls the growthrate of the volume of metric balls in a manifold.
A tensor which has the same components in all rotated coordinate systems. All rank-0 tensors (scalars) are isotropic, but no rank-1 tensors (vectors) are. The unique rank-2 isotropic tensor is the Kronecker delta, and the unique rank-3 isotropic tensor is the permutation symbol (Goldstein 1980, p. 172).The number of isotropic tensors of rank 0, 1, 2, ... are 1, 0, 1, 1, 3, 6, 15, 36, 91, 232, ... (OEIS A005043). These numbers are called the Motzkin sum numbers and are given by the recurrence relation(1)with and . Closed forms for are given by(2)(3)The terms have the generating function(4)(5)(6)Starting at rank 5, syzygies play a role in restricting the number of isotropic tensors. In particular, syzygies occur at rank 5, 7, 8, and all higher ranks.
Given a general second tensor rank tensor and a metric , define(1)(2)(3)where is the Kronecker delta and is the permutation symbol. Then(4)where , , and are tensors of tensor rank 0, 1, and 2.
An affine tensor is a tensor that corresponds to certain allowable linear coordinate transformations, , where the determinant of is nonzero. This transformation takes the rectangular coordinate system into the coordinate system having oblique axes. In this way an affine tensor can be seen as a special kind of Cartesian tensor.These tensors have the Jacobians,(1)(2)(3)(4)The transformation laws for affine contravariant (tangent) tensors are(5)(6)(7)and so on, and the transformation laws for affine covariants (covectors) tensors are(8)(9)(10)and so on.The transformation laws for mixed affine tensors are(11)(12)
The index associated to a symmetric, non-degenerate, and bilinear over a finite-dimensional vector space is a nonnegative integer defined bywhere the set is defined to beAs a concrete example, a pair consisting of a smooth manifold with a symmetric tensor field is said to be a Lorentzian manifold if and only if and the index associated to the quadratic form satisfies for all (Sachs and Wu 1977). This particular definition succinctly conveys the fact that Lorentzian manifolds have indefinite metric tensors of signature (or equivalently ) without having to make precise any definitions related to metric signatures, quadratic form signatures, etc.The above example also illustrates the deep connection between the index of a quadratic form and the notion of the index of a metric tensor defined on a smooth manifold . In particular, the index of a metric tensor is defined to be the quadratic form index associated to for any element . Because of this connection,..
The indices of a covariant tensor can be raised, forming a contravariant tensor , by multiplication by a so-called metric tensor , e.g.,
The plane spanned by the three points , , and on a curve as . Let be a point on the osculating plane, thenwhere denotes the scalar triple product. The osculating plane passes through the tangent. The intersection of the osculating plane with the normal plane is known as the (principal) normal vector. The vectors and (tangent vector and normal vector) span the osculating plane.
A curve whose centrode revolves about a fixed axis with constant angle and speed when the curve is traversed with unit speed. The tangent indicatrix of a curve of constant precession is a spherical helix. An arc length parameterization of a curve of constant precession with natural equations(1)(2)is(3)(4)(5)where(6)and , and are constant. This curve lies on a circular one-sheeted hyperboloid(7)The curve is closed iff is rational.
The "temperature" of a curve is defined aswhere is the length of and is the length of the perimeter of the convex hull. The temperature of a curve is 0 only if the curve is a straight line, and increases as the curve becomes more "wiggly."
Let be a smooth curve in a manifold from to with and . Then where is the tangent space of at . The length of with respect to the Riemannian structure is given by
For a plane curve, the tangential angle is defined by(1)where is the arc length and is the radius of curvature. The tangential angle is therefore given by(2)where is the curvature. For a plane curve , the tangential angle can also be defined by(3)Gray (1997) calls the turning angle instead of the tangential angle.
