We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. In fact, we identify the quantum isometry group with the universal object in a bigger category, namely the category of 'quantum families of smooth isometries', defined along the line of Woronowicz and Soltan. We also construct a spectral triple on the Hilbert space of forms on a noncommutative manifold which is equivariant with respect to a natural unitary representation of the quantum isometry group. We give explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in [11] as the universal quantum group of holomorphic isometries of the noncommutative torus.

The real sphere S N −1 R appears as increasing union, over d ∈ {1, . . . , N }, of its "polygonal" versions S N −1,d−1Motivated by general classification questions for the undeformed noncommutative spheres, smooth or not, we study here the quantum isometries of S N −1,d−1 R , and of its various noncommutative analogues, obtained via liberation and twisting. We discuss as well a complex version of these results, with S N −1 R replaced by the complex sphere S N −1 C

We formulate a quantum group analogue of the group of orientation-preserving Riemannian isometries of a compact Riemannian spin manifold, more generally, of a (possibly R-twisted and of compact type) spectral triple. The main advantage of this formulation, which is directly in terms of the Dirac operator, is that it does not need the existence of any 'good' Laplacian as in our previous works on quantum isometry groups. Several interesting examples, including those coming from Rieffel-type deformation as well as the equivariant spectral triples on SU μ (2) and S 2 μ,c are discussed.

In this follow-up of [4], where quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum isometry group of an isospectral deformation of a (classical or noncommutative) manifold is a suitable deformation of the quantum isometry group of the original (undeformed) manifold.

We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces (X, d) which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on (X, d). In fact, our existence theorem applies to a larger class, namely for any compact metric space (X, d) which admits a one-to-one continuous map f : X → R n for some n such that d 0 (f (x), f (y)) = φ(d(x, y)) (where d 0 is the Euclidean metric) for some homeomorphism φ of R + .As concrete examples, we obtain Wang's quantum permutation group S + n and also the free wreath product of Z 2 by S + n as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in [13].