Noncommutative Geometry and Geometric Structures

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In a fundamental paper Rennie claims to characterize closed Riemannian spin manifolds as spectral triples with a commutative C*-algebra plus some additional algebraic data. Firstly, the status of this result has to be clarified. Secondly, versions for manifolds with additional geometric structure are to be characterized. Moreover, compact manifolds with additional geometric structure (e.g. Kähler metrics or, more generally, metrics with restricted holonomy) are to be characterized. Moreover, compact manifolds with boundary and suitable noncompact manifolds should be incorporated. Finally, one will try to pass from the Riemannian to the Lorentzian case.