J. A. Thorpe

Surface Area and Volume

We consider now the problem of how to find the volume (area when n = 2) of an n-surface in ℝ n +1. As with the length of plane curves, this is done in two steps. First we define the volume of a parametrized n-surface and then we define the volume of an n-surface in terms of local parametrizations.

The Weingarten Map

We shall now consider the local behavior of curvature on an n-surface. The way in which an n-surface curves around in ℝ n +1 is measured by the way the normal direction changes as we move from point to point on the surfacc. In order to measure the rate of changc of the normal direction, we need to be able to differentiate vector fields on n-surfaces.

The Gauss Map

An oriented n-surface in ℝ n +l is more than just an n-surface S, it is an n-surface S together with a smooth unit normal vector field N on S. The function N:S → ℝ n +1 associated with the vector field N by N(p) = (p, N(p)), p ∈ S, actually maps S into the unit n-sphere S n ⊂ ℝ n +1 since ∥N(p)∥ = 1 for all p ∈ S. Thus, associated to each oriented n-surface S is a smooth map N: S → S n . called the Gauss map. N may be thought of as the map which assigns to each point p ∈ S the point in ℝ n +1 obtained by “translating” the unit normal vector N(p) to the origin (see Figure 6.1).

The Tangent Space

Let f: U → ℝ be a smooth function, where U ⊂ ℝ n +1 is an open set. let c ∈ ℝ be such that f−1(c) is non-empty, and let p ∈ f−1(c). A vector at p is said to be tangent to the level set f−1(c) if it is a velocity vector of a parametrized curve in ℝ n +1 whose image is contained in f−1(c) (see Figure 3.1).

Curvature of Surfaces

Let S be an n-surfacc in ℝ n +1, oriented by the unit normal vcctor field N, and let p ∈ S. The Weingarten map L p : S p → S p , defined by L p (v) = − ∇vN for v ∈ S p , measures the turning of the normal as one moves in S through p with various velocities v. Thus L p measures the way S curves in ℝ n +1 at p. For n = 1, we have seen that L p is just multiplication by a number K(p) the curvature of S at p. We shall now analyze L p when n > 1.

Curvature of Plane Curves

Let C = f−1(c), where f: U → ℝ, be a plane curve in the open set U ⊂ ℝ2, oriented by N = ∇f/∥∇f∥. Then, for each p ∈ C. the Weingartcn map Lp, is a linear transformation on the 1-dimensional spacc Cp. Sincc every linear transformation from a 1-dimensional space to itself is multiplication by a real number, there exists, for each p ∈ C, a real number K(p) such that

Local Equivalence of Surfaces and Parametrized Surfaces

{L_p}\left( v \right)\, = \,\kappa \left( p \right)\nu \,for\,all\,\nu \, \in \,{C_p}

The Exponential Map

. K(p) is called the curvature of C at p.