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In particular, the length of a tangent vector is given by |
and the angle between two vectors and is calculated by |
The surface area is another numerical quantity which should depend only on the surface itself, and not on how it is parameterized. If the surface is parameterized by the function over the domain in the -plane, then the surface area of is given by the integral |
where denotes the cross product, and the absolute value denotes the length of a vector in Euclidean space. By Lagrange's identity for the cross product, the integral can be written |
Let be a smooth manifold of dimension ; for instance a surface (in the case ) or hypersurface in the Cartesian space . At each point there is a vector space , called the tangent space, consisting of all tangent vectors to the manifold at the point . A metric tensor at is a function which takes as inputs a pair of tangent vectors and at , and produces as an output a real number (scalar), so that the following conditions are satisfied: |
A metric tensor field on assigns to each point of a metric tensor in the tangent space at in a way that varies smoothly with . More precisely, given any open subset of manifold and any (smooth) vector fields and on , the real function |
The components of the metric in any basis of vector fields, or frame, are given by |
The functions form the entries of an symmetric matrix, . If |
are two vectors at , then the value of the metric applied to and is determined by the coefficients () by bilinearity: |
Denoting the matrix by and arranging the components of the vectors and into column vectors and , |
where T and T denote the transpose of the vectors and , respectively. Under a change of basis of the form |
for some invertible matrix , the matrix of components of the metric changes by as well. That is, |
or, in terms of the entries of this matrix, |
For this reason, the system of quantities is said to transform covariantly with respect to changes in the frame . |
A system of real-valued functions , giving a local coordinate system on an open set in , determines a basis of vector fields on |
The metric has components relative to this frame given by |
Relative to a new system of local coordinates, say |
the metric tensor will determine a different matrix of coefficients, |
This new system of functions is related to the original by means of the chain rule |
Or, in terms of the matrices and , |
where denotes the Jacobian matrix of the coordinate change. |
Associated to any metric tensor is the quadratic form defined in each tangent space by |
If is positive for all non-zero , then the metric is positive-definite at . If the metric is positive-definite at every , then is called a Riemannian metric. More generally, if the quadratic forms have constant signature independent of , then the signature of is this signature, and is called a pseudo-Riemannian metric. If is connected, then the signature of does not depend on . |
By Sylvester's law of inertia, a basis of tangent vectors can be chosen locally so that the quadratic form diagonalizes in the following manner |
for some between 1 and . Any two such expressions of (at the same point of ) will have the same number of positive signs. The signature of is the pair of integers , signifying that there are positive signs and negative signs in any such expression. Equivalently, the metric has signature if the matrix of the metric has positive and negative eigenvalues. |
Certain metric signatures which arise frequently in applications are: |
Let be a basis of vector fields, and as above let be the matrix of coefficients |
One can consider the inverse matrix , which is identified with the inverse metric (or "conjugate" or "dual metric"). The inverse metric satisfies a transformation law when the frame is changed by a matrix via |
The inverse metric transforms "contravariantly", or with respect to the inverse of the change of basis matrix . Whereas the metric itself provides a way to measure the length of (or angle between) vector fields, the inverse metric supplies a means of measuring the length of (or angle between) covector fields; that is, fields of linear functionals. |
To see this, suppose that is a covector field. To wit, for each point , determines a function defined on tangent vectors at so that the following linearity condition holds for all tangent vectors and , and all real numbers and : |
As varies, is assumed to be a smooth function in the sense that |
is a smooth function of for any smooth vector field . |
Any covector field has components in the basis of vector fields . These are determined by |
Denote the row vector of these components by |
Under a change of by a matrix , changes by the rule |
That is, the row vector of components transforms as a "covariant" vector. |
For a pair and of covector fields, define the inverse metric applied to these two covectors by |
The resulting definition, although it involves the choice of basis , does not actually depend on in an essential way. Indeed, changing basis to gives |
So that the right-hand side of equation () is unaffected by changing the basis to any other basis whatsoever. Consequently, the equation may be assigned a meaning independently of the choice of basis. The entries of the matrix are denoted by , where the indices and have been raised to indicate the transformation law (). |
