Let R be a Riemann surface
defined upon an oriented surface S smoothly immersed in E3. This paper studies
holomorphic quadratic differentials on R which are related to the geometry on S,
especially those of the form
where Λ= Adx2+ 2Bdxdy + Cdy2 is a smooth linear combination Λ=fI +ĝΠ of
the fundamental forms on S, and z = x + iy is any conformal parameter on R. Most
results deal with the case in which R = RΛ is determined on S by some smooth
positive definite linear combination Λ = fI + gII on S. It is shown, for example, that
S is isothermal with respect to Λ if and only if RΛ supports a holomorphic ΩΛ≢0 in
some neighborhood of any nonumbilic point. By way of contrast, another result states
that a holomorphic ΩΛ≢0 is automatically available in the neighborhood of any
nonumbilic point p, unless R coincides at p with some RΛ. The paper closes with a
study of surfaces which support an RΛ on which both ΩI≢0 and ΩII≢0 are
holomorphic.
