DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

CHAPTER 2Surfaces: Local Theory1. Parametrized Surfaces and the First Fundamental FormLet U be an open set in R 2 .Afunction f : U → R m (for us, m =1and 3 will be most common)is called C 1 if f and its partial derivatives ∂f ∂fand are all continuous. We will ordinarily use∂u ∂v(u, v) ascoordinates in our parameter space, and (x, y, z) ascoordinates in R 3 . Similarly, for anyk ≥ 2, we say f is C k if all its partial derivatives of order up to k exist and are continuous. We say fis smooth if f is C k for every positive integer k. Wewill henceforth assume all our functions are C kfor k ≥ 3. One of the crucial results for differential geometry is that if f is C 2 , then(and similarly for higher-order derivatives).Notation: We will often also use subscripts to indicate partial derivatives, as follows:f uf vf uuf uv =(f u ) v↔↔↔↔∂f∂u∂f∂v∂ 2 f∂u 2∂ 2 f∂v∂u∂2 f∂u∂v =Definition. A regular parametrization of a subset M ⊂ R 3 isa(C 3 ) one-to-one functionx: U → M ⊂ R 3 so that x u × x v ≠ 0∂2 f∂v∂ufor some open set U ⊂ R 2 . 1 A connected subset M ⊂ R 3 is called a surface if each point has aneighborhood that is regularly parametrized.We might consider the curves on M obtained by fixing v = v 0 and varying u, called a u-curve,and obtained by fixing u = u 0 and varying v, called a v-curve; these are depicted in Figure 1.1. Atthe point P = x(u 0 ,v 0 ), we see that x u (u 0 ,v 0 )istangent to the u-curve and x v (u 0 ,v 0 )istangentto the v-curve. We are requiring that these vectors span a plane, whose normal vector is given byx u × x v .Example 1. We give some basic examples of parametrized surfaces. Note that our parametersdo not necessarily range over an open set of values.1 For technical reasons with which we shall not concern ourselves in this course, we should also require that theinverse function x −1 : x(U) → U be continuous.34