In the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
field of
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
, 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 as ...
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, ofte ...
on a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
allows defining distances and angles there. More precisely, a metric tensor at a point of is a
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
defined on the
tangent space at (that is, a
bilinear function 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 ...
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s), and a metric field 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
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
. 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, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
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 (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. Conversely, the metric tensor itself is the
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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
Gauss
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
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 associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
were understood by, in particular,
Gregorio Ricci-Curbastro and
Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a
tensor field.
The components of a metric tensor in a
coordinate basis take on the form of a
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
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
Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
in his 1827 ''
Disquisitiones generales circa superficies curvas'' (''General investigations of curved surfaces'') considered a surface
parametrically, with the
Cartesian coordinates , , 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
A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
:
depending on an
ordered pair of real variables , and defined in an
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
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 can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
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 measure of a region's size on a 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 open surface or the boundary of a three-di ...
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 length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
of that curve is given by the
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
:
where
represents the
Euclidean norm. Here the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
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). P ...
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
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of . It permits the calculation of curvature and ...
of . Intuitively, it represents the
principal part 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
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
relates , , and to , , and via the
matrix 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 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 associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
. 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 ...
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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
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
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
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 (symbol ''A'') 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 d ...
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 for the cross product, the integral can be written
:
where is the
determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
.
Definition
Let be a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
(
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
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
, . If
:
are two vectors at , then the value of the metric applied to and is determined by the coefficients () by bilinearity:
: