In this section, we take a closer look at the curvature at a point of a curve on a surface . Assuming that is parameterized by arc length, we will see that the vector (which is equal to , where is the principal normal to the curve at , and is the curvature) can be written as
where is the normal to the surface at , and is a tangential component normal to the curve.
The component is called the normal curvature. Computing it will lead to the second fundamental form, another very important quadratic form associated with a surface. The component is called the geodesic curvature.
It turns out that it only depends on the first fundamental form, but computing it is quite complicated, and this will lead to the Christoffel symbols.
Definition 1. Given a surface , given any curve on , for any point on , the orthonormal frame is defined such that
where is the normal vector to the surface at . The vector is called the geodesic normal vector.
Observe that is the unit normal vector to the curve contained in the tangent space at . If we use the frame , we will see shortly that can be written as
The component is the orthogonal projection of onto the normal direction , and for this reason is called the normal curvature of at . The component is the orthogonal projection of onto the tangent space at .
We now show how to compute the normal curvature. This will uncover the second fundamental form. Since , using chain rule we get
In order to decompose into its normal component (along ) and its tangential component, we use a neat trick suggested by Eugenio Calabi. Recall that
Using this identity we have
Since is a unit vector we can write
Thus, it is clear that the normal component is
and the normal curvature is given by
using the Lagrange identity
we see that
and can be writtne as
where is the determinant of three vectors. Some authors (including Gauss himself and Darboux) use the notation
and we also have
These expressions were used by Gauss to prove his famous Theorema Egregium.
Definition 2. Given a surface , for any point on , letting
where is the unit normal at , the quadratic form
is called the second fundamental form of at . It is often denoted as and in matrix form, we have
For a curve on the surface (parameterized by arc length), the quantity given by the formula
is called the normal curvature of at .
The second fundamental form was introduced by Gauss in 1827. Unlike the first fundamental form, the second fundamental form is not necessarily positive or definite.
Gaussian curvature. The Gaussian curvature of a surface is given by
Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.