Metric tensor

basic tensor, fundamental tensor

A twice covariant symmetric tensor field $g=g(X,Y)$ on an $n$-dimensional differentiable manifold $M^n$, $n\geq2$. The assignment of a metric tensor on $M^n$ introduces a scalar product $\langle X,Y\rangle$ of contravariant vectors $X,Y\in M_p^n$ on the tangent space $M_p^n$ of $M^n$ at $p\in M^n$, defined as the bilinear function $g_p(X,Y)$, where $g_p$ is the value of the field $g$ at the point $p$. In coordinate notation:

$$\langle X,Y\rangle=g_{ij}(p)X^iY^j,\quad X=\{X^i\},\quad Y=\{Y^j\},\quad0\leq i,j\leq n.$$

The metric in $M_p^n$ with this scalar product is regarded as infinitesimal for the metric of the manifold $M^n$, which is expressed by the choice of the quadratic differential form

$$ds^2=g_{ij}(p)dx^idx^j\label{*}\tag{*}$$

as the square of the differential of the arc length of curves in $M^n$, going from $p$ in the direction $dx^1,\dots,dx^n$. With respect to its geometric meaning the form $\eqref{*}$ is called the metric form or first fundamental form on $M^n$, corresponding to the metric tensor $g$. Conversely, if a symmetric quadratic form $\eqref{*}$ on $M^n$ is given, then there is a twice covariant tensor field $g(X,Y)=g_{ij}X^iY^j$ associated with it and whose corresponding metric form is $g$. Thus, the specification of a metric tensor $g$ on $M^n$ is equivalent to the specification of a metric form on $M^n$ with a quadratic line element of the form $\eqref{*}$. The metric tensor completely determines the intrinsic geometry of $M^n$.

The collection of metric tensors $g$, and the metric forms defined by them, is divided into two classes, the degenerate metrics, when $\det(g_{ij})=0$, and the non-degenerate metrics, when $\det(g_{ij})\neq0$. A manifold $M^n$ with a degenerate metric form $\eqref{*}$ is called isotropic. Among the non-degenerate metric tensors, in their turn, are distinguished the Riemannian metric tensors, for which the quadratic form $\eqref{*}$ is positive definite, and the pseudo-Riemannian metric tensors, when $\eqref{*}$ has variable sign. A Riemannian (pseudo-Riemannian) metric introduced on $M^n$ via a Riemannian (pseudo-Riemannian) metric tensor defines on $M^n$ a Riemannian (respectively, pseudo-Riemannian) geometry.

Usually a metric tensor, without special indication, means a Riemannian metric tensor; but if one wishes to stress that the discussion is about Riemannian and not about pseudo-Riemannian metric tensors, then one speaks of a proper Riemannian metric tensor. A proper Riemannian metric tensor can be introduced on any paracompact differentiable manifold.

References

[1]	L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)
[2]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[3]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)