Metric Tensor[转载]


Roughly speaking, the metric tensor g_(ij) is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements dx_i in a generalized Pythagorean theorem:


In Euclidean space, g_(ij)=delta_(ij) where delta is the Kronecker delta (which is 0 for i!=j and 1 for i=j), reproducing the usual form of the Pythagorean theorem


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 M, whereby a metric tensor is essentially a geometrical object g=g(·,·) taking two vector inputs and calculating either the squared length g(v,v) of a single vector v or a scalar product g(u,v) of two different vectors u!=v (Misner et al. 1978). In this analogy, the inputs in question are most commonly tangent vectors lying in the tangent space T_pM for some pointp in M, a fact which facilitates the more common definition of metric tensor as an assignment of differentiable inner products to the collection of all tangent spaces of a differentiable manifold M (O’Neill 1967). For this reason, some literature defines a metric tensor on a differentiable manifold M 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 (0,2) tensor field g on a smooth manifold M so that, for all x in M, gx is non-degenerate and index(gx)=I for some nonnegative integer I (Sachs and Wu 1977). Here, Iis called the index of g and the expression index(·) 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 1967). Manifolds endowed with metric tensors are sometimes called geometric manifolds (O’Neill 1967), while a pair (X,G) consisting of a real vector space Xand a metric tensor G:X×X->R is called a metric vector space (Dodson and Poston 1991). Symbolically, metric tensors are most often denoted by g or g_(ij), although the notations ds^2 (O’Neill 1967), g^->^-> (Fleisch 2012), and G(Dodson and Poston 1991) are also sometimes used.

When defined as a differentiable inner product of every tangent space of a differentiable manifold M, the inner product associated to a metric tensor is most often assumed to be symmetric, non-degenerate, and bilinear, i.e., it is most often assumed to take two vectors v,w as arguments and to produce a real number <v,w> such that


Note, however, that the inner product need not be positive definite, i.e., the condition


with equality if and only if v=0 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 ds^2=dx_1^2+dx_2^2+... 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 (1,n-1) which induces the standard Lorentzian Inner Product on n-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 metric tensors whose associated quadratic forms fail to be symmetric, though this is far less common.

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




where eta_(alphabeta) is the matrix of metric coefficients. One example of identity (0) comes from special relativity where eta_(alphabeta) is the matrix of metric coefficients for the Minkowski metric of signature (1,3), i.e.

 eta_(alphabeta)=[-1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1].

Generally speaking, identities (3), (2), and (1) can be succinctly written as



D_(alphamu) = (partialxi^alpha)/(partialx^mu)
D_(alphamu)^(T) = D_(mualpha).

What’s more,




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


The orthogonality of contravariant and covariant metrics stipulated by


for i=1,2,3,...,n gives n linear equations relating the 2n quantities g_(ij) and g^(ij). Therefore, if n metrics are known, the others can be determined, a fact summarized by saying that the existence of metric tensors gives a geometrical way of changing from contravariant tensors to covariant ones and vice versa (Dodson and Poston 1991).

In two-space,

g^(11) = (g_(22))/g
g^(12) = g^(21)=-(g_(12))/g
g^(22) = (g_(11))/g.

Therefore, if g is symmetric,

g_(alphabeta) = g_(betaalpha)
g^(alphabeta) = g^(betaalpha).

In any symmetric space (e.g., in Euclidean space),


and so


The angle phi between two parametric curves is given by






In arbitrary (finite) dimension, the line element can be written


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


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


Moreover, because g_(ij)=0 for i!=j when working with respect to orthogonal coordinate systems, the line elementfor three-space becomes

ds^2 = g_(11)dq_1^2+g_(22)dq_2^2+g_(33)dq_3^2
= (h_1dq_1)^2+(h_2dq_2)^2+(h_3dq_3)^2,

where h_i=sqrt(g_(ii)) are called the scale factors. Many of these notions can be generalized to higher dimensions and to more general contexts.

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