In a basis of vector fields , any smooth tangent vector field can be written in the form |
for some uniquely determined smooth functions . Upon changing the basis by a nonsingular matrix , the coefficients change in such a way that equation () remains true. That is, |
Consequently, . In other words, the components of a vector transform "contravariantly" (that is, inversely or in the opposite way) under a change of basis by the nonsingular matrix . The contravariance of the components of is notationally designated by placing the indices of in the upper position. |
A frame also allows covectors to be expressed in terms of their components. For the basis of vector fields define the dual basis to be the linear functionals such that |
That is, , the Kronecker delta. Let |
Under a change of basis for a nonsingular matrix , transforms via |
Any linear functional on tangent vectors can be expanded in terms of the dual basis |
where denotes the row vector . The components transform when the basis is replaced by in such a way that equation () continues to hold. That is, |
whence, because , it follows that . That is, the components transform "covariantly" (by the matrix rather than its inverse). The covariance of the components of is notationally designated by placing the indices of in the lower position. |
Now, the metric tensor gives a means to identify vectors and covectors as follows. Holding fixed, the function |
of tangent vector defines a linear functional on the tangent space at . This operation takes a vector at a point and produces a covector . In a basis of vector fields , if a vector field has components , then the components of the covector field in the dual basis are given by the entries of the row vector |
Under a change of basis , the right-hand side of this equation transforms via |
so that : transforms covariantly. The operation of associating to the (contravariant) components of a vector field T the (covariant) components of the covector field , where |
To "raise the index", one applies the same construction but with the inverse metric instead of the metric. If are the components of a covector in the dual basis , then the column vector |
Consequently, the quantity does not depend on the choice of basis in an essential way, and thus defines a vector field on . The operation () associating to the (covariant) components of a covector the (contravariant) components of a vector given is called raising the index. In components, () is |
Let be an open set in , and let be a continuously differentiable function from into the Euclidean space , where . The mapping is called an immersion if its differential is injective at every point of . The image of is called an immersed submanifold. More specifically, for , which means that the ambient Euclidean space is , the induced metric tensor is called the first fundamental form. |
Suppose that is an immersion onto the submanifold . The usual Euclidean dot product in is a metric which, when restricted to vectors tangent to , gives a means for taking the dot product of these tangent vectors. This is called the induced metric. |
Suppose that is a tangent vector at a point of , say |
where are the standard coordinate vectors in . When is applied to , the vector goes over to the vector tangent to given by |
(This is called the pushforward of along .) Given two such vectors, and , the induced metric is defined by |
It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields is given by |
The notion of a metric can be defined intrinsically using the language of fiber bundles and vector bundles. In these terms, a metric tensor is a function |
from the fiber product of the tangent bundle of with itself to such that the restriction of to each fiber is a nondegenerate bilinear mapping |
The mapping () is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether can support such a structure. |
Metric as a section of a bundle. |
By the universal property of the tensor product, any bilinear mapping () gives rise naturally to a section of the dual of the tensor product bundle of with itself |
The section is defined on simple elements of by |
and is defined on arbitrary elements of by extending linearly to linear combinations of simple elements. The original bilinear form is symmetric if and only if |
Since is finite-dimensional, there is a natural isomorphism |
so that is regarded also as a section of the bundle of the cotangent bundle with itself. Since is symmetric as a bilinear mapping, it follows that is a symmetric tensor. |
More generally, one may speak of a metric in a vector bundle. If is a vector bundle over a manifold , then a metric is a mapping |
from the fiber product of to which is bilinear in each fiber: |
Using duality as above, a metric is often identified with a section of the tensor product bundle . (See metric (vector bundle).) |
The metric tensor gives a natural isomorphism from the tangent bundle to the cotangent bundle, sometimes called the musical isomorphism. This isomorphism is obtained by setting, for each tangent vector , |
the linear functional on which sends a tangent vector at to . That is, in terms of the pairing between and its dual space , |
