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In the mathematical field of
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, a metric tensor (or simply metric) is an additional
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
(such as a surface) that allows defining distances and angles, just as the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a
bilinear function In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
that maps pairs of
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
s to
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
between and can be defined as the infimum of the lengths of all such curves; this makes a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
. Conversely, the metric tensor itself is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the distance function (taken in a suitable manner). While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
were understood by, in particular, Gregorio Ricci-Curbastro and
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made signi ...
, who first codified the notion of a tensor. The metric tensor is an example of a
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
. The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.


Introduction

Carl Friedrich Gauss in his 1827 '' Disquisitiones generales circa superficies curvas'' (''General investigations of curved surfaces'') considered a surface parametrically, with the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, , and of points on the surface depending on two auxiliary variables and . Thus a parametric surface is (in today's terms) a vector-valued function :\vec(u,\,v) = \bigl( x(u,\,v),\, y(u,\,v),\, z(u,\,v) \bigr) depending on an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
of real variables , and defined in an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
in the -plane. One of the chief aims of Gauss's investigations was to deduce those features of the surface which could be described by a function which would remain unchanged if the surface underwent a transformation in space (such as bending the surface without stretching it), or a change in the particular parametric form of the same geometrical surface. One natural such invariant quantity is the length of a curve drawn along the surface. Another is the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
between a pair of curves drawn along the surface and meeting at a common point. A third such quantity is the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...
of a piece of the surface. The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor. The metric tensor is \begin E & F \\ F & G \end in the description below; E, F, and G in the matrix can contain any number as long as the matrix is positive definite.


Arc length

If the variables and are taken to depend on a third variable, , taking values in an interval , then will trace out a parametric curve in parametric surface . The
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
of that curve is given by the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
: \begin s &= \int_a^b\left\, \frac\vec(u(t),v(t))\right\, \,dt \\ pt &= \int_a^b \sqrt\, dt \,, \end where \left\, \cdot \right\, represents the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. Here the chain rule has been applied, and the subscripts denote
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s: :\vec_u = \frac\,, \quad \vec_v = \frac\,. The integrand is the restriction to the curve of the square root of the ( quadratic) differential where The quantity in () is called the line element, while is called the first fundamental form of . Intuitively, it represents the
principal part In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function. Laurent series definition The principal part at z=a of a function : f(z) = \sum_^\infty a_ ...
of the square of the displacement undergone by when is increased by units, and is increased by units. Using matrix notation, the first fundamental form becomes :ds^2 = \begin du & dv \end \begin E & F \\ F & G \end \begin du \\ dv \end


Coordinate transformations

Suppose now that a different parameterization is selected, by allowing and to depend on another pair of variables and . Then the analog of () for the new variables is The chain rule relates , , and to , , and via the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
equation where the superscript T denotes the matrix transpose. The matrix with the coefficients , , and arranged in this way therefore transforms by the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of the coordinate change : J = \begin \frac & \frac \\ \frac & \frac \end\,. A matrix which transforms in this way is one kind of what is called a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
. The matrix :\begin E & F \\ F & G \end with the transformation law () is known as the metric tensor of the surface.


Invariance of arclength under coordinate transformations

first observed the significance of a system of coefficients , , and , that transformed in this way on passing from one system of coordinates to another. The upshot is that the first fundamental form () is ''invariant'' under changes in the coordinate system, and that this follows exclusively from the transformation properties of , , and . Indeed, by the chain rule, :\begin du \\ dv \end = \begin \dfrac & \dfrac \\ \dfrac & \dfrac \end \begin du' \\ dv' \end so that :\begin ds^2 &= \begin du & dv \end \begin E & F \\ F & G \end \begin du \\ dv \end \\ pt &= \begin du' & dv' \end \begin \dfrac & \dfrac \\ pt \dfrac & \dfrac \end^\mathsf \begin E & F \\ F & G \end \begin \dfrac & \dfrac \\ pt \dfrac & \dfrac \end \begin du' \\ dv' \end \\ pt &= \begin du' & dv' \end \begin E' & F' \\ F' & G' \end \begin du' \\ dv' \end\\ pt &= (ds')^2 \,. \end