A natural equation is an equation which specifies a curve independent of any choice of coordinates or parameterization. The study of natural equations began with the following problem: given two functions of one parameter, find the space curve for which the functions are the curvature and torsion.Euler gave an integral solution for plane curves (which always have torsion ). Call the angle between the tangent line to the curve and the x-axis the tangential angle, then(1)where is the curvature. Then the equations(2)(3)where is the torsion, are solved by the curve with parametric equations(4)(5)The equations and are called the natural (or intrinsic) equations of the space curve. An equation expressing a plane curve in terms of and radius of curvature (or ) is called a Cesàro equation, and an equation expressing a plane curve in terms of and is called a Whewell equation. The natural parametric equations of a curve parametrize it in terms..
where is the tangent vector defined by
The line integral of a vector field on a curve is defined by(1)where denotes a dot product. In Cartesian coordinates, the line integral can be written(2)where(3)For complex and a path in the complex plane parameterized by ,(4)Poincaré's theorem states that if in a simply connected neighborhood of a point , then in this neighborhood, is the gradient of a scalar field ,(5)for , where is the gradient operator. Consequently, the gradient theorem gives(6)for any path located completely within , starting at and ending at .This means that if (i.e., is an irrotational field in some region), then the line integral is path-independent in this region. If desired, a Cartesian path can therefore be chosen between starting and ending point to give(7)If (i.e., is a divergenceless field, a.k.a. solenoidal field), then there exists a vector field such that(8)where is uniquely determined up to a gradient field (and which can be chosen so that )...
The point on the positive ray of the normal vector at a distance , where is the radius of curvature. It is given by(1)(2)where is the normal vector and is the tangent vector. It can be written in terms of explicitly as(3)For a curve represented parametrically by ,(4)(5)(Lawrence 1972, p. 25).
Let the speed of a closed curve on the unit sphere never vanish. Then the tangent indicatrix, also called the tantrix,is another closed curve on .If immerses in , then so will .
Let a space curve have line elements , , and along the normal, tangent, and binormal vectors respectively, then(1)where(2)(3)(4)and and are the curvature and torsion, respectively.
The semiperimeter on a figure is defined as(1)where is the perimeter. The semiperimeter of polygons appears in unexpected ways in the computation of their areas. The most notable cases are in the altitude, exradius, and inradius of a triangle, the Soddy circles, Heron's formula for the area of a triangle in terms of the legs , , and (2)and Brahmagupta's formula for the areaof a quadrilateral(3)The semiperimeter also appears in the beautiful l'Huilier'stheorem about spherical triangles.For a triangle, the following identities hold,(4)(5)(6)Now consider the above figure. Let be the incenter of the triangle , with , , and the tangent points of the incircle. Extend the line with . Note that the pairs of triangles , , are congruent. Then(7)(8)(9)(10)(11)(12)(13)Furthermore,(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(Dunham 1990). These equations are some of the building blocks of Heron's derivationof Heron's formula...
An Cesàro equation is a natural equation which expresses a curve in terms of its arc length function and radius of curvature (or equivalently, the curvature ). Note that while the Cesàro equation is said to be intrinsic because it is invariant under transformations that preserve length and angle, it is not intrinsic to a curve because it depends on the starting point from which arc length is measured and hence on the parametrization (see the table below for examples).The following table summarizes the Cesàro equations for certain parametrizations of a number of curves (cf. Lawrence 1972, p. 5 and Yates 1952, p. 126).curveparametrizationCesàro equationastroidcardioidcardioidcatenarycirclecircle involutecycloidcycloiddeltoidnephroidtractrixTschirnhausen cubic..
If two single-valued continuous functions (curvature) and (torsion) are given for , then there exists exactly one space curve, determined except for orientation and position in space (i.e., up to a Euclidean motion), where is the arc length, is the curvature, and is the torsion.
Let and be differentiable scalar functions defined at all points on a surface . In computer graphics, the functions and often represent texture coordinates for a 3-dimensional polygonal model. A rendering technique known as bump mapping orients the basis vectors of the tangent plane at any point so that they are aligned with the direction in which the derivative of or is zero. In this context, the tangent vector is specifically defined to be the unit vector lying in the tangent plane for which and is positive. The bitangent vector is defined to be the unit vector lying in the tangent plane for which and is positive. The vectors and are not necessarily orthogonal and may not exist for poorly conditioned functions and .The vector given byis a unit normal to the surface at the point . For a closed surface , this normal vector can be characterized as outward-facing or inward-facing. The basis vectors of the local tangent space at the point are defined to be..
