Vol. 54, No. 1, 1974

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Mappings by parallel normals preserving principal directions

Dimitri Koutroufiotis

Vol. 54 (1974), No. 1, 191–200

Two smooth surfaces S,S with positive Gaussian curvature and with the same closed hemisphere as spherical image can be mapped onto each other by parallel normals. It is assumed, in addition, that principal directions at every point on S are mapped into principal directions at the image point on S. Let ki(= 1,2) be the principal curvatures of S,ki the corresponding principal curvatures of S. Via the spherical image mapping, one may consider the function φ = (k11 k11). (k21 k21) as being defined on the unit sphere Σ. We show: If φ does not change sign and appropriate boundary conditions are satisfied, then S differs from S by a translation. Since the spherical image mapping always preserves principal directions, one obtains in particular characterizations of the hemisphere. Further results for ovaloids S,S within this class of mappings: If k1 k1,k2 k2 everywhere, then a translate of S fits inside S; if S and S have the same total mean curvature, then Σφdω 0 with equality if and only if S is a translate of S.

Mathematical Subject Classification 2000
Primary: 53A05
Received: 5 March 1973
Published: 1 September 1974
Dimitri Koutroufiotis