Length and angle

Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of
tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
s to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
(non-euclidean geometry) of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface can be written in the form :\mathbf = p_1\vec_u + p_2\vec_v for suitable real numbers and . If two tangent vectors are given: :\begin \mathbf &= a_1\vec_u + a_2\vec_v \\ \mathbf &= b_1\vec_u + b_2\vec_v \end then using the bilinearity of the dot product, :\begin \mathbf \cdot \mathbf &= a_1 b_1 \vec_u\cdot\vec_u + a_1b_2 \vec_u\cdot\vec_v + a_2b_1 \vec_v\cdot\vec_u + a_2 b_2 \vec_v\cdot\vec_v \\ pt &= a_1 b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G. \\ pt &= \begin a_1 & a_2 \end \begin E & F \\ F & G \end \begin b_1 \\ b_2 \end \,. \end This is plainly a function of the four variables , , , and . It is more profitably viewed, however, as a function that takes a pair of arguments and which are vectors in the -plane. That is, put :g(\mathbf, \mathbf) = a_1b_1 E + a_1b_2 F + a_2b_1 F + a_2b_2G \,. This is a symmetric function in and , meaning that :g(\mathbf, \mathbf) = g(\mathbf, \mathbf)\,. It is also bilinear, meaning that it is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in each variable and separately. That is, :\begin g\left(\lambda\mathbf + \mu\mathbf', \mathbf\right) &= \lambda g(\mathbf, \mathbf) + \mu g\left(\mathbf', \mathbf\right),\quad\text \\ g\left(\mathbf, \lambda\mathbf + \mu\mathbf'\right) &= \lambda g(\mathbf, \mathbf) + \mu g\left(\mathbf, \mathbf'\right) \end for any vectors , , , and in the plane, and any real numbers and . In particular, the length of a tangent vector is given by : \left\, \mathbf \right\, = \sqrt and the angle between two vectors and is calculated by :\cos(\theta) = \frac \,.


Area

The
surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...
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 :\iint_D \left, \vec_u \times \vec_v\\,du\,dv where denotes the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
, and the absolute value denotes the length of a vector in Euclidean space. By
Lagrange's identity In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is: \begin \left( \sum_^n a_k^2\right) \left(\sum_^n b_k^2\right) - \left(\sum_^n a_k b_k\right)^2 & = \sum_^ \sum_^n \left(a_i b_j - a_j b_i\right)^2 \\ & \left(= \frac \sum_^n ...
for the cross product, the integral can be written :\begin &\iint_D \sqrt\,du\,dv \\ pt = &\iint_D \sqrt\,du\,dv\\ pt = &\iint_D \sqrt\, du\, dv \end where is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
.


Definition

Let be a smooth manifold of dimension ; for instance a surface (in the case ) or hypersurface in the Cartesian space \R^. At each point there is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
, 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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
( scalar), so that the following conditions are satisfied: * is bilinear. A function of two vector arguments is bilinear if it is linear separately in each argument. Thus if , , are three tangent vectors at and and are real numbers, then \begin g_p(aU_p + bV_p, Y_p) &= ag_p(U_p, Y_p) + bg_p(V_p, Y_p) \,, \quad \text \\ g_p(Y_p, aU_p + bV_p) &= ag_p(Y_p, U_p) + bg_p(Y_p, V_p) \,. \end * is symmetric. A function of two vector arguments is symmetric provided that for all vectors and , g_p(X_p, Y_p) = g_p(Y_p, X_p)\,. * is nondegenerate. A bilinear function is nondegenerate provided that, for every tangent vector , the function Y_p \mapsto g_p(X_p,Y_p) obtained by holding constant and allowing to vary is not identically zero. That is, for every there exists a such that . 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 g(X, Y)(p) = g_p(X_p, Y_p) is a smooth function of .


Components of the metric

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 :v = \sum_^n v^iX_i \,, \quad w = \sum_^n w^iX_i are two vectors at , then the value of the metric applied to and is determined by the coefficients () by bilinearity: :g(v, w) = \sum_^n v^iw^jg\left(X_i,X_j\right) = \sum_^n v^iw^jg_ mathbf/math> Denoting the
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
by and arranging the components of the vectors and into column vectors and , :g(v,w) = \mathbf mathbf\mathsf G mathbf\mathbf mathbf= \mathbf mathbf\mathsf G mathbfmathbf mathbf/math> where T and T denote the transpose of the vectors and , respectively. Under a change of basis of the form :\mathbf\mapsto \mathbf' = \left(\sum_k X_ka_,\dots,\sum_k X_ka_\right) = \mathbfA for some invertible matrix , the matrix of components of the metric changes by as well. That is, :G mathbfA= A^\mathsf G mathbf or, in terms of the entries of this matrix, :g_ mathbfA= \sum_^n a_g_ mathbf_ \, . For this reason, the system of quantities is said to transform covariantly with respect to changes in the frame .