The term rectification is sometimes used to refer to the determination of the length of a curve.Rectification also refers to the operation which converts the midpoints of the edges of a regular polyhedron to the vertices of the related "rectified" polyhedron. Rectified forms are bounded by a combination of rectified cells and vertex figures. Therefore, a rectified polychoron is bounded by s and s. For example, is bounded by 600 truncated tetrahedra (truncated cells) and 120 icosahedra (vertex figures). A rectified polyhedron is indicated by prepending an "r" to the Schläfli symbol.polyhedronSchläfli symbolrectified polyhedronSchläfli symboltetrahedronoctahedronoctahedroncuboctahedroncubecuboctahedronicosahedronicosidodecahedrondodecahedronicosidodecahedron16-cell24-cellRectification of the six regular polychora gives five (not six) new polychora since the rectified..
Also known as the Serret-Frenet formulas, these vector differential equations relate inherent properties of a parametrized curve. In matrix form, they can be writtenwhere is the unit tangent vector, is the unit normal vector, is the unit binormal vector, is the torsion, is the curvature, and denotes .
The Gauss map is a function from an oriented surface in Euclidean space to the unit sphere in . It associates to every point on the surface its oriented unit normal vector. Since the tangent space at a point on is parallel to the tangent space at its image point on the sphere, the differential can be considered as a map of the tangent space at into itself. The determinant of this map is the Gaussian curvature, and negative one-half of the trace is the mean curvature.Another meaning of the Gauss map is the function(Trott 2004, p. 44), where is the floor function, plotted above on the real line and in the complex plane.The related function is plotted above, where is the fractional part.The plots above show blowups of the absolute values of these functions (a version of the left figure appears in Trott 2004, p. 44)...
Abstractly, the tensor direct product is the same as the vector space tensor product. However, it reflects an approach toward calculation using coordinates, and indices in particular. The notion of tensor product is more algebraic, intrinsic, and abstract. For instance, up to isomorphism, the tensor product is commutative because . Note this does not mean that the tensor product is symmetric.For two first-tensor rank tensors (i.e., vectors), the tensor direct product is defined as(1)which is a second-tensor rank tensor. The tensor contraction of a direct product of first-tensor rank tensors is the scalar(2)For second-tensor rank tensors,(3)(4)In general, the direct product of two tensors is a tensor of rank equal to the sum of the two initial ranks. The direct product is associative, but not commutative.The tensor direct product of two tensors and can be implemented in the Wolfram Language as TensorDirectProduct[a_List, b_List] :=..
Given metric spaces , with metrics respectively, the product metric is a metric on the Cartesian product defined asThis definition can be extended to the product of countably many metric spaces.If for all , and is the Euclidean metric of the real line, the product metric induces the Euclidean topology of the -dimensional Euclidean space . It does not coincide with the Euclidean metric of , but it is equivalent to it.
A four-vector is said to be timelike if its four-vector norm satisfies .One should note that the four-vector norm is nothing more than a special case of the more general Lorentzian inner product on -dimensional Lorentzian space with metric signature . In this more general environment, the inner product of two vectors and has the formwhereby one defines a vector to be timelike precisely when .Geometrically, the collection of all timelike vectors lie in the open subset of formed by the interior of the light cone: In particular, the upper half of the interior consists of vectors which are positive timelike whereas the lower half consists of all negative timelike vectors.
The Minkowski metric, also called the Minkowski tensor or pseudo-Riemannian metric, is a tensor whose elements are defined by the matrix(1)where the convention is used, and the indices run over 0, 1, 2, and 3, with the time coordinate and the space coordinates.The Euclidean metric(2)gives the line element(3)(4)while the Minkowski metric gives its relativistic generalization, the proper time(5)(6)The Minkowski metric is fundamental in relativity theory, and arises in the definitionof the Lorentz transformation as(7)where is a Lorentz tensor. It also satisfies(8)(9)(10)The metric of Minkowski space is diagonalwith(11)and so satisfies(12)The necessary and sufficient conditions for a metric to be equivalent to the Minkowski metric are that the Riemann tensor vanishes everywhere () and that at some point has three positive and one negative eigenvalues...
Also known as the first fundamental form,In the principal axis frame for three dimensions,At ordinary points on a surface, the line elementis positive definite.
The taxicab metric, also called the Manhattan distance, is the metricof the Euclidean plane defined byfor all points and . This number is equal to the length of all paths connecting and along horizontal and vertical segments, without ever going back, like those described by a car moving in a lattice-like street pattern.
