SIGMA 9 (2013), 071, 9 pages      arXiv:1307.3775      https://doi.org/10.3842/SIGMA.2013.071
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

Levi-Civita's Theorem for Noncommutative Tori

Jonathan Rosenberg
Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Received July 26, 2013, in final form November 19, 2013; Published online November 21, 2013; Proposition 3.4 corrected January 20, 2015

Abstract
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.

Key words: noncommutative torus; noncommutative vector field; Riemannian metric; Levi-Civita connection; Riemannian curvature; Gauss-Bonnet theorem.

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References

1. Bratteli O., Elliott G.A., Jorgensen P.E.T., Decomposition of unbounded derivations into invariant and approximately inner parts, J. Reine Angew. Math. 346 (1984), 166-193.
2. Connes A., C^* algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), A599-A604, hep-th/0101093.
3. Connes A., Moscovici H., Modular curvature for noncommutative two-tori, J. Amer. Math. Soc. 27 (2014), 639-684, arXiv:1110.3500.
4. Connes A., Tretkoff P., The Gauss-Bonnet theorem for the noncommutative two torus, in Noncommutative Geometry, Arithmetic, and Related Topics, Johns Hopkins Univ. Press, Baltimore, MD, 2011, 141-158, arXiv:0910.0188.
5. do Carmo M.P., Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1992.
6. Elliott G.A., The diffeomorphism group of the irrational rotation C^*-algebra, C. R. Math. Rep. Acad. Sci. Canada 8 (1986), 329-334.
7. Fathizadeh F., Khalkhali M., The Gauss-Bonnet theorem for noncommutative two tori with a general conformal structure, J. Noncommut. Geom. 6 (2012), 457-480, arXiv:1005.4947.
8. Fathizadeh F., Khalkhali M., Scalar curvature for the noncommutative two torus, J. Noncommut. Geom. 7 (2013), 1145-1183, arXiv:1110.3511.
9. Fathizadeh F., Khalkhali M., Scalar curvature for noncommutative four-tori, arXiv:1301.6135.
10. Levi-Civita T., Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana, Rend. Circ. Mat. Palermo 42 (1917), 172-205.
11. Rosenberg J., Noncommutative variations on Laplace's equation, Anal. PDE 1 (2008), 95-114, arXiv:0802.4033.