Riemannian metric

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 = g_adx^adx^b. It is techincally a pseudo-Riemannan metric, since the metric tensor is found not to be positive definite.