In -dimensional Lorentzian space , the light cone is defined to be the subset consisting of all vectors(1)whose squared (Lorentzian) norm is identically zero:(2)Alternatively, is the collection of all lightlike vectors in . The decomposition of into Lorentzian space of signature leads to a natural decomposition of such a vector into its component and its -subvector . Using this notation, the squared norm of can be expressed as(3)whereby one can also define the light cone to be the collection of all vectors satisfying(4)This particular perspective makes natural the distinction between positiveand negative lightlike vectors.The open subset of formed by the interior of the light cone consists of all timelike vectors; the open subset formed by the exterior of consists of all vectors which are spacelike...
A four-vector is said to be spacelike if its four-vector norm satisfies .One should note that the four-vector norm is nothing more than a special case of the more general Lorentzian inner product on -dimensional Lorentzian space with metric signature . In this more general environment, the inner product of two vectors and has the formwhereby one defines a vector to be spacelike precisely when .Geometrically, the collection of all spacelike vectors lie in the open subset of formed by the exterior of the light cone.
A topology that is "potentially" a metric topology, in the sense that one can define a suitable metric that induces it. The word "potentially" here means that although the metric exists, it may be unknown.In fact, there are sufficient criteria on the topology that assure the existence of such a metric even if this is not explicitly given. An example of an existence theorem of this kind is due to Urysohn (Kelley 1955, p. 125), who proved that a regular T1-space whose topology has a countable basis is metrizable.Conversely, a metrizable space is always and regular, but the condition on the basis has to be weakened since in general, it is only true that the topology has a basis which is formed by countably many locally finite families of open sets.Special metrizability criteria are known for T2-spaces. A compact -space is metrizable iff the set of all elements of is a zero set (Willard 1970, p. 163). The continuous image..
For a diagonal metric tensor , where is the Kronecker delta, the scale factor for a parametrization , , ..., is defined by(1)(2)The line element (first fundamentalform) is then given by(3)(4)The scale factor appears in vector derivatives of coordinates in curvilinearcoordinates.
A metric on a bunch of segments with a common endpoint , which defines the distance between two points and as the length of the shortest path connecting them inside this geometric configuration. If and lie on the same segment, this is the length of segment . Otherwise, it is the sum of the lengths of segment and segment .
Suppose for every point in a manifold , an inner product is defined on a tangent space of at . Then the collection of all these inner products is called the Riemannian metric. In 1870, Christoffel and Lipschitz showed how to decide when two Riemannian metrics differ by only a coordinate transformation.
Roughly speaking, the metric tensor is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements in a generalized Pythagorean theorem:(1)In Euclidean space, where is the Kronecker delta (which is 0 for and 1 for ), reproducing the usual form of the Pythagorean theorem(2)In this way, the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be "arithmetized" by way of introducing a sort of generalized coordinate system (Borisenko and Tarapov 1979).In the above simplification, the space in question is most often a smooth manifold , whereby a metric tensor is essentially a geometrical object taking two vector inputs and calculating either the squared length of a single vector or a scalar product of two different vectors (Misner et al. 1978). In this..
The Ricci flow equation is the evolution equationfor a Riemannian metric , where is the Ricci curvature tensor. Hamilton (1982) showed that there is a unique solution to this equation for an arbitrary smooth metric on a closed manifold over a sufficiently short time. Hamilton (1982, 1986) also showed that Ricci flow preserves positivity of the Ricci curvature tensor in three dimensions and the curvature operator in all dimensions (Perelman 2002).
Two metrics and defined on a space are called equivalent if they induce the same metric topology on . This is the case iff, for every point of , every ball with center at defined with respect to :(1)contains a ball with center with respect to :(2)and conversely.Every metric on has uncountably many equivalent metrics. For every positive real number , a "scaled" metric can be defined such that for all ,(3)In fact, for all :(4)Another metric equivalent to is defined by(5)for all . In fact,(6)and(7)In the Euclidean plane , the metric(8)with circular balls can be defined in addition to the Euclidean metric. An equivalent more general metric for all positive real numbers and can be defined as(9)with elliptic balls, and the taxicab metric(10)can be defined with square "balls." All these are equivalent to the Euclidean metric...
