Metric Tensor - Arclength and The Line Element

Arclength and The Line Element

Suppose that g is a Riemannian metric on M. In a local coordinate system xi, i = 1,2,…,n, the metric tensor appears as a matrix, denoted here by G, whose entries are the components g_ij of the metric tensor relative to the coordinate vector fields.

Let γ(t) be a piecewise differentiable parametric curve in M, for a ≤t ≤ b. The arclength of the curve is defined by

In connection with this geometrical application, the quadratic differential form

is called the first fundamental form associated to the metric, while ds is the line element. When ds2 is pulled back to the image of a curve in M, it represents the square of the differential with respect to arclength.

For a pseudo-Riemannian metric, the length formula above is not always defined, because the term under the square root may become negative. We generally only define the length of a curve when the quantity under the square root is always of one sign or the other. In this case, define

Note that, while these formulas use coordinate expressions, they are in fact independent of the coordinates chosen; they depend only on the metric, and the curve along which the formula is integrated.