Abstract
We discuss classical Riemannian geometry and its noncommutative geometric counterparts. At first the definition and properties of the Hodge Laplacian and the Dirac operator are given. We also derive the characterizations of isometries (resp. orientation preserving isometries) in terms of the Laplacian (resp. Dirac operator). This is followed by discussion on noncommutative manifolds given by spectral triples, including the definitions of noncommutative space of forms and the Laplacian in this set up. The last section of this chapter deals with the quantum group equivariance in noncommutative geometry where we discuss some natural examples of equivariant spectral triples on the Podles’ spheres.
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