Metric in coordinates

A system of real-valued functions , giving a local coordinate system on an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
in , determines a basis of vector fields on :\mathbf = \left(X_1 = \frac, \dots, X_n = \frac\right) \,. The metric has components relative to this frame given by :g_\left mathbf\right= g\left(\frac, \frac\right) \,. Relative to a new system of local coordinates, say :y^i = y^i(x^1, x^2, \dots, x^n),\quad i=1,2,\dots,n the metric tensor will determine a different matrix of coefficients, :g_\left mathbf'\right= g\left(\frac, \frac\right). This new system of functions is related to the original by means of the chain rule :\frac = \sum_^n \frac\frac so that :g_\left mathbf'\right= \sum_^n \frac g_\left mathbf\rightfrac. Or, in terms of the matrices and , :G\left mathbf'\right= \left((Dy)^\right)^\mathsf G\left mathbf\right(Dy)^ where denotes the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
of the coordinate change.


Signature of a metric

Associated to any metric tensor is the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
defined in each tangent space by :q_m(X_m) = g_m(X_m,X_m) \,, \quad X_m\in T_mM. 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 A signature (; from la, signare, "to sign") is a Handwriting, handwritten (and often Stylization, stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and ...
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 :q_m\left(\sum_i\xi^iX_i\right) = \left(\xi^1\right)^2+\left(\xi^2\right)^2+\cdots+\left(\xi^p\right)^2 - \left(\xi^\right)^2-\cdots-\left(\xi^n\right)^2 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
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s. Certain metric signatures which arise frequently in applications are: * If has signature , then is a Riemannian metric, and is called a Riemannian manifold. Otherwise, is a pseudo-Riemannian metric, and is called a pseudo-Riemannian manifold (the term semi-Riemannian is also used). * If is four-dimensional with signature or , then the metric is called Lorentzian. More generally, a metric tensor in dimension other than 4 of signature or is sometimes also called Lorentzian. * If is -dimensional and has signature , then the metric is called ultrahyperbolic.


Inverse metric

Let be a basis of vector fields, and as above let be the matrix of coefficients :g_ mathbf= g\left(X_i,X_j\right) \,. 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 In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
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 : :\alpha_p \left(aX_p + bY_p\right) = a\alpha_p \left(X_p\right) + b\alpha_p \left(Y_p\right)\,. As varies, is assumed to be a
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
in the sense that :p \mapsto \alpha_p \left(X_p\right) 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 :\alpha_i = \alpha \left(X_i\right)\,,\quad i = 1, 2, \dots, n\,. Denote the row vector of these components by :\alpha mathbf= \big\lbrack\begin \alpha_1 & \alpha_2 & \dots & \alpha_n \end\big\rbrack \,. Under a change of by a matrix , changes by the rule :\alpha mathbfA= \alpha mathbf \,. 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 :\begin &\alpha mathbfAG mathbfA \beta mathbfA\mathsf \\ = &\left(\alpha mathbf\right) \left(A^G mathbf \left(A^\right)^\mathsf\right) \left(A^\mathsf\beta mathbf\mathsf\right) \\ = &\alpha mathbfG mathbf \beta mathbf\mathsf. \end 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 ().


Raising and lowering indices

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, :X = \mathbfv mathbf= \mathbfv mathbf,. 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 :\theta^i mathbfX_j) = \begin 1 & \mathrm\ i=j\\ 0&\mathrm\ i\not=j.\end That is, , the Kronecker delta. Let :\theta mathbf= \begin\theta^1 mathbf\\ \theta^2 mathbf\\ \vdots \\ \theta^n mathbfend. Under a change of basis for a nonsingular matrix , transforms via :\theta mathbfA= A^\theta mathbf 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, :\alpha = a mathbfAtheta mathbfA= a mathbftheta mathbf/math> 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 :g_p(X_p, -) : Y_p \mapsto g_p(X_p, Y_p) 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 :a mathbf= v mathbf\mathsf G mathbf Under a change of basis , the right-hand side of this equation transforms via : v mathbfA\mathsf G mathbfA= v mathbf\mathsf \left(A^\right)^\mathsf A^\mathsf G mathbf = v mathbf\mathsf G mathbf 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 :a_i mathbf= \sum_^n v^k mathbf_ mathbf/math> is called lowering the index. 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 has components which transform contravariantly: :v mathbfA= A^v mathbf 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 :v^i mathbf= \sum_^n g^ mathbfa_k mathbf


