The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure

Fathizadeh, Farzad; Khalkhali, Masoud

doi:10.4171/jncg/97

Cited by 64 publications

(119 citation statements)

References 4 publications

Supporting

Mentioning

114

Contrasting

Unclassified

Order By: Relevance

“…The result holds for all smooth metrics in two dimensions, but Gauss' proof only covers analytic metrics. Since conformal classes of metrics on a two torus are parametrized by the upper half plane modulo the action of the modular group, this justifies the initial choice of metrics for noncommutative tori by Connes and Cohen in their Gauss-Bonnet theorem in [22], and for general conformal structures in our paper [30]. By all chances, in the noncommutatve case one needs to go beyond the class of localy conformally flat metrics.…”

Section: 1mentioning

confidence: 76%

“…Thus ζ(0) is a topological invariant, and, in particular, it remains invariant under the conformal perturbation g → e h g of the metric. This result was then extended to all conformal classes in the upper half plane in our paper [30]. After the work of Gauss, a decisive giant step was taken by Riemann in his epoch-making paper Ueber die Hypothesen, welche der Geometrie zu Grunde liegen, which is a text of his Habilitationsvortrag of June 1854.…”

Section: 1mentioning

confidence: 79%

“…Using the symmetries of these functions describing the term a 2 ( τ,h ) integrates to 0, hence the analog of the Gauss-Bonnet theorem. This result was proved in [22,30] in a kind of simpler manner as by exploiting the trace property of ϕ 0 from the beginning of the symbolic calculations, only a one variable function was necessary to describe ϕ 0 (a 2 ( τ,h )). However, for the description of a 2 one needs both one and two variable functions, which are given by (19) and (20).…”

Section: Gauss-bonnet Theorem and Curvature For Noncommutative 2-torimentioning

confidence: 91%

“…This episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Cohen for a curved noncommutative torus in [22] (see also the MPI preprint [8] where many ideas are already laid out). This paper was immediately followed in [30] where the Gauss-Bonnet was proved for general conformal structures. The metric structure of a noncommutative space is encoded in a (twisted) spectral triple.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Curvature in noncommutative geometry

Fathizadeh

Khalkhali

2019

Advances in Noncommutative Geometry

Self Cite

View full text Add to dashboard Cite

Dedicated to Alain Connes with admiration, affection, and much appreciation Contents 1. Introduction 2. Curvature in noncommutative geometry 2.1. A brief history of curvature 2.2. Laplace type operators and Gilkey's theorem 2.3. Noncommutative Chern-Weil theory 2.4. From spectral geometry to spectral triples 3. Pseudodifferential calculus and heat expansion 3.1. Classical pseudodifferential calculus 3.2. Small-time heat kernel expansion 3.3. Pseudodifferential calculus and heat kernel expansion for noncommutative tori 4. Gauss-Bonnet theorem and curvature for noncommutative 2-tori 4.1. Scalar curvature and Gauss-Bonnet theorem for T 2 θ 4.2. The Laplacian on (1, 0)-forms on T 2θ with curved metric 5. Noncommutative residues for noncommutative tori and curvature of noncommutative 4-tori 5.1. Noncommutative residues 5.2. Scalar curvature of the noncommutative 4-torus 6.The Riemann curvature tensor and the term a 4 for noncommutative tori 6.1. Functional relations 6.2. A partial differential system associated with the functional relations 6.3. Action of cyclic groups in the differential system, invariant expressions and simple flow of the system 6.4. Gradient calculations leading to functional relations 6.5. The term a 4 for non-conformally flat metrics on noncommutative four tori 7. Twisted spectral triples and Chern-Gauss-Bonnet theorem for ergodic C * -dynamical systems 7.1. Twisted spectral triples 7.2. Conformal perturbation of a spectral triple 7.3. Conformal perturbation of the flat metric on T 2 θ 7.4. Conformally twisted spectral triples for C * -dynamical systems 7.5. The Chern-Gauss-Bonnet theorem for C * -dynamical systems

show abstract

Section: 1mentioning

confidence: 76%

Section: 1mentioning

confidence: 79%

Section: Gauss-bonnet Theorem and Curvature For Noncommutative 2-torimentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Curvature in noncommutative geometry

Fathizadeh

Khalkhali

2019

Advances in Noncommutative Geometry

Self Cite

View full text Add to dashboard Cite

show abstract

“…In a series of papers first [3,4] and [7][8][9][10][11][12] a conformally rescaled metric has been proposed and studied for the noncommutative two and four-tori. This led to the expressions of Gauss-Bonnet theorem and formulae for the noncommutative counterpart of scalar curvature.…”

Section: Introductionmentioning

confidence: 99%

Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus

Sitarz

2014

J. Pseudo-Differ. Oper. Appl.

View full text Add to dashboard Cite

show abstract

Traces of Holomorphic Families of Operators on the Noncommutative Torus and on Hilbert Modules

Azzali¹,

Lévy²,

Neira-Jiménez³

et al. 2015

Trends in Mathematics

View full text Add to dashboard Cite

We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks; the noncommutative torus and Hilbert modules. These traces are meromorphic functions whose residues at the poles as well as the constant term of the Laurent expansion at zero (the latter when the family at zero is a differential operator) can be expressed in terms of Wodzicki residues and extended Wodzicki residues involving logarithmic operators. They are therefore local and contain geometric information. For holomorphic families leading to zeta regularised traces, they relate to the heat-kernel asymptotic coefficients via an inverse Mellin mapping theorem. We revisit Atiyah's L 2index theorem by means of the (extended) Wodzicki residue and interpret the scalar curvature on the noncommutative two torus as an (extended) Wodzicki residue.

show abstract

The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure

Cited by 64 publications

References 4 publications

Curvature in noncommutative geometry

Curvature in noncommutative geometry

Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus

Traces of Holomorphic Families of Operators on the Noncommutative Torus and on Hilbert Modules

Contact Info

Product

Resources

About