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 = (k1−1−k1−1). (k2−1−k2−1) 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.
