# Almost-Einstein manifolds with nonnegative isotropic curvature

• [1] Indian Institute of Science Department of Mathematics Bangalore 560012 (India)
• Volume: 60, Issue: 7, page 2493-2501
• ISSN: 0373-0956

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## Abstract

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Let $\left(M,g\right)$, $n\ge 4$, be a compact simply-connected Riemannian $n$-manifold with nonnegative isotropic curvature. Given $0<l\le L$, we prove that there exists $\epsilon =\epsilon \left(l,L,n\right)$ satisfying the following: If the scalar curvature $s$ of $g$ satisfies$l\le s\le L$and the Einstein tensor satisfies$\left|\mathrm{Ric}\phantom{\rule{0.166667em}{0ex}}-\frac{s}{n}g\right|\le \epsilon$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.

## How to cite

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Seshadri, Harish. "Almost-Einstein manifolds with nonnegative isotropic curvature." Annales de l’institut Fourier 60.7 (2010): 2493-2501. <http://eudml.org/doc/116343>.

abstract = {Let $(M,g)$, $n \ge 4$, be a compact simply-connected Riemannian $n$-manifold with nonnegative isotropic curvature. Given $0&lt;l\le L$, 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.},
affiliation = {Indian Institute of Science Department of Mathematics Bangalore 560012 (India)},
journal = {Annales de l’institut Fourier},
keywords = {Almost-Einstein manifolds; non-negative isotropic curvature; almost-Einstein manifolds},
language = {eng},
number = {7},
pages = {2493-2501},
publisher = {Association des Annales de l’institut Fourier},
title = {Almost-Einstein manifolds with nonnegative isotropic curvature},
url = {http://eudml.org/doc/116343},
volume = {60},
year = {2010},
}

TY - JOUR
TI - Almost-Einstein manifolds with nonnegative isotropic curvature
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 7
SP - 2493
EP - 2501
AB - Let $(M,g)$, $n \ge 4$, be a compact simply-connected Riemannian $n$-manifold with nonnegative isotropic curvature. Given $0&lt;l\le L$, 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.
LA - eng
KW - Almost-Einstein manifolds; non-negative isotropic curvature; almost-Einstein manifolds
UR - http://eudml.org/doc/116343
ER -

## References

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1. Simon Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric Zbl1189.53042MR2573825
2. Simon Brendle, Richard Schoen, Manifolds with $1/4$-pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009), 287-307 Zbl1251.53021MR2449060
3. Norihito Koiso, Rigidity and stability of Einstein metrics—the case of compact symmetric spaces, Osaka J. Math. 17 (1980), 51-73 Zbl0426.53037MR558319
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), 199-227 Zbl0661.53027MR924677
5. Mario J. Micallef, McKenzie Y. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), 649-672 Zbl0804.53058MR1253619
6. Peter Petersen, Terence Tao, Classification of almost quarter-pinched manifolds, Proc. Amer. Math. Soc. 137 (2009), 2437-2440 Zbl1168.53020MR2495279
7. A. Petrunin, W. Tuschmann, Diffeomorphism finiteness, positive pinching, and second homotopy, Geom. Funct. Anal. 9 (1999), 736-774 Zbl0941.53026MR1719602
8. H. Seshadri, Manifolds with nonnegative isotropic curvature Zbl1197.53047MR2601346
9. Peter Topping, Lectures on the Ricci flow, 325 (2006), Cambridge University Press, Cambridge Zbl1105.58013MR2265040

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