1. Find a complete system of invariants, or 2. Decide when two metrics differ only by a coordinatetransformation. The most common statement of the problem is, "Given metrics and , does there exist a coordinate transformation from one to the other?" Christoffel (1869) and Lipschitz (1870) showed how to decide this question for two Riemannian metrics.The solution by É. Cartan requires computation of the 10th order covariant derivatives. The demonstration was simplified by A. Karlhede using the tetrad formalism so that only seventh order covariant derivatives need be computed. however, in many common cases, the first or second-order derivatives are sufficient to answer the question.
The metric defined on a nonempty set by(1)(2)if for all .It follows that the open ball of radius and center at (3)is(4)The metric topology induced is the discrete topology.
A distance on a set that fulfils the same properties as a metric except relaxes the definition to allow the distance between two different points to be zero.An example of pseudometric on the set of all functions is defined by . It is nonnegative, symmetric, fulfils the triangle inequality and the condition , but it is also true that .
Given a metric , the discriminant is defined by(1)(2)(3)Let be the discriminant and the transformed discriminant, then(4)(5)where(6)(7)
A nonnegative function describing the "distance" between neighboring points for a given set. A metric satisfies the triangle inequality(1)and is symmetric, so(2)A metric also satisfies(3)as well as the condition that implies . If this latter condition is dropped, then is called a pseudometric instead of a metric.A set possessing a metric is called a metric space. When viewed as a tensor, the metric is called a metric tensor.
The geodesics in a complete Riemannian metric go on indefinitely, i.e., each geodesic is isometric to the real line. For example, Euclidean space is complete, but the open unit disk is not complete since any geodesic ends at a finite distance. Whether or not a manifold is complete depends on the metric.For instance, the punctured plane is not complete with the usual metric. However, with the Riemannian Metric , the punctured plane is the infinite (flat) cylinder, which is complete. The figure above illustrates a geodesic which can only go a finite distance because it reaches a hole in the punctured plane, exemplifying that the punctured plane with the usual metric is not complete. The path is a geodesic parametrized by arc length.Any metric on a compact manifold is complete. Consequently, the pullback metric on the universal cover of a compact manifold is complete...
The interior product is a dual notion of the wedge product in an exterior algebra , where is a vector space. Given an orthonormal basis of , the forms(1)are an orthonormal basis for . They define a metric on the exterior algebra, . The interior product with a form is the adjoint of the wedge product with . That is,(2)for all . For example,(3)and(4)where the are orthonormal, are two interior products.An inner product on gives an isomorphism with the dual vector space . The interior product is the composition of this isomorphism with tensor contraction.
A geodesic on a paraboloid(1)(2)(3)has differential parameters defined by(4)(5)(6)(7)(8)(9)The geodesic is then given by solving the Euler-Lagrangedifferential equation(10)As given by Weinstock (1974), the solution simplifies to(11)
The great sphere on the surface of a hypersphere is the three-dimensional analog of the great circle on the surface of a sphere. Let be the number of reflecting spheres, and let great spheres divide a hypersphere into four-dimensional tetrahedra. Then for the polytope with Schläfli symbol ,
An ellipsoid can be specified parametrically by(1)(2)(3)The geodesic parameters are then(4)(5)(6)When the coordinates of a point are on the quadric(7)and expressed in terms of the parameters and of the confocal quadrics passing through that point (in other words, having , , , and , , for the squares of their semimajor axes), then the equation of a geodesic can be expressed in the form(8)with an arbitrary constant, and the arc length element is given by(9)where upper and lower signs are taken together.
The comass of a differential p-form is the largest value of on a vector of -volume one,
The Christoffel symbols are tensor-like objects derived from a Riemannian metric . They are used to study the geometry of the metric and appear, for example, in the geodesic equation. There are two closely related kinds of Christoffel symbols, the first kind , and the second kind . Christoffel symbols of the second kind are also known as affine connections (Weinberg 1972, p. 71) or connection coefficients (Misner et al. 1973, p. 210).It is always possible to pick a coordinate system on a Riemannian manifold such that the Christoffel symbol vanishes at a chosen point. In general relativity, Christoffel symbols are "gravitational forces," and the preferred coordinate system referred to above would be one attached to a body in free fall.
where is the torsion, is the curvature, is the tangent vector, and is the binormal vector.
