Vol. 16, No. 1, 1966

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On the determination of conformal imbedding

Tilla Weinstein

Vol. 16 (1966), No. 1, 113–119

Two imbedding fundamental forms determine (up to motions) the smooth imbedding of an oriented surface in E3. The situation is, however, substantially different for the sufficiently smooth conformal imbedding of a Riemann surface R in E3. Conventionally such an imbedding is achieved by a conformal correspondence between R and the Riemann surface R1 determined on a smoothly imbedded oriented surface S in E3 by its first fundamental from I. We show that except where H K = 0 on S, such an R1 conformal imbedding of R in E3 is determined (up to motions) by the second fundamental form II on S, expressed as a form on R. In particular, I is determined by II on R1, where H K0 on S.

Similar remarks are valid for two less standard methods of conformal imbedding. If an oriented surface S is smoothly imbedded in E3 so that H > 0 and K > 0, then II defines a Riemann surface R2 on S. And, if S is imbedded so that K < 0, then II given by

H′II′ = KI − HII


      ∘ -------
H ′ = − H2 − K

defines a Riemann surface R2on S. Thus a conformal correspondence between R and R2 (or R2) is called an R2 (or R2) conformal imbedding of R in E3. We show that I on S, expressed as a form on R, determines the R2 or (wherever H0 and sign H is know) the R2imbedding of R in E3 (up to motions). In particular, I determines II on R2, and (where H0, and sign H is known) on R2as well. Finally, we give restatements of the fundamental theorem of surface theory in forms appropriate to R1, R2 and R2conformal imbeddings in E3.

Mathematical Subject Classification
Primary: 53.01
Secondary: 53.25
Received: 10 January 1964
Published: 1 January 1966
Tilla Weinstein