Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T21:45:03.906Z Has data issue: false hasContentIssue false
  • English
  • Français

SOME SPHERE THEOREMS FOR SUBMANIFOLDS WITH POSITIVE BIORTHOGONAL CURVATURE

Published online by Cambridge University Press:  07 February 2017

ELZIMAR RUFINO
ELZIMAR RUFINO*
Affiliation:
Departamento de Matemática, Universidade Federal de Roraima, UFRR, Campus Paricarana, Bloco V. Av. Cap. Ene Garcez, 2413, Bairro Aeroporto, 69310-000, Boa Vista, Roraima, Brazil e-mail: elzimar.rufino@ufrr.br, fenix.elzy@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

53C2153C2353C25
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 3 , September 2017 , pp. 717 - 728
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

1. Asperti, A. and Costa, E., Vanishing of homology groups, Ricci estimate for submanifolds and applications, Kodai Math. J. 24 (2001), 313–328.CrossRefGoogle Scholar
2. Bettiol, R., Positive biorthogonal curvature on $\mathbb{S}$ 2 × $\mathbb{S}$ 2 , Proc. Amer. Math. Soc. 142 (2014), 4341–4353.CrossRefGoogle Scholar
3. Berger, M., Sur quelques variétés riemanniennes suffisamment pincées, Bull. Soc. Math. France 88 (1960), 57–71.CrossRefGoogle Scholar
4. Brendle, S., Einstein manifolds with nonnegative isotropic curvature are locally symmetric, Duke Math. J. 151 (2010), 1–21.CrossRefGoogle Scholar
5. Brendle, S., A general convergence result for the Ricci flow in higher dimensions, Duke Math. J. 145 (2008), 585–601, MR 2462114 Zbl 1161.53052.CrossRefGoogle Scholar
6. Brendle, S. and Schoen, R., Classification of manifolds with weakly 1/4-pinched curvatures, Acta Math. 200 (2008), 1–13, MR 2386107.CrossRefGoogle Scholar
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), 49–84.Google Scholar
8. Böhm, C. and Wilking, B., Manifolds with positive curvature operators are space forms, Ann. Math. 167 (2) (2008), 1079–1097.Google Scholar
9. Costa, E. and Ribeiro, E. Jr, Four-dimensional compact manifolds with nonnegative biorthogonal curvature, Michigan Math. J. 63 (2014), 673–688.CrossRefGoogle Scholar
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).Google Scholar
11. Costa, E., Ribeiro, E. Jr and Santos, A., Modified Yamabe problem on four-dimensional compact manifolds, Houston J. Math. 42 (4) (2016), 1141–1152.Google Scholar
12. Chang-Yu Xia, H., A sphere theorem for submanifolds in a manifold with pinched positive curvature, Monast. für Math. 124 (1997), 365–368.Google Scholar
13. Gu, J.-R. and Xu, H.-W., The sphere theorems for manifolds with positive scalar curvature, J. Differ. Geom. 92 (2012), 507–545.CrossRefGoogle Scholar
14. Hamilton, R., Four-manifolds with positive curvature operator, J. Differ. Geom. 24 (1986), 153–179.CrossRefGoogle Scholar
15. Hamilton, R., Three-manifolds with positive Ricci curvature, J. Differ. Geom. 17 (1982), 255–306.CrossRefGoogle Scholar
16. Micallef, M. and Wang, M., Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), 649–672.CrossRefGoogle Scholar
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), 199–227.CrossRefGoogle Scholar
18. Noronha, M. H., Positively curved 4-manifold and the nonnegativity of isotropic curvatures, Michigan Math. J. 44 (1997).CrossRefGoogle Scholar
19. Lawson, H. and Simons, J., On stable currents and their applications to global problems in real and complex geometry, Ann. Math. 98 (1973), 427–450.CrossRefGoogle Scholar
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), 305–311.CrossRefGoogle Scholar
21. Ribeiro, E. Jr, Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor, Annali di Matematica Pura ed. Appl. 195 (2016), 2171–2181.CrossRefGoogle Scholar
22. Seaman, W., Orthogonally pinched curvature tensors and aplications, Math. Scand. 69 (1991), 5–14.CrossRefGoogle Scholar
23. Tashibana, S., A theorem on Riemannian manifolds with positive curvature operator, Proc. Japan Acad. 50 (1974), 301–302.Google Scholar
24. Xin, Y., An application of integral currents to the vanishing theorems, Sci. Sinica Ser. A 27 (1984), 233–241.Google Scholar
25. Xu, H. and Zhao, E., Topological and differentiable sphere theorems for complete submanifolds, Comm. Analys. Geom. 17 (2009), 565–585.CrossRefGoogle Scholar