We derive an explicit formula for the scalar curvature over a two-torus with a Dirac operator conformally rescaled by a globally diagonalizable matrix. We show that the Gauss-Bonnet theorem holds and extends the result to all Riemann surfaces with Dirac operators modified in the same way.
REFERENCES
1.
Arnlind
, J.
and Wilson
, M.
, “On the Chern-Gauss-Bonnet theorem for the noncommutative 4-sphere
,” J. Geom. Phys.
111
, 126
–141
(2017
).2.
Brzezinski
, T.
, Ciccoli
, N.
, Dąbrowski
, L.
, and Sitarz
, A.
, “Twisted reality condition for Dirac operators
,” Math. Phys. Anal. Geom.
19
(3
), 16
(2016
).3.
Connes
, A.
and Fathizadeh
, F.
, “The term a4 in the heat kernel expansion of noncommutative tori
,” e-print arXiv:1611.09815.4.
Connes
, A.
and Tretkoff
, P.
, “The Gauss–Bonnet theorem for the noncommutative two torus
,” in Noncommutative Geometry, Arithmetic, and Related Topics
(Johns Hopkins University Press
, Baltimore, MD
, 2011
), pp. 141
–158
.5.
Connes
, A.
and Moscovici
, H.
, “Modular curvature for noncommutative two-tori
,” J. Am. Math. Soc.
27
, 639
(2014
).6.
Dąbrowski
, L.
and Sitarz
, A.
, “Curved noncommutative torus and Gauss–Bonnet
,” J. Math. Phys.
54
, 013518
(2013
).7.
Dąbrowski
, L.
and Sitarz
, A.
, “An Asymmetric noncommutative torus
,” SIGMA Symmetry Integrability Geom. Methods Appl.
11
, 075
(2015
).8.
Eckstein
, M.
, Sitarz
, A.
, and Wulkenhaar
, R.
, “The Moyal sphere
,” J. Math. Phys.
57
, 112301
(2016
).9.
Fathi
, A.
and Khalkhali
, M.
, “On certain spectral invariants of Dirac operators on noncommutative tori
,” e-print arXiv:1504.01174v1.10.
Fathi
, A.
, Ghorbanpour
, A.
, and Khalkhali
, M.
, “The curvature of the determinant line bundle on the noncommutative two torus
,” Math. Phys. Anal. Geom.
20
(2
), 4
(2017
).11.
Fathizadeh
, F.
, “On the scalar curvature for the noncommutative four torus
,” J. Math. Phys.
56
(6
), 062303
(2015
).12.
Fathizadeh
, F.
and Khalkhali
, M.
, “The Gauss-Bonnet theorem for noncommutative two tori with a general conformal structure
,” J. Noncommut. Geom.
6
(3
), 457
–480
(2012
).13.
Fathizadeh
, F.
and Khalkhali
, M.
, “Scalar curvature for the noncommutative two torus
,” J. Noncommut. Geom.
7
, 1145
–1183
(2013
).14.
Fathizadeh
, F.
and Olivier
, G.
, “On the Chern-Gauss-Bonnet theorem and conformally twisted spectral triples for C*-dynamical systems
,” SIGMA Symmetry Integrability Geom. Methods Appl.
12
, 016
(2016
).15.
Floricel
, R.
, Ghorbanpour
, A.
, and Khalkhali
, M.
, “The Ricci curvature in noncommutative geometry
,” e-print arXiv:1612.06688.16.
Friedman
, G.
and Park
, E.
, “Unitary equivalence of normal matrices over topological spaces
,” J. Topol. Anal.
8
(2
), 313
–348
(2016
).17.
Guillemin
, V. W.
, “A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues
,” Adv. Math.
55
(2
), 131
–160
(1985
).18.
Iochum
, B.
and Masson
, T.
, “Heat trace for Laplace type operators with non-scalar symbols
,” J. Geom. Phys.
116
, 90
–118
(2017
).19.
Kalau
, W.
and Walze
, M.
, “Gravity, noncommutative geometry and the Wodzicki residue
,” J. Geom. Phys.
16
, 327
–344
(1995
).20.
Khalkhali
, M.
, Moatadelro
, A.
, and Sadeghi
, S.
, “A scalar curvature formula for the noncommutative 3-torus
,” e-print arXiv:1610.04740.21.
Lesch
, M.
, “Divided differences in noncommutative geometry: Rearrangement lemma, functional calculus and expansional formula
,” J. Noncommut. Geom.
11
(1
), 193
–223
(2017
).22.
23.
Loring
, T.
, “The torus and noncommutative topology
,” Ph.D. dissertation (University of California
, Berkeley
, 1986
).24.
Nazaikinskii
, V.
, Savin
, A.
, Schulze
, B.-W.
, and Sternin
, B.
, “Pseudodifferential operators
,” a draft version of Chapter VII of the book Differential Operators on Manifolds with Singularities. Analysis and Topology
(to be published), see https://publishup.uni-potsdam.de/opus4-ubp/frontdoor/deliver/index/docId/2469/file/2004_06.pdf.25.
Pflaum
, M.
, “The Normal symbol on Riemannian manifolds
,” New York J. Math.
4
, 97
–125
(1998
), available at http://nyjm.albany.edu/j/1998/4-8.html.26.
Rosenberg
, J.
, “Levi-Civita’s theorem for noncommutative tori
,” SIGMA
9
, 071
(2013
).27.
Wodzicki
, M.
, “Local invariants of spectral asymmetry
,” Invent. Math.
75
(1
), 143
–178
(1995
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.
