This book offers an up-to-date overview of the recently proposed theory of quantum isometry
groups. Written by the founders it is the first book to present the research on the ¿quantum
isometry group¿ highlighting the interaction of noncommutative geometry and quantum groups
which is a noncommutative generalization of the notion of group of isometry of a classical
Riemannian manifold. The motivation for this generalization is the importance of isometry
groups in both mathematics and physics. The framework consists of Alain Connes¿ ¿noncommutative
geometry¿ and the operator-algebraic theory of ¿quantum groups¿. The authors prove the
existence of quantum isometry group for noncommutative manifolds given by spectral triples
under mild conditions and discuss a number of methods for computing them. One of the most
striking and profound findings is the non-existence of non-classical quantum isometry groups
for arbitrary classical connected compact manifolds and by using this the authors explicitly
describe quantum isometry groups of most of the noncommutative manifolds studied in the
literature. Some physical motivations and possible applications are also discussed.
