Although we do not know whether each $g(\sigma^i,\sigma^i)$ is $+1$ or $-1$, it is easily seen that the number of plus signs ($p$) and minus signs ($m$) is basis independent. We define the signature of the metric $g$ to be \begin{equation} s = m \end{equation} and therefore the signature is just the number of minus signs. 1)

Of particular interest are the cases $s=0$, for which $g$ is positive definite, which gives rise to Riemannian geometry, and $s=1$, for which there is precisely one minus sign, which is called Lorentzian geometry, and which is the geometric arena for (both special and general) relativity.

Particular examples are Euclidean 2-space, usually denoted $\RR^2$ (rather than the more accurate $\EEE^2$), with line element \begin{equation} ds^2 = dx^2 + dy^2 \end{equation} and Minkowski 2-space, $\MM^2$, with line element \begin{equation} ds^2 = dx^2 - dt^2 \end{equation}