Metric Tensor

A metric tensor for a Riemannian metric determines infinitesimal lengths and angles on a smooth manifold. For this positive definite case, the distance between two points in the same connected component is the infimum of the lengths of curves joining them. More generally, a metric tensor may be indefinite, in which case this construction does not define a distance. In two coordinates, its components enter the line element as

(1)

In Cartesian coordinates on Euclidean space, where is the Kronecker delta (which is 0 for and 1 for ), reproducing the usual form of the Pythagorean theorem

(2)

In this way, the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be "arithmetized" by way of introducing a sort of generalized coordinate system (Borisenko and Tarapov 1979).

In the above simplification, the space in question is most often a smooth manifold , whereby a metric tensor is essentially a geometrical object taking two vector inputs and calculating either the associated quadratic form on a single vector or the bilinear form on two different vectors (Misner et al. 1973). In this analogy, the inputs in question are most commonly manifold tangent vectors lying in the tangent space for some point , a fact which facilitates the more common definition of metric tensor as an assignment of differentiable symmetric bilinear forms to the collection of all tangent spaces of a smooth manifold (O'Neill 2006). For this reason, some literature defines a metric tensor on a differentiable manifold to be nothing more than a symmetric non-degenerate bilinear form (Dodson and Poston 1991).

An equivalent definition can be stated using the language of tensor fields and indices thereon. Along these lines, some literature defines a metric tensor to be a symmetric tensor field on a smooth manifold so that, for all , is non-degenerate and for some nonnegative integer (Sachs and Wu 1977). Here, is called the index of and the expression refers to the index of the respective quadratic form. This definition seems to occur less commonly than those stated above.

Metric tensors have a number of synonyms across the literature. In particular, metric tensors are sometimes called fundamental tensors (Fleisch 2012) or geometric structures (O'Neill 2006). Two-dimensional Riemannian manifolds are also called geometric surfaces (O'Neill 2006, pp. 322-323 and 326), while a pair consisting of a real vector space and a metric tensor is called a metric vector space (Dodson and Poston 1991). Symbolically, metric tensors are most often denoted by or , although the notations (O'Neill 2006), (Fleisch 2012), and (Dodson and Poston 1991) are also sometimes used.

On each tangent space of the differentiable manifold , the metric tensor is a non-degenerate symmetric bilinear form that takes two vectors as arguments and produces a real number satisfying

(3)

(4)

(5)

(6)

Note, however, that this symmetric bilinear form need not be positive definite, i.e., the condition

(7)

with equality iff need not always be satisfied. When the metric tensor is positive definite, it is called a Riemannian metric or, more precisely, a weak Riemannian metric; otherwise, it is called non-Riemannian, (weak) pseudo-Riemannian, or semi-Riemannian, though the latter two terms are sometimes used differently in different contexts. The simplest example of a Riemannian metric is the Euclidean metric discussed above; the simplest example of a non-Riemannian metric is the Minkowski metric of special relativity, the four-dimensional version of the more general metric of signature which induces the standard Lorentzian inner product on -dimensional Lorentzian space. In some literature, the condition of non-degeneracy is varied to include either weak or strong non-degeneracy (Marsden et al. 2002); one may also consider nonsymmetric tensor fields, although these are not metric tensors in the usual sense.

In coordinate notation (with respect to a chosen vector basis), the metric tensor and its inverse satisfy a number of fundamental identities, e.g.,

(8)

(9)

and

(10)

where is the matrix of metric coefficients. One example of identity (10) comes from special relativity where is the matrix of metric coefficients for the Minkowski metric of metric signature , i.e.,

(11)

The coordinate transformation identity for can be succinctly written in matrix form as

(12)

where is the Jacobian matrix, and its transpose satisfies . What's more,

(13)

gives

(14)

and hence yields a quantitative relationship between a metric tensor and its inverse.

In the event that the metric is positive definite, the metric discriminants are positive. For a metric in two-space, this fact can be expressed quantitatively by the inequality

(15)

The orthogonality of contravariant and covariant metrics stipulated by

(16)

for all gives component equations stating that the matrices and are inverses. Thus all components of either matrix determine the other, providing a geometrical way of changing from contravariant tensors to covariant tensors and vice versa (Dodson and Poston 1991).

In two-space,

			(17)
			(18)
			(19)

Therefore, if is symmetric,

			(20)
			(21)

The mixed-index components satisfy

(22)

but symmetry alone does not make corresponding diagonal entries reciprocal. If the metric matrix is diagonal in the chosen coordinates, then

(23)

with no sum over . For a parametric surface in three-dimensional Euclidean space, let and . The angle between the coordinate curves is given by

(24)

so

(25)

and

(26)

In Euclidean space of arbitrary finite dimension, the line element can be written

(27)

where Einstein summation has been used. In three dimensions, this yields

(28)

and so it follows that the metric tensor in three-space can be written as

(29)

Moreover, because for when working with respect to orthogonal coordinate systems, the line element for three-space becomes

			(30)

where are called the scale factors. Many of these notions can be generalized to higher dimensions and to more general contexts.