This book is based on lectures delivered over the years by the author at the
Universit´
e Pierre et Marie Curie, Paris, at the University of Stuttgart, and at
City University of Hong Kong. Its two-fold aim is to give thorough introduc-
tions to the basic theorems of differential geometry and to elasticity theory in
curvilinear coordinates.
The treatment is essentially self-contained and proofs are complete. The
prerequisites essentially consist in a working knowledge of basic notions of anal-
ysis and functional analysis, such as differential calculus, integration theory
and Sobolev spaces, and some familiarity with ordinary and partial differential
equations.
In particular, no apriori knowledge of differential geometry or of elasticity
theory is assumed.
In the first chapter, we review the basic notions, such as the metric tensor
and covariant derivatives, arising when a three-dimensional open set is equipped
with curvilinear coordinates. We then prove that the vanishing of the Riemann
curvature tensor is sufficient for the existence of isometric immersions from a
simply-connected open subset of Rn equipped with a Riemannian metric into
a Euclidean space of the same dimension. We also prove the corresponding
uniqueness theorem, also called rigidity theorem.
In the second chapter, study basic notions about surfaces, such their