Riemannian Geometry

The subject of differential geometry has developed so extensively since T.J. Willmore's book "Introduction to Differential Geometry" appeared over 30 years ago, that a new book with particular emphasis on recent work in Riemannian geometry was necessary. The methods used include classical tensor calculus, the suffix-free calculus and the method of moving frames involving the exterior calculus of differential forms, each method having advantages in particular situations. This book includes an elementary account of twistor spaces which should be of interest to applied mathematicians and physicists. It contains recent developments in the theory of harmonic spaces, commutative spaces and mean-value theorems previously available only in research papers. The final chapter gives the only account available in book form of manifolds known in the literature as "Willmore surfaces", a topic which has received much attention during the last 15 years. These surfaces are illustrated by a series of computer-generated pictures.