Geodesic
Spin Geometry of Dirac and Noncommutative Geometry of Connes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and its applications, Tome 298 (2017), pp. 276-314

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The review is devoted to the interpretation of the Dirac spin geometry in terms of noncommutative geometry. In particular, we give an idea of the proof of the theorem stating that the classical Dirac geometry is a noncommutative spin geometry in the sense of Connes, as well as an idea of the proof of the converse theorem stating that any noncommutative spin geometry over the algebra of smooth functions on a smooth manifold is the Dirac spin geometry. 
@article{TM_2017_298_a16,
     author = {A. G. Sergeev},
     title = {Spin {Geometry} of {Dirac} and {Noncommutative} {Geometry} of {Connes}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {276--314},
     publisher = {mathdoc},
     volume = {298},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_298_a16/}
}
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A. G. Sergeev. Spin Geometry of Dirac and Noncommutative Geometry of Connes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and its applications, Tome 298 (2017), pp. 276-314. http://geodesic.mathdoc.fr/item/TM_2017_298_a16/