The primary examples of noncommutative dierential geometry are the non-commutative tori [1]. The algebra in these examples can be constructed from the algebra of smooth functions on the ordinary torus Tl by deforming the product with rotation action of the torus on itself. There is also an action of Tl on the resulting deformed algebra. Beginning with a spin structure on Tl there is a Dirac operator 6Dacting on the Hilbert space H of spinors, which is invariant under the lifted action of Tl. Thus the deformed algebra is also represented on this Hilbert space, resulting in a spectral triple with the same Dirac operator. So we have an isospectral deformation of the canonincal commutative spectral triple (C1(Tl);H;D6 ). This example was generalized by Connes and Landi [2] by deforming any compact Riemannian manifold M that admits an isometric action of the torus Tl, for l  2. This method was further rened by Connes and Dubois-Violette [3] to include the noncompact case but still with a torus action. In all these cases, the Dirac operator is invariant under the lifted action and (C1(M);H; 6D) is a noncommutative spin geometry that is isospectral to the undeformed case, because D6 is unchanged. In this presentation we will go through these results and discuss some possible directions for studying certain noncommutative geometries in this setting. References [1] A. Connes, Noncommutative Geometry. Academic Press, London, 1994. [2] A. Connes and G. Landi, Noncommutative manifolds, the insanton algebra and isospectral deformation, Commun. Math. Phys. 221(2001), 141-159. [3] A. Connes and M. Dubois-Violette, Noncommutative nite-dimensional manifolds. I. Spherical manifolds and related examples, Commun. Math. Phys. 230 (2002), 539-579. 1

Isospectral deformations of Riemannian manifolds I