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Index of Dirac operator and scalar curvature almost non-negative manifolds

Fuquan Fang

2003
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Asian Journal of Mathematics
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A manifold M is called scalar curvature almost non-negative if for any constant £ > 0, there is a Riemannian metric g on M such that Sg • diam(M,^) 2 > -e and the sectional curvature Secg < 1, where Sg (resp. diam(M,g)) is the scalar curvature (resp. diameter) of (M,g). Among others we prove that for a scalar curvature almost non-negative manifold M with A(M) T£ 0 (resp. A(M) 6 Z2 nonzero and x(M) ^ 0 if n = 2(mod 8)), there is a constant £(n) > 0 such that, if the scalar curvature SM > -efa),
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... vature SM > -efa), then (i) the fundamental group 7ri(M) is finite; (ii) M admits a real analytic Ricci flat metric go such that its Riemannian universal covering M is isometric to the product of Ricci flat Kahler-Einstein manifolds and/or Joyce manifolds of dimension 8 with special holonomy group Spin(7). Introduction. Let M be a closed Spin manifold. It is well-known that M admits a metric with positive scalar curvature only if the index of the Dirac operator vanishes, by the classical Lichnerowicz formula and the Atiyah-Singer index theorem (cf. [LM]). The Gromov-Lawson conjecture, confirmed by Stolz [St], asserts the converse for simply connected Spin manifolds of dimension at least 5. There are many manifolds with non-negative scalar curvature but do not accept metrics of positive scalar curvature, e.g. torus, JCs-surfaces, etc. A basic theorem of Bourguignon shows that such a Spin manifold must be Ricci flat. Starting from [St], Futaki [Fu] characterized all simply connected Spin manifolds of dimension at least 5 with non-negative scalar curvature. It turns out that such a manifold either admits a metric with positive scalar curvature, or it is the product of Ricci flat Kahler-Einstein manifolds and/or Joyce manifold of dimension 8 with Spin(7)-holonomy (manifolds in the latter class are called rigidly scalar flat). A keypoint involved is that, every harmonic spinor must be parallel and therefore, the holonomy group must be special (cf. [Fu] [Wa]). Recall that a manifold is called almost flat (resp. almost Ricci flat) if for any positive constant e > 0, there is a Riemannian metric g on M such that \SeCg • diam(M,#) 2 | < e (resp. \RiCg • diam (M,g) 2 \ < e), where SeCg (resp. Ric 5 ) is the sectional (resp. Ricci) curvature. The celebrated Gromov theorem asserts that an almost flat manifold must be an infra-nilmanifold, i.e. a finite regular cover must be a nilmanifold. By the Bochner technique it is also well-known that an almost Ricci flat manifold has the first Betti number at most n, where n is the dimension. In contrast, however, very recently Lohkamp [Lo] proved that for every compact manifold M of dimension at least 3 and for any given positive constant e, there is a metric g so that its scalar curvature satisfies that \sg • diam(M,p) 2 | < s. We call a manifold M is of scalar curvature almost non-negative if for any constant e > 0, there is a Riemannian metric g on M such that s g • diam(M,^) 2 > -s and the sectional curvature Sec g < 1, where s^ (resp. diam(M, g)) is the scalar curvature *

doi:10.4310/ajm.2003.v7.n1.a3
fatcat:nhev43yehvghxjod2elrwbp2wm