In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there.
The study of these invariants of a surface led Gauss to introduce the predecessor of the modern notion of the metric tensor.
Here the chain rule has been applied, and the subscripts denote partial derivatives: The integrand is the restriction[1] to the curve of the square root of the (quadratic) differential where The quantity ds in (1) is called the line element, while ds2 is called the first fundamental form of M. Intuitively, it represents the principal part of the square of the displacement undergone by r→(u, v) when u is increased by du units, and v is increased by dv units.
Using matrix notation, the first fundamental form becomes Suppose now that a different parameterization is selected, by allowing u and v to depend on another pair of variables u′ and v′.
The upshot is that the first fundamental form (1) is invariant under changes in the coordinate system, and that this follows exclusively from the transformation properties of E, F, and G. Indeed, by the chain rule, so that Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors 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(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 M can be written in the form for suitable real numbers p1 and p2.
Let M be a smooth manifold of dimension n; for instance a surface (in the case n = 2) or hypersurface in the Cartesian space
At each point p ∈ M there is a vector space TpM, called the tangent space, consisting of all tangent vectors to the manifold at the point p. A metric tensor at p is a function gp(Xp, Yp) which takes as inputs a pair of tangent vectors Xp and Yp at p, and produces as an output a real number (scalar), so that the following conditions are satisfied: A metric tensor field g on M assigns to each point p of M a metric tensor gp in the tangent space at p in a way that varies smoothly with p. More precisely, given any open subset U of manifold M and any (smooth) vector fields X and Y on U, the real function
That is, or, in terms of the entries of this matrix, For this reason, the system of quantities gij[f] is said to transform covariantly with respect to changes in the frame f. A system of n real-valued functions (x1, ..., xn), giving a local coordinate system on an open set U in M, determines a basis of vector fields on U The metric g 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 gij(f) by means of the chain rule so that Or, in terms of the matrices G[f] = (gij[f]) and G[f′] = (gij[f′]), where Dy denotes the Jacobian matrix of the coordinate change.
[4] If M is connected, then the signature of qm does not depend on m.[5] By Sylvester's law of inertia, a basis of tangent vectors Xi can be chosen locally so that the quadratic form diagonalizes in the following manner for some p between 1 and n. Any two such expressions of q (at the same point m of M) will have the same number p of positive signs.
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 A.
The contravariance of the components of v[f] is notationally designated by placing the indices of vi[f] in the upper position.
The components ai transform when the basis f is replaced by fA in such a way that equation (8) continues to hold.
The covariance of the components of a[f] is notationally designated by placing the indices of ai[f] in the lower position.
More specifically, for m = 3, which means that the ambient Euclidean space is ℝ3, the induced metric tensor is called the first fundamental form.
In these terms, a metric tensor is a function from the fiber product of the tangent bundle of M with itself to R such that the restriction of g to each fiber is a nondegenerate bilinear mapping The mapping (10) is required to be continuous, and often continuously differentiable, smooth, or real analytic, depending on the case of interest, and whether M can support such a structure.
That is, in terms of the pairing [−, −] between TpM and its dual space T∗pM, for all tangent vectors Xp and Yp.
Furthermore, Sg is a symmetric linear transformation in the sense that for all tangent vectors Xp and Yp.
The inverse of Sg is a mapping T*M → TM which, analogously, gives an abstract formulation of "raising the index" on a covector field.
The inverse S−1g defines a linear mapping which is nonsingular and symmetric in the sense that for all covectors α, β.
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 g is a Riemannian metric on M. In a local coordinate system xi, i = 1, 2, …, n, the metric tensor appears as a matrix, denoted here by G, whose entries are the components gij of the metric tensor relative to the coordinate vector fields.
When ds2 is pulled back to the image of a curve in M, 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.
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 E can be seen to directly correspond to the kinetic energy of a point particle moving on the surface of a manifold.
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.
A measure can be defined, by the Riesz representation theorem, by giving a positive linear functional Λ on the space C0(M) of compactly supported continuous functions on M. More precisely, if M is a manifold with a (pseudo-)Riemannian metric tensor g, then there is a unique positive Borel measure μg such that for any coordinate chart (U, φ),
Here det g is the determinant of the matrix formed by the components of the metric tensor in the coordinate chart.
That Λ is well-defined on functions supported in coordinate neighborhoods is justified by Jacobian change of variables.
With coordinates we can write the metric as where G (inside the matrix) is the gravitational constant and M represents the total mass–energy content of the central object.