Many problems in pure and applied mathematics boil down to determining the
shape of a surface in space or constructing surfaces with prescribed geometric
properties. These problems range from classical problems in geometry,
elasticity, and capillarity to problems in computer vision, medical imaging,
and graphics. There has been a sustained effort to understand these questions,
but many problems remain open or only partially solved.
This book describes how to use quaternions and spinors to study conformal
immersions of Riemann surfaces into $\Bbb R^3$. The first part develops the
necessary quaternionic calculus on surfaces, its application to surface theory
and the study of conformal immersions and spinor transforms. The integrability
conditions for spinor transforms lead naturally to Dirac spinors and their
application to conformal immersions. The second part presents a complete spinor
calculus on a Riemann surface, the definition of a conformal Dirac operator,
and a generalized Weierstrass representation valid for all surfaces. This
theory is used to investigate first, to what extent a surface is determined by
its tangent plane distribution, and second, to what extent curvature determines
the shape.
The book is geared toward graduate students and researchers
interested in differential geometry and geometric analysis and
their applications in computer vision and computer graphics.
Readership
Graduate students and research mathematicians interested in
differential geometry and geometric analysis and its applications, computer
science, computer vision, and computer graphics.