Blaga P. Lectures on the differential geometry of - tiera.ru

4.1 Parameterized surfaces (patches)
CHAPTER 4
General theory of surfaces
Definition. A regular parameterized surface (patch) in R3 is a smooth map r : U → R3 ,
(u, v) → r(u, v), where U is a domain (an open, connected subset) in R2 , while r is
subject to
r ′
u × r ′
v � 0. (4.1.1)
The condition (4.1.1) is called the regularity condition.
A parameterized surface is usually denoted by (U, r), (U, r = r(u, v)) or just r =
r(u, v).
Definition. The set r(U) ⊂ R 3 is called the support of the parameterized surface (U, r).
Remark. Usually, one and the same point of the support of a parameterized surface
(U, r) may correspond to several distinct pairs (u, v), since the map r is not assumed to
be injective.
Definition. Two parameterized surfaces (U, r) and (V, r1) are called equivalent if there
is a diffeomorphism λ : U → V such that r = r1 ◦ λ.
Remark. The supports of two equivalent parameterized surfaces always coincide.