Conformal invariants of twisted Dirac operators and positive scalar curvature

JOURNAL OF GEOMETRY AND PHYSICS

卷:70

Article

Benameur, Moulay Tahar^1,2  Mathai, Varghese³ 

[1] Univ Metz, Lab & Dept Mathemat, UMR 7122, F-57045 Metz 1, France

[2] CNRS, F-57045 Metz 1, France

[3] Univ Adelaide, Dept Math, Adelaide, SA 5005, Australia

关键词: Twisted Dirac rho invariant;    Twisted Dirac eta invariant;    Conformal invariants;    Twisted Dirac operator;    Positive scalar curvature;    Manifolds with boundary;   

DOI  :  10.1016/j.geomphys.2013.03.010

来源: Elsevier

【 摘 要 】

For a closed, spin, odd dimensional Riemannian manifold (Y, g), we define the rho invariant rho(spin)(Y, epsilon, H, [g]) for the twisted Dirac operator (sic)(H)(epsilon) on Y, acting on sections of a flat Hermitian vector bundle epsilon over Y, where H = Sigma i(j+1)H(2j+1) is an odd-degree closed differential form on Y and H2j+1 is a real-valued differential form of degree 2j+1. We prove that it only depends on the conformal class [g] of the metric g. In the special case when H is a closed 3-form, we use a Lichnerowicz-Weitzenbock formula for the square of the twisted Dirac operator, which in this case has no first order terms, to show that rho(spin),(Y, 8, H, [g]) = rho(spin),(Y, epsilon, [g]) for all vertical bar H vertical bar small enough, whenever g is conformally equivalent to a Riemannian metric of positive scalar curvature. When H is a top-degree form on an oriented three dimensional manifold, we also compute rho(spin)(Y, epsilon, H). (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free