By George Kamberov, Peter Norman, Franz Pedit, Ulrich Pinkall

Many difficulties in natural and utilized arithmetic boil all the way down to making a choice on the form of a floor in area or developing surfaces with prescribed geometric houses. those difficulties variety from classical difficulties in geometry, elasticity, and capillarity to difficulties in laptop imaginative and prescient, clinical imaging, and pix. there was a sustained attempt to appreciate those questions, yet many difficulties stay open or basically in part solved. those comprise identifying the form of a floor from its metric and suggest curvature (Bonnet's problem), deciding on an immersion from the projectivised Gauss map (Christoffel's challenge) and its purposes to the pc imaginative and prescient challenge on improving form from shading, the development of surfaces with prescribed curvature homes, developing extremal surfaces and interfaces, and representing floor deformations.This publication reviews those questions by way of proposing a thought using to either international and native questions and emphasizing conformal immersions instead of isometric immersions. The e-book bargains: a unified and finished presentation of the quaternionic and spinor method of the idea of floor immersions in 3 and 4 dimensional area; new geometric invariants of surfaces in area and new open difficulties; a brand new viewpoint and new effects at the classical geometric difficulties of floor and floor form popularity and floor illustration; a resource of difficulties to encourage learn and dissertations; functions in machine imaginative and prescient and special effects; and proofs of many effects awarded by way of the authors at colloquia, meetings, and congresses during the last years.This e-book describes how one can use quaternions and spinors to check conformal immersions of Riemann surfaces into $\Bbb R^3$. the 1st half develops the mandatory quaternionic calculus on surfaces, its program to floor concept and the research of conformal immersions and spinor transforms. The integrability stipulations for spinor transforms lead evidently to Dirac spinors and their software to conformal immersions. the second one half provides an entire spinor calculus on a Riemann floor, the definition of a conformal Dirac operator, and a generalized Weierstrass illustration legitimate for all surfaces. This conception is used to enquire first, to what quantity a floor is dependent upon its tangent airplane distribution, and moment, to what volume curvature determines the form. The e-book is aimed at graduate scholars and study mathematicians attracted to differential geometry and geometric research and its purposes, machine technology, laptop imaginative and prescient, and special effects