8.1 Gaussian Curvature of a Surface

We will give a brief introduction to the curvature of a surface for heuristic intuition.

Suppose is a region, and let . Consider a surface parameterized via

where . If never vanishes for , then defines a smooth surface . For a fixed , the vectors and form the basis of the tangent space (a plane) of at , denoted by .

The square of the length of the vector infinitesimal , or

is known as the first fundamental form of , where , , and .

Let be near . It follows that . The distance between and is , where . By application of the multivariate Taylor’s Theorem, we have

and therefore,

The first two linear terms vanish by properties of the triple scalar product. The second fundamental form of is defined as

where , , and . Since and , by differentiation, we have

It follows that , , and . Because ,

The second fundamental form, in a rough sense, measures the curvature of the surface at (refer to Figure 19). Both the first and second fundamental forms are geometric invariants; they are independent of the parameterization of . The first fundamental form is also referred to as the intrinsic metric (we will not delve into the metric tensor here) of , and the second fundamental form is an extrinsic property of as it is invariant up to the orientation of the surface (consequent direction of the normal vector).

Let be a curve parameterized by arc length, . Then the unit tangent vector at is

Consequently,

where the last two terms are in . Because for all by the arc-length parameterization, we have

Hence, and are orthogonal and is a normal to the curve . Let be the unit normal to at . The normal curvature of at in is defined as

The quotient

varies depending on the curve traversing (and ultimately, depending on the direction induced by and ). On , the two representations are equivalent since . The maximum and minimum values of are known as the principal curvatures and of at , achieved along the principal directions of the (unit) tangent vectors at .

The mean curvature of at is defined to be . Let be the radii of curvature corresponding to and . The product of the two principal curvatures is known as the Gaussian curvature of at , denoted by . We will now heuristically derive the explicit formulas for and in terms of .

Suppose . Adopt the matrix notation of , as in

to reduce to the optimization problem of

We may restrict so that the denominator is always , aiming to optimize the numerator. By the method of Lagrange multipliers, we write

The equation for can then be decomposed into (where ):

The first two equations can be written as

Let the matrix on the left be denoted by . In order for non-trivial to exist, we must have . That is,

This is a quadratic giving two solutions for . From

it is apparent that the roots . Moreover, from (8.1.3) we have

Hence, the two roots are precisely the principal curvatures. Vieta’s formulas give that

Now, assume a parameterization of by (thrice continuously differentiable) such that

(which we will later formalize as a conformal metric). Then there is an alternate representation of the Gaussian curvature in terms of .

By definition, while . Moreover,

Similarly,

By differentiation of the equations

we have

and

Substituting (8.1.5) into (8.1.4) then gives

Differentiating these give

Differentiating the inner two expressions of (8.1.5), we have

It follows that

where here is . Then

and