First Fundamental Form

The metric E, F, G measuring length, angle, and area on a surface.

The idea

The first fundamental form records the measurements that can be made on a surface from within it: lengths, angles, and areas. A quantity determined by such internal measurements alone, with no reference to the surrounding space, is called intrinsic.

At a point of a surface parametrized by $\mathbf{x}(u,v)$, a small step $(du, dv)$ in the parameters displaces the point by approximately $\mathbf{x}_u\,du + \mathbf{x}_v\,dv$ along the surface. Expanding the squared length of this displacement produces three coefficients, $E = \mathbf{x}_u \cdot \mathbf{x}_u, \quad F = \mathbf{x}_u \cdot \mathbf{x}_v, \quad G = \mathbf{x}_v \cdot \mathbf{x}_v,$ and the squared length itself is $ds^{2} = E\,du^{2} + 2F\,du\,dv + G\,dv^{2}.$

Every intrinsic quantity follows from these three functions: lengths of curves by integrating $ds$, angles from the dot products the coefficients encode, and area from the element $\sqrt{EG - F^{2}}\,du\,dv$. The coefficients depend on the chosen coordinates, as the components of a vector depend on a basis; the lengths, angles, and areas they compute do not. Two surfaces carrying the same first fundamental form are indistinguishable by internal measurement, however differently they sit in space: a flat sheet and the cylinder it rolls into are the standard example, since rolling bends the sheet without stretching it.

Ways to work on it

• Walkthrough. Build the first fundamental form's coefficients for the plane and read off the metric and area element.
• Practice. Compute a coefficient of the first fundamental form from tangent vectors.
• Hardest. Use the first fundamental form to compute an arc length on a surface.

Not sure where to start? Take the ten-question placement test.