A symplectic form on a smooth manifold is a smooth closed 2-form on which is nondegenerate such that at every point , the alternating bilinear form on the tangent space is nondegenerate.A symplectic form on a vector space over is a function (defined for all and taking values in ) which satisfies(1)(2)and(3) is called non-degenerate if for all implies that . Symplectic forms can exist on (or ) only if (or ) is even-dimensional. An example of a symplectic form over a vector space is the complex Hilbert space with inner product given by(4)
A calibration form on a Riemannian manifold is a differential p-form such that 1. is a closed form. 2. The comass of ,(1)defined as the largest value of on a vector of -volume one, equals 1. A -dimensional submanifold is calibrated when restricts to give the volume form.It is not hard to see that a calibrated submanifold minimizes its volume among objects in its homology class. By Stokes' theorem, if represents the same homology class, then(2)Since(3)and(4)it follows that the volume of is less than or equal to the volume of .A simple example is on the plane, for which the lines are calibrated submanifolds. In fact, in this example, the calibrated submanifolds give a foliation. On a Kähler manifold, the Kähler form is a calibration form, which is indecomposable. For example, on(5)the Kähler form is(6)On a Kähler manifold, the calibrated submanifolds are precisely the complex submanifolds. Consequently, the complex submanifolds..
A map between the symplectic manifolds and which is a diffeomorphism and , where is the pullback map induced by (i.e., the derivative of the diffeomorphism acting on tangent vectors). A symplectic diffeomorphism is also known as a symplectomorphism or canonical transformation.
On a Riemannian manifold , tangent vectors can be moved along a path by parallel transport, which preserves vector addition and scalar multiplication. So a closed loop at a base point , gives rise to a invertible linear map of , the tangent vectors at . It is possible to compose closed loops by following one after the other, and to invert them by going backwards. Hence, the set of linear transformations arising from parallel transport along closed loops is a group, called the holonomy group.Since parallel transport preserves the Riemannian metric, the holonomy group is contained in the orthogonal group . Moreover, if the manifold is orientable, then it is contained in the special orthogonal group. A generic Riemannian metric on an orientable manifold has holonomy group , but for some special metrics it can be a subgroup, in which case the manifold is said to have special holonomy.A Kähler manifold is a -dimensional manifold whose holonomy lies..
A Hermitian metric on a complex vector bundle assigns a Hermitian inner product to every fiber bundle. The basic example is the trivial bundle , where is an open set in . Then a positive definite Hermitian matrix defines a Hermitian metric bywhere is the complex conjugate of . By a partition of unity, any complex vector bundle has a Hermitian metric.In the special case of a complex manifold, the complexified tangent bundle may have a Hermitian metric, in which case its real part is a Riemannian metric and its imaginary part is a nondegenerate alternating multilinear form . When is closed, i.e., in this case a symplectic form, then is a Kähler form.On a holomorphic vector bundle with a Hermitian metric , there is a unique connection compatible with and the complex structure. Namely, it must be , where in a trivialization...
For a curve with first fundamental form(1)the Gaussian curvature is(2)where(3)(4)For a patch where , the Gaussian curvature is given by(5)(6)
Gauge theory studies principal bundle connections, called gauge fields, on a principal bundle. These connections correspond to fields, in physics, such as an electromagnetic field, and the Lie group of the principal bundle corresponds to the symmetries of the physical system. The base manifold to the principal bundle is usually a four-dimensional manifold which corresponds to space-time. In the case of an electromagnetic field, the symmetry group is the unitary group . The other two groups that arise in physical theories are the special unitary groups and . Also, a group representation of the symmetry group, called internal space, gives rise to an associated vector bundle.Actually,the principal bundle connections which minimize an energy functional are the only ones of physical interest. For example, the Yang-Mills connections minimize the Yang-Mills functional. These connections are useful in low-dimensional topology. In fact,..
A ruled surface is said to be a binormal developable of a curve if can be parameterized by , where is the binormal vector.
An asymptotic direction at a point of a regular surface is a direction in which the normal curvature of vanishes. 1. There are no asymptotic directions at an ellipticpoint. 2. There are exactly two asymptotic directions at a hyperbolicpoint. 3. There is exactly one asymptotic direction at aparabolic point. 4. Every direction is asymptotic at a planar point.