Induced metric

Let be an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
in , and let be a
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
function from into the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, where . The mapping is called an immersion if its differential is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
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 mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
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 :v = v^1\mathbf_1 + \dots + v^n\mathbf_n where are the standard coordinate vectors in . When is applied to , the vector goes over to the vector tangent to given by :\varphi_*(v) = \sum_^n \sum_^m v^i\frac\mathbf_a\,. (This is called the pushforward of along .) Given two such vectors, and , the induced metric is defined by :g(v,w) = \varphi_*(v)\cdot \varphi_*(w). It follows from a straightforward calculation that the matrix of the induced metric in the basis of coordinate vector fields is given by :G(\mathbf) = (D\varphi)^\mathsf(D\varphi) where is the Jacobian matrix: :D\varphi = \begin \frac & \frac & \dots & \frac \\ ex \frac & \frac & \dots & \frac \\ \vdots & \vdots & \ddots & \vdots \\ \frac & \frac & \dots & \frac \end.


Intrinsic definitions of a metric

The notion of a metric can be defined intrinsically using the language of fiber bundles and
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s. 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 :g_p : \mathrm_pM\times \mathrm_pM \to \mathbf. The mapping () is required to be
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
, and often
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
, smooth, or
real analytic In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...
, 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 :g_\otimes \in \Gamma\left((\mathrmM \otimes \mathrmM)^*\right). The section is defined on simple elements of by :g_\otimes(v \otimes w) = g(v, w) 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 :g_\otimes \circ \tau = g_\otimes where :\tau : \mathrmM \otimes \mathrmM \stackrel TM \otimes TM is the braiding map. Since is finite-dimensional, there is a natural isomorphism :(\mathrmM \otimes \mathrmM)^* \cong \mathrm^*M \otimes \mathrm^*M, so that is regarded also as a section of the bundle of the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
with itself. Since is symmetric as a bilinear mapping, it follows that is a symmetric tensor.


Metric in a vector bundle

More generally, one may speak of a metric in a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
. If is a vector bundle over a manifold , then a metric is a mapping :g : E\times_M E\to \mathbf from the fiber product of to which is bilinear in each fiber: :g_p : E_p \times E_p\to \mathbf. Using duality as above, a metric is often identified with a section of the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
bundle . (See metric (vector bundle).)


Tangent–cotangent isomorphism

The metric tensor gives a natural isomorphism from the tangent bundle to the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
, sometimes called the
musical isomorphism In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a pseudo-Riemannian manifold induced b ...
. This isomorphism is obtained by setting, for each tangent vector , :S_gX_p\, \stackrel\text\, g(X_p, -), the linear functional on which sends a tangent vector at to . That is, in terms of the pairing between and its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
, : _gX_p, Y_p= g_p(X_p, Y_p) 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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
in the sense that : _gX_p, Y_p= _gY_p, X_p for all tangent vectors and . Conversely, any linear isomorphism defines a non-degenerate bilinear form on by means of :g_S(X_p, Y_p) = X_p, Y_p,. 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. The inverse defines a linear mapping :S_g^ : \mathrm^*M \to \mathrmM which is nonsingular and symmetric in the sense that :\left _g^\alpha, \beta\right= \left _g^\beta, \alpha\right/math> for all covectors , . Such a nonsingular symmetric mapping gives rise (by the tensor-hom adjunction) to a map :\mathrm^*M \otimes \mathrm^*M \to \mathbf or by the double dual isomorphism to a section of the tensor product :\mathrmM \otimes \mathrmM.


Arclength and the line element

Suppose that is a Riemannian metric on . In a local coordinate system , , the metric tensor appears as a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
, 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 :L = \int_a^b \sqrt\,dt \,. In connection with this geometrical application, the quadratic
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
:ds^2 = \sum_^n g_(p) dx^i dx^j 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 :L = \int_a^b \sqrt\,dt \, . 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.


