Seshadri: Almost-Einstein manifolds with nonnegative isotropic curvature

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Harish Seshadri
Almost-Einstein manifolds with nonnegative isotropic curvature
(Variétés presque Einstein à courbure isotrope positive ou nulle)
Annales de l'institut Fourier, 60 no. 7 (2010), p. 2493-2501, doi: 10.5802/aif.2616
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Class. Math.: 53C21
Keywords: Almost-Einstein manifolds, non-negative isotropic curvature

Résumé - Abstract

Let $(M,g)$, $n \ge 4$, be a compact simply-connected Riemannian $n$-manifold with nonnegative isotropic curvature. Given $0<l\le L$, we prove that there exists $\varepsilon = \varepsilon (l,L,n)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies

$$ l \le s \le L $$

and the Einstein tensor satisfies

$$ \Bigl \vert \mbox {\rm Ric}\, - \frac{s}{n}g \Bigr \vert \le \varepsilon $$

then $M$ is diffeomorphic to a symmetric space of compact type.

This is related to the result of S. Brendle on the metric rigidity of Einstein manifolds with nonnegative isotropic curvature.

Bibliography

[1] Simon Brendle, “Einstein manifolds with nonnegative isotropic curvature are locally symmetric”, to appear in Duke Mathematical Journal MR 2573825 | Zbl 1189.53042
[2] Simon Brendle & Richard Schoen, “Manifolds with $1/4$-pinched curvature are space forms”, J. Amer. Math. Soc. 22 (2009) no. 1, p. 287-307
Article | MR 2449060 | Zbl pre05859406
[3] Norihito Koiso, “Rigidity and stability of Einstein metrics—the case of compact symmetric spaces”, Osaka J. Math. 17 (1980) no. 1, p. 51-73
Article | MR 558319 | Zbl 0426.53037
[4] Mario J. Micallef & John Douglas Moore, “Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes”, Ann. of Math. (2) 127 (1988) no. 1, p. 199-227
Article | MR 924677 | Zbl 0661.53027
[5] Mario J. Micallef & McKenzie Y. Wang, “Metrics with nonnegative isotropic curvature”, Duke Math. J. 72 (1993) no. 3, p. 649-672
Article | MR 1253619 | Zbl 0804.53058
[6] Peter Petersen & Terence Tao, “Classification of almost quarter-pinched manifolds”, Proc. Amer. Math. Soc. 137 (2009) no. 7, p. 2437-2440
Article | MR 2495279 | Zbl 1168.53020
[7] A. Petrunin & W. Tuschmann, “Diffeomorphism finiteness, positive pinching, and second homotopy”, Geom. Funct. Anal. 9 (1999) no. 4, p. 736-774
Article | MR 1719602 | Zbl 0941.53026
[8] H. Seshadri, “Manifolds with nonnegative isotropic curvature”, To appear in Communications in Analysis and Geometry, http://www.math.iisc.ernet.in/~harish/papers/pic-cag.pdf MR 2601346
[9] Peter Topping, Lectures on the Ricci flow, London Mathematical Society Lecture Note Series 325, Cambridge University Press, 2006 MR 2265040 | Zbl 1105.58013