The distribution parameter of a noncylindricalruled surface parameterized by(1)where is the striction curve and the director curve, is the function defined by(2)The Gaussian curvature of a ruledsurface is given in terms of its distribution parameter by(3)
Given a regular surface , an asymptotic curve is formally defined as a curve on such that the normal curvature is 0 in the direction for all in the domain of . The differential equation for the parametric representation of an asymptotic curve is(1)where , , and are coefficients of the second fundamental form. The differential equation for asymptotic curves on a Monge patch is(2)and on a polar patch is(3)The images below show asymptotic curves for the elliptic helicoid, funnel, hyperbolic paraboloid, and monkey saddle.
The area element for a surface with firstfundamental formiswhere is the wedge product.
A differential ideal on a manifold is an ideal in the exterior algebra of differential k-forms on which is also closed under the exterior derivative . That is, for any differential -form and any form , then 1. , and 2. For example, is a differential ideal on .A smooth map is called an integral of if the pullback map of all forms in vanish on , i.e., .
The angular velocity is the time derivative of the angular distance with direction perpendicular to the plane of angular motion,
A metric space is isometric to a metric space if there is a bijection between and that preserves distances. That is, . In the context of Riemannian geometry, two manifolds and are isometric if there is a diffeomorphism such that the Riemannian metric from one pulls back to the metric on the other. Since the geodesics define a distance, a Riemannian metric makes the manifold a metric space. An isometry between Riemannian manifolds is also an isometry between the two manifolds, considered as metric spaces.Isometric spaces are considered isomorphic. For instance, the circle of radius one around the origin is isometric to the circle of radius one around .
The complex structure of a point in the plane is defined by the linear map (1)and corresponds to a counterclockwise rotation by . This map satisfies(2)(3)(4)where is the identity map.More generally, if is a two-dimensional vector space, a linear map such that is called a complex structure on . If , this collapses to the previous definition.
Let a particle travel a distance as a function of time (here, can be thought of as the arc length of the curve traced out by the particle). The speed (the scalar norm of the vector velocity) is then given by(1)The acceleration is defined as the time derivative of the velocity, so the scalar acceleration is given by(2)(3)(4)(5)(6)The vector acceleration is given by(7)(8)(9)where is the unit tangent vector, the curvature, the arc length, and the unit normal vector.Let a particle move along a straight line so that the positions at times , , and are , , and , respectively. Then the particle is uniformly accelerated with acceleration iff(10)is a constant (Klamkin 1995, 1996).Consider the measurement of acceleration in a rotating reference frame. Apply therotation operator(11)twice to the radius vector and suppress the body notation,(12)(13)(14)(15)(16)Grouping terms and using the definitions of the velocity and angular velocity gives the expression(17)Now,..
A subset is called a regular surface if for each point , there exists a neighborhood of in and a map of an open set onto such that 1. is differentiable, 2. is a homeomorphism, and 3. Each map is a regular patch. Any open subset of a regular surface is also a regular surface.
A regular patch is a patch for which the Jacobian has rank 2 for all . A patch is said to be regular at a point provided that its Jacobian has rank 2 at . For example, the points at in the standard parameterization of the sphere are not regular.An example of a patch which is regular but not injective is the cylinder defined parametrically by with and . However, if is an injective regular patch, then maps diffeomorphically onto .
Given a geodesic triangle (a triangle formedby the arcs of three geodesics on a smooth surface),Given the Euler characteristic ,so the integral curvature of a closed surface is not altered by a topological transformation.
A parameterization of a surface in and is regular if the tangent vectorsare always linearly independent.
An injective patch is a patch such that implies that and . An example of a patch which is injective but not regular is the function defined by for . However, if is an injective regular patch, then maps diffeomorphically onto .
The Weingarten equations express the derivatives of the normal vector to a surface using derivatives of the position vector. Let be a regular patch, then the shape operator of is given in terms of the basis by(1)(2)where is the normal vector, , , and the coefficients of the first fundamental form(3)and , , and the coefficients of the second fundamental form given by(4)(5)(6)(7)(8)(9)(10)(11)
A tangent vector is a principal vector iffwhere , , and are coefficients of the first fundamental form and , , of the second fundamental form.
