Curvature and the Shape Operator

The First Fundamental Form

In a biography published several years after Riemann's death, fellow German mathematician Hans Freundenthal (1905-1990) described this Riemann surface as one that

posesses shortest lines, now called geodesics, which resemble ordinary straight lines. In fact, at first approximation in a geodesic coordinate system such a metric is flat in the same way that a curved surface up to higher-order terms looks like its tangent plane. Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras' theorem.

In order to calculate the length of a straight line on a plane, thus, we may use the equation $$A^2+B^2=C^2$$.

In order to calculate the length of a line on a curve, however, we use an equation referred to as the first fundamental form:

ds^2 = E\left(du^2\right) + 2F\left(du\cdot dv\right) + G\left(dv^2\right)

where the coefficients are given by:

E = |\frac{\partial x}{\partial u}|^2, F = \frac{\partial x}{\partial u} \cdot \frac{\partial x}{\partial v}, G = |\frac{\partial x}{\partial v}|^2

For our purposes, it isn't important that we memorize this equation. Rather, simply realize that it exists, and understand what it is used for.