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SOME SPHERE THEOREMS FOR SUBMANIFOLDS WITH POSITIVE BIORTHOGONAL CURVATURE

  • ELZIMAR RUFINO (a1)

Abstract

The purpose of this paper is to investigate sphere theorems for submanifolds with positive biorthogonal (sectional) curvature. We provide some upper bounds for the full norm of the second fundamental form under which a compact submanifold must be diffeomorphic to a sphere.

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1. Asperti, A. and Costa, E., Vanishing of homology groups, Ricci estimate for submanifolds and applications, Kodai Math. J. 24 (2001), 313328.
2. Bettiol, R., Positive biorthogonal curvature on $\mathbb{S}$ 2 × $\mathbb{S}$ 2 , Proc. Amer. Math. Soc. 142 (2014), 43414353.
3. Berger, M., Sur quelques variétés riemanniennes suffisamment pincées, Bull. Soc. Math. France 88 (1960), 5771.
4. Brendle, S., Einstein manifolds with nonnegative isotropic curvature are locally symmetric, Duke Math. J. 151 (2010), 121.
5. Brendle, S., A general convergence result for the Ricci flow in higher dimensions, Duke Math. J. 145 (2008), 585601, MR 2462114 Zbl 1161.53052.
6. Brendle, S. and Schoen, R., Classification of manifolds with weakly 1/4-pinched curvatures, Acta Math. 200 (2008), 113, MR 2386107.
7. Brendle, S. and Schoen, R., Sphere theorems in geometry, Surveys in differential geometry, Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, Surv. Differ. Geom., vol. 13 (International Press, Somerville, MA, 2009), 4984.
8. Böhm, C. and Wilking, B., Manifolds with positive curvature operators are space forms, Ann. Math. 167 (2) (2008), 10791097.
9. Costa, E. and Ribeiro, E. Jr, Four-dimensional compact manifolds with nonnegative biorthogonal curvature, Michigan Math. J. 63 (2014), 673688.
10. Costa, E. and Ribeiro, E. Jr, Minimal volume invariants, topological sphere theorems and biorthogonal curvature on 4-manifolds, avaliable at: arXiv:1504.06212 [math.DG] (2015).
11. Costa, E., Ribeiro, E. Jr and Santos, A., Modified Yamabe problem on four-dimensional compact manifolds, Houston J. Math. 42 (4) (2016), 11411152.
12. Chang-Yu Xia, H., A sphere theorem for submanifolds in a manifold with pinched positive curvature, Monast. für Math. 124 (1997), 365368.
13. Gu, J.-R. and Xu, H.-W., The sphere theorems for manifolds with positive scalar curvature, J. Differ. Geom. 92 (2012), 507545.
14. Hamilton, R., Four-manifolds with positive curvature operator, J. Differ. Geom. 24 (1986), 153179.
15. Hamilton, R., Three-manifolds with positive Ricci curvature, J. Differ. Geom. 17 (1982), 255306.
16. Micallef, M. and Wang, M., Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), 649672.
17. Micallef, M. and Moore, J., Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. Math. 127 (1988), 199227.
18. Noronha, M. H., Positively curved 4-manifold and the nonnegativity of isotropic curvatures, Michigan Math. J. 44 (1997).
19. Lawson, H. and Simons, J., On stable currents and their applications to global problems in real and complex geometry, Ann. Math. 98 (1973), 427450.
20. Leung, P., On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold, Proc. Edinburgh Math. Soc. 28 (1985), 305311.
21. Ribeiro, E. Jr, Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor, Annali di Matematica Pura ed. Appl. 195 (2016), 21712181.
22. Seaman, W., Orthogonally pinched curvature tensors and aplications, Math. Scand. 69 (1991), 514.
23. Tashibana, S., A theorem on Riemannian manifolds with positive curvature operator, Proc. Japan Acad. 50 (1974), 301302.
24. Xin, Y., An application of integral currents to the vanishing theorems, Sci. Sinica Ser. A 27 (1984), 233241.
25. Xu, H. and Zhao, E., Topological and differentiable sphere theorems for complete submanifolds, Comm. Analys. Geom. 17 (2009), 565585.

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SOME SPHERE THEOREMS FOR SUBMANIFOLDS WITH POSITIVE BIORTHOGONAL CURVATURE

  • ELZIMAR RUFINO (a1)

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