A theorem on surface area

where D* is the unit disc u2 +?v2 < 1 and T* is continuous on D*. If the Lebesgue area of this second surface is denoted by A (T*), then the question arises: How should the two representations T and T* be related in order for A (T) to equal A (T*) ? 0.2. It is well known that A (T) =A (T*) if T can be obtained from T* by a change of parameters, in precise terms, if there exists a homeomorphism H(D) =D* such that T(p) = T*H(p) for every point p in D. More generally, A (T) =A (T*) if T is F-equivalent (equivalent in the Frechet sense) to T*, that is, if for every positive number e there exists a homeomorphism H1(D) =D* such that the distance in Euclidean 3-space between the points T(p) and T*He(p) is less than e for every point p in D. Since two F-equivalent representations yield the same Frechet surface, this is merely a restatement of the fact that the Lebesgue area is independent of the choice of representation for the surface (see 0.1). In the sequel we shall write T-T*(F) to indicate that T and T* are F-equivalent. We shall also speak of the F-invariance of A (T), meaning that T-T*(F) implies A (T) = A (T*). 0.3. The principal result of this paper is to establish the relation A(T) =A (T*) in a still more general situation; namely, when T is K-equivalent to T* (in symbols, T-T*(K)), that is, if T and T* possess simultaneous monotone-light factorizations T=LM and T* =LM*, where Mand M* are continu-