Riemannian Geometry of a Discretized Circle and Torus

@article{Bochniak2020RiemannianGO,
  title={Riemannian Geometry of a Discretized Circle and Torus},
  author={Arkadiusz Bochniak and Andrzej Sitarz and P. Zalecki},
  journal={Symmetry, Integrability and Geometry: Methods and Applications},
  year={2020}
}
We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and metric compatibility condition in general and show that there are several classes of solutions, out of which only special ones are compatible with a metric that gives a Hilbert C∗-module structure on the space of the one-forms. We compute curvature and scalar… 

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References

SHOWING 1-10 OF 44 REFERENCES

A new look at Levi-Civita connection in noncommutative geometry

We prove the existence and uniqueness of Levi-Civita connections for a noncommutative pseudo-Riemannian metric on a class of centered bimodule of one forms. As an application, we compute the Ricci

An Asymmetric Noncommutative Torus

We introduce a family of spectral triples that describe the curved noncommuta- tive two-torus. The relevant family of new Dirac operators is given by rescaling one of two terms in the flat Dirac

Linear connections in non-commutative geometry

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalization of the Leibniz rules of commutative geometry and uses the bimodule structure

On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a

Matrix geometries and fuzzy spaces as finite spectral triples

A class of real spectral triples that are similar in structure to a Riemannian manifold but have a finite-dimensional Hilbert space is defined and investigated, determining a general form for the

Metric on quantum spaces

We introduce the analogue of the metric tensor in the case ofq-deformed differential calculus. We analyse the consequences of the existence of the metric, showing that this enforces severe

The Ricci curvature in noncommutative geometry

Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented

Finite spectral triples for the fuzzy torus

Finite real spectral triples are defined to characterise the non-commutative geometry of a fuzzy torus. The geometries are the non-commutative analogues of flat tori with moduli determined by integer

On the Koszul formula in noncommutative geometry

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul

Compatible Connections in Noncommutative Riemannian Geometry