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Adiabatic limits, vanishing theorems and the noncommutative residue. (English) Zbl 1193.53120

Summary: In this paper, we compute the adiabatic limit of the scalar curvature and prove several vanishing theorems by taking adiabatic limits. As an application, we give a Kastler-Kalau-Walze type theorem for foliations.

MSC:

53C27 Spin and Spin\({}^c\) geometry
51H25 Geometries with differentiable structure
46L87 Noncommutative differential geometry
53C12 Foliations (differential geometric aspects)
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[1] Connes A. Cyclic cohomology and the transverse fundamental class of a foliation. In: Arkai H, Effors G, eds. Geometric Methods in Operator Algebras, a Pitman Research Notes in Math Series, Vol. 123. Harlow: Longman, 1986, 52–144 · Zbl 0647.46054
[2] Liu K, Zhang W. Adiabatic limits and foliations. In: Topology, Geometry, and Algebra: Interactions and New Directions, Contemp Math, Vol. 279. Providence, RI: American Mathematical Society, 2001. 195–208 · Zbl 0987.53022
[3] Liu K, Ma X, Zhang W. On elliptic genera and foliations. Math Res Lett, 8(3): 361–376 (2001) · Zbl 0986.57023
[4] Kordyukov Y. Adiabatic limits and spectral geometry of foliations. Math Ann, 313(4): 763–783 (1999) · Zbl 0930.58017 · doi:10.1007/s002080050281
[5] López J A, Kordyukov Y A. Adiabatic limits and spectral sequences for Riemannian foliations. Geom Funct Anal, 10(5): 977–1027 (2000) · Zbl 0965.57024 · doi:10.1007/PL00001653
[6] Rumin M. Sub-Riemannian limit of the differential form spectrum of contact manifolds. Geom Funct Anal, 10(2): 407–452 (2000) · Zbl 1008.53033 · doi:10.1007/s000390050013
[7] Bismut J M. A local index theorem for non-Kähler manifolds. Math Ann, 284(4): 681–699 (1989) · Zbl 0666.58042 · doi:10.1007/BF01443359
[8] Wodzicki M. Local invariants of spectral asymmetry. Invent Math, 75(1): 143–177 (1984) · Zbl 0538.58038 · doi:10.1007/BF01403095
[9] Figueroa H, Gracia-Bondía J, Várilly J. Elements of Noncommutative Geometry. Birkhäuser: Boston, 2001 · Zbl 0958.46039
[10] Blair D. Contact Manifolds in Riemannian Geometry. In: Lecture Notes in Mathematics, Vol. 509. Berlin-New York: Springer-Verlag, 1976 · Zbl 0319.53026
[11] Petit R. Spinc-structures and Dirac operators on contact manifolds. Differential Geom Appl, 22(2): 229–252 (2005) · Zbl 1074.53042 · doi:10.1016/j.difgeo.2005.01.003
[12] Lawson H, Michelsohn M. Spin Geometry. In: Princeton Mathematical Series, Vol. 38. Princeton: Princeton University Press, 1989
[13] Kastler D. The Dirac operator and gravitation. Comm Math Phys, 166(3): 633–643 (1995) · Zbl 0823.58046 · doi:10.1007/BF02099890
[14] Kalau W, Walze M. Gravity, non-commutative geometry and the Wodzicki residue. J Geom Phys, 16(4): 327–344 (1995) · Zbl 0826.58008 · doi:10.1016/0393-0440(94)00032-Y
[15] Ackermann T. A note on the Wodzicki residue. J Geom Phys, 20(4): 404–406 (1996) · Zbl 0864.58057 · doi:10.1016/S0393-0440(95)00061-5
[16] Kordyukov Y. Noncommutative spectral geometry of Riemannian foliations. Manuscripta Math, 94(1): 45–73 (1997) · Zbl 0896.58065 · doi:10.1007/BF02677838
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