Abstract
Let C be a cone in the space of algebraic curvature tensors. Moreover, let (M,g) be a compact Einstein manifold with the property that the curvature tensor of (M,g) lies in the interior of the cone C at each point on M. We show that (M,g) has constant sectional curvature if the cone C satisfies certain structure conditions.
Mathematics Subject Classification (2010) Primary 53C25. Secondary 53C24, 53C44.
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References
Berger, M.: Sur les variétés d’Einstein compactes, Ann. Mat. Pura Appl. 53, 89–95 (1961)
Besse, A.: Einstein Manifolds. Classics in Mathematics, Springer-Verlag, Berlin (2008)
Brendle, S.: Einstein manifolds with nonnegative isotropic curvature are locally symmetric. Duke Math. J. 151, 1–21 (2010)
Brendle, S.: Ricci Flow and the Sphere Theorem. Graduate Studies in Mathematics vol. 111, American Mathematical Society (2010)
Brendle, S., Schoen, R.: Manifolds with 1/4-pinched curvature are space forms. J. Amer. Math. Soc. 22, 287–307 (2009)
Hamilton, R.: Four-manifolds with positive curvature operator. J. Diff. Geom. 24, 153–179 (1986)
Micallef, M., Wang, M.: Metrics with nonnegative isotropic curvature. Duke Math. J. 72,649–672 (1993)
Nguyen, H.: Isotropic curvature and the Ricci flow. Internat. Math. Res. Notices no. 3, 536–558 (2010)
Tachibana, S.: A theorem on Riemannian manifolds with positive curvature operator. Proc. Japan Acad. 50, 301–302 (1974)
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© 2011 Springer-Verlag Berlin Heidelberg
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Brendle, S. (2011). Einstein metrics and preserved curvature conditions for the Ricci flow. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_4
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DOI: https://doi.org/10.1007/978-3-642-20300-8_4
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Online ISBN: 978-3-642-20300-8
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