Geodesic
Levi-Civita's Theorem for Noncommutative Tori
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas.
Keywords: noncommutative torus; noncommutative vector field; Riemannian metric; Levi-Civita connection; Riemannian curvature; Gauss–Bonnet theorem. 
@article{SIGMA_2013_9_a70,
     author = {Jonathan Rosenberg},
     title = {Levi-Civita's {Theorem} for {Noncommutative} {Tori}},
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     volume = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a70/}
}
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Jonathan Rosenberg. Levi-Civita's Theorem for Noncommutative Tori. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a70/

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