for all tangent vectors and . The mapping is a linear transformation from to . It follows from the definition of non-degeneracy that the kernel of is reduced to zero, and so by the rank–nullity theorem, is a linear isomorphism. Furthermore, is a symmetric linear transformation in the sense that |
Conversely, any linear isomorphism defines a non-degenerate bilinear form on by means of |
This bilinear form is symmetric if and only if is symmetric. There is thus a natural one-to-one correspondence between symmetric bilinear forms on and symmetric linear isomorphisms of to the dual . |
As varies over , defines a section of the bundle of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same smoothness as : it is continuous, differentiable, smooth, or real-analytic according as . The mapping , which associates to every vector field on a covector field on gives an abstract formulation of "lowering the index" on a vector field. The inverse of is a mapping which, analogously, gives an abstract formulation of "raising the index" on a covector field. |
which is nonsingular and symmetric in the sense that |
for all covectors , . Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map |
or by the double dual isomorphism to a section of the tensor product |
Suppose that is a Riemannian metric on . In a local coordinate system , , the metric tensor appears as a matrix, denoted here by , whose entries are the components of the metric tensor relative to the coordinate vector fields. |
Let be a piecewise-differentiable parametric curve in , for . The arclength of the curve is defined by |
In connection with this geometrical application, the quadratic differential form |
is called the first fundamental form associated to the metric, while is the line element. When is pulled back to the image of a curve in , it represents the square of the differential with respect to arclength. |
For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. In this case, define |
Note that, while these formulas use coordinate expressions, they are in fact independent of the coordinates chosen; they depend only on the metric, and the curve along which the formula is integrated. |
Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve: |
This usage comes from physics, specifically, classical mechanics, where the integral can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of Maupertuis' principle, the metric tensor can be seen to correspond to the mass tensor of a moving particle. |
In many cases, whenever a calculation calls for the length to be used, a similar calculation using the energy may be done as well. This often leads to simpler formulas by avoiding the need for the square-root. Thus, for example, the geodesic equations may be obtained by applying variational principles to either the length or the energy. In the latter case, the geodesic equations are seen to arise from the principle of least action: they describe the motion of a "free particle" (a particle feeling no forces) that is confined to move on the manifold, but otherwise moves freely, with constant momentum, within the manifold. |
In analogy with the case of surfaces, a metric tensor on an -dimensional paracompact manifold gives rise to a natural way to measure the -dimensional volume of subsets of the manifold. The resulting natural positive Borel measure allows one to develop a theory of integrating functions on the manifold by means of the associated Lebesgue integral. |
A measure can be defined, by the Riesz representation theorem, by giving a positive linear functional on the space of compactly supported continuous functions on . More precisely, if is a manifold with a (pseudo-)Riemannian metric tensor , then there is a unique positive Borel measure such that for any coordinate chart , |
for all supported in . Here is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart. That is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables. It extends to a unique positive linear functional on by means of a partition of unity. |
If is also oriented, then it is possible to define a natural volume form from the metric tensor. In a positively oriented coordinate system the volume form is represented as |
where the are the coordinate differentials and denotes the exterior product in the algebra of differential forms. The volume form also gives a way to integrate functions on the manifold, and this geometric integral agrees with the integral obtained by the canonical Borel measure. |
The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean metric tensor. In the usual coordinates, we can write |
The length of a curve reduces to the formula: |
The Euclidean metric in some other common coordinate systems can be written as follows. |
In general, in a Cartesian coordinate system on a Euclidean space, the partial derivatives are orthonormal with respect to the Euclidean metric. Thus the metric tensor is the Kronecker delta δ"ij" in this coordinate system. The metric tensor with respect to arbitrary (possibly curvilinear) coordinates is given by |
The unit sphere in comes equipped with a natural metric induced from the ambient Euclidean metric, through the process explained in the induced metric section. In standard spherical coordinates , with the colatitude, the angle measured from the -axis, and the angle from the -axis in the -plane, the metric takes the form |
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