The energy, variational principles and geodesics

Given a segment of a curve, another frequently defined quantity is the (kinetic) energy of the curve: :E = \frac \int_a^b \sum_^ng_(\gamma(t)) \left(\fracx^i \circ \gamma(t)\right)\left(\fracx^j \circ \gamma(t)\right)\,dt \,. This usage comes from
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, specifically,
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, where the integral can be seen to directly correspond to the
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
of a point particle moving on the surface of a manifold. Thus, for example, in Jacobi's formulation of
Maupertuis' principle In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of ''path'' and ''length''). It is a special case of ...
, 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 equation In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s 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.


Canonical measure and volume form

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 Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of subsets of the manifold. The resulting natural positive
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
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 function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s on . More precisely, if is a manifold with a (pseudo-)Riemannian metric tensor , then there is a unique positive
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
such that for any coordinate chart , \Lambda f = \int_U f \, d\mu_g = \int_ f \circ \varphi^(x) \sqrt\,dx for all supported in . Here is the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
, then it is possible to define a natural
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
from the metric tensor. In a positively oriented coordinate system the volume form is represented as \omega = \sqrt \, dx^1 \wedge \cdots \wedge dx^n where the are the
coordinate differential In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is s ...
s and denotes the exterior product in the algebra of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s. 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.


Examples


Euclidean metric

The most familiar example is that of elementary Euclidean geometry: the two-dimensional Euclidean metric tensor. In the usual coordinates, we can write :g = \begin 1 & 0 \\ 0 & 1\end \,. The length of a curve reduces to the formula: :L = \int_a^b \sqrt \,. The Euclidean metric in some other common coordinate systems can be written as follows.
Polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
: :\begin x &= r \cos\theta \\ y &= r \sin\theta \\ J &= \begin\cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta\end \,. \end So :g = J^\mathsfJ = \begin \cos^2\theta + \sin^2\theta & -r\sin\theta \cos\theta + r\sin\theta\cos\theta \\ -r\cos\theta\sin\theta + r\cos\theta\sin\theta & r^2 \sin^2\theta + r^2\cos^2\theta \end = \begin 1 & 0 \\ 0 & r^2 \end by trigonometric identities. In general, in a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
on a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the partial derivatives are
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
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 :g_ = \sum_\delta_\frac \frac = \sum_k\frac\frac.


The round metric on a sphere

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 In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. Here Southern latitudes are defined to be negative, and as a result the colatitude is a no ...
, the angle measured from the -axis, and the angle from the -axis in the -plane, the metric takes the form :g = \begin 1 & 0 \\ 0 & \sin^2 \theta\end \,. This is usually written in the form :ds^2 = d\theta^2 + \sin^2\theta\,d\varphi^2\,.


Lorentzian metrics from relativity

In flat Minkowski space (
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
), with coordinates :r^\mu \rightarrow \left(x^0, x^1, x^2, x^3\right) = (ct, x, y, z) \, , the metric is, depending on choice of metric signature, :g = \begin 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end \quad \text \quad g = \begin -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end \,. For a curve with—for example—constant time coordinate, the length formula with this metric reduces to the usual length formula. For a timelike curve, the length formula gives the proper time along the curve. In this case, the spacetime interval is written as :ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 = dr^\mu dr_\mu = g_ dr^\mu dr^\nu\,. The
Schwarzschild metric In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assump ...
describes the spacetime around a spherically symmetric body, such as a planet, or a black hole. With coordinates :\left(x^0, x^1, x^2, x^3\right) = (ct, r, \theta, \varphi) \,, we can write the metric as :g_ = \begin \left(1 - \frac\right) & 0 & 0 & 0 \\ 0 & -\left(1 - \frac\right)^ & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -r^2 \sin^2 \theta \end\,, where (inside the matrix) is the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
and represents the total mass-energy content of the central object.


See also

* Basic introduction to the mathematics of curved spacetime * Clifford algebra *
Finsler manifold In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve ...
* List of coordinate charts *
Ricci calculus In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
*
Tissot's indicatrix In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 ...
, a technique to visualize the metric tensor


Notes


References

* * . * translated by A. M. Hiltebeitel and J. C. Morehead
"Disquisitiones generales circa superficies curvas"
''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' Vol. VI (1827), pp. 99–146. * . * . * . * . * (''to appear''). * * * * * {{Manifolds Riemannian geometry Tensors Concepts in physics Differential geometry *1