For a curve with radius vector , the unit tangent vector is defined by(1)(2)(3)where is a parameterization variable, is the arc length, and an overdot denotes a derivative with respect to , . For a function given parametrically by , the tangent vector relative to the point is therefore given by(4)(5)To actually place the vector tangent to the curve, it must be displaced by . It is also true that(6)(7)(8)where is the normal vector, is the curvature, is the torsion, and is the scalar triple product.
In general, there are two important types of curvature: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction.After the curvature of two- and three-dimensional curves was studied, attention turned to the curvature of surfaces in three-space. The main curvatures that emerged from this scrutiny are the mean curvature, Gaussian curvature, and the shape operator. Mean curvature was the most important for applications at the time and was the most studied, but Gauss was the first to recognize the importance of the Gaussian curvature.Because Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface,..
Arc length is defined as the length along a curve,(1)where is a differential displacement vector along a curve . For example, for a circle of radius , the arc length between two points with angles and (measured in radians) is simply(2)Defining the line element , parameterizing the curve in terms of a parameter , and noting that is simply the magnitude of the velocity with which the end of the radius vector moves gives(3)In polar coordinates,(4)so(5)(6)In Cartesian coordinates,(7)(8)(9)(10)Therefore, if the curve is written(11)then(12)If the curve is instead written(13)then(14)In three dimensions,(15)so(16)The arc length of the polar curve is given by(17)
An inflection point is a point on a curve at which the sign of the curvature (i.e., the concavity) changes. Inflection points may be stationary points, but are not local maxima or local minima. For example, for the curve plotted above, the point is an inflection point.The first derivative test can sometimes distinguish inflection points from extrema for differentiable functions .The second derivative test is also useful. A necessary condition for to be an inflection point is . A sufficient condition requires and to have opposite signs in the neighborhood of (Bronshtein and Semendyayev 2004, p. 231).
Minkowski space is a four-dimensional space possessing a Minkowskimetric, i.e., a metric tensor having the formAlternatively (though less desirably), Minkowski space can be considered to have a Euclidean metric with imaginary time coordinate where is the speed of light (by convention is normally used) and where i is the imaginary number . Minkowski space unifies Euclidean three-space plus time (the "fourth dimension") in Einstein's theory of special relativity.In equation (5) above, the metric signature is assumed; under this assumption, Minkowski space is typically written . One may also express equation (5) with respect to the metric signature by reversing the order of the positive and negative squared terms therein, in which case Minkowski space is denoted .The Minkowski metric induces an inner product, the four-dimensional Lorentzian inner product (sometimes referred to as the Minkowski inner product), which fails..
A homogeneous space is a space with a transitive group action by a Lie group. Because a transitive group action implies that there is only one group orbit, is isomorphic to the quotient space where is the isotropy group . The choice of does not affect the isomorphism type of because all of the isotropy groups are conjugate.Many common spaces are homogeneous spaces, such as the hypersphere,(1)and the complex projective space(2)The real Grassmannian of -dimensional subspaces in is(3)The projection makes a principal bundle on with fiber . For example, is a bundle, i.e., a circle bundle, on the sphere. The subgroup(4)acts on the right, and does not affect the first column so is well-defined.
There does not exist an everywhere nonzero tangent vector field on the 2-sphere . This implies that somewhere on the surface of the Earth, there is a point with zero horizontal wind velocity. The theorem can be generalized to the statement that the -sphere has a nonzero tangent vector field iff is odd.
The term perimeter refers either to the curve constituting the boundary of a laminaor else to the length of this boundary.The perimeter of a circle is called the circumference, although that term is used by some authors to refer to the perimeter of an arbitrary curved geometric figure.The perimeter of a region is implemented in the WolframLanguage as Perimeter[reg].The perimeters of some common laminae are summarized in the table below. In the table, is the eccentricity of an ellipse, is its semimajor axis, and is a complete elliptic integral of the second kind.laminaperimetercircleellipserectanglesquaretriangle
A ket is a vector living in a dual vector space to that containing bras . Bras and kets are commonly encountered in quantum mechanics. Bras and kets can be considered as 1-vectors and 1-forms (or vice versa), although this is almost always done only in a finite-dimensional vector space.Considered as an inner product, the bra and ket form an angle bracket (bra+ket = bracket) .