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
:
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
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 ...
:
where
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:
:
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
:
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
:
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
:
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,
:
so that
:
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
:
for suitable real numbers and . If two tangent vectors are given:
:
then using the
bilinearity of the dot product,
:
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
:
This is a
symmetric function in and , meaning that
:
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,
:
for any vectors , , , and in the plane, and any real numbers and .
In particular, the length of a tangent vector is given by
:
and the angle between two vectors and is calculated by
:
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
:
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
:
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 . 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
* is
symmetric. A function of two vector arguments is symmetric provided that for all vectors and ,
* is
nondegenerate. A bilinear function is nondegenerate provided that, for every tangent vector , the function
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
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
:
are two vectors at , then the value of the metric applied to and is determined by the coefficients () by bilinearity:
: