What you use for measuring distance in Riemann's scheme of geometry. If z is your coordinate, then you would write the metric in the following from:
    ds = F(z)dz
What is meant by this is that if you have some curve z(t), you can measure the distance along it as:
    distance = (integral) ds = (integral) F(z)dz = (integral from t0 to t1) F(z(t)) (dz/dt) dt
(the curve goes from z(t0) to z(t1)). F(z) is required to be positive definite so that distance do not become negative.

Example: in Euclidean 3-space, ds2 = dx2+dy2+dz2, in agreement with Pythagoras's Theorem.

In General Relativity the metric is expressed in terms of the metric tensor, ds2 = gadxadxb. It is techincally a pseudo-Riemannan metric, since the metric tensor is found not to be positive definite.

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