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The even dimensional pinching problem and SU(3)/T

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Abstract

A notion of optimal pinching for positively curved manifolds in introduced, and a corresponding rigidity problem is discussed. For the lowest dimensional non-standard homogeneous manifold of positive curvature, SU(3)/T, an estimate is given.

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References

  1. Aloff, S. and Wallach, N. R., ‘An Infinite Family of Distinct 7-Manifolds admitting Positively Curved Riemannian Structures’,Bull. Amer. Math. Soc. 81 (1975), 93–97.

    Article  MathSciNet  MATH  Google Scholar 

  2. Berard-Bergery, L., ‘Les Variétés Riemanniennes homogènes simplement connexes de dimension impair à courbure strictement positive’,J. Math. Pures Appl. 55 (1976), 47–68.

    MathSciNet  MATH  Google Scholar 

  3. Berger, M., ‘Les Variétés Riemanniennes homogènes normales simplement connexes à courbure strictement positive’,Ann. Scuola Norm. Sup. Pisa 15 (1961), 179–246.

    MathSciNet  MATH  Google Scholar 

  4. Berger, M., ‘Sur les Variétés Riemanniennes pincées just au-dessous de 1/4’,Ann. Inst. Fourier, Grenoble 33 (1983), 135–150.

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheeger, J., ‘Finiteness Theorems for Riemannian Manifolds’,Amer. J. Math. XCII (1970), 61–74.

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheeger, J. and Ebin, D. G., ‘Comparison Theorems in Riemannian Geometry’,North Holland Math. Library 9 (1975).

  7. Eliasson, H. I., ‘Die Krümmung des Raumes Sp(2)/SU(2) von Berger’,Math. Ann. 164 (1966), 317–323.

    Article  MathSciNet  MATH  Google Scholar 

  8. Eschenburg, J.-H., ‘New Examples of Manifolds with Strictly Positive Curvature’,Invent. Math. 66 (1982), 469–480.

    Article  MathSciNet  MATH  Google Scholar 

  9. Eschenburg, J.-H., ‘Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen’,Schrift. Math. Inst. Uni. Münster 32 (1984).

  10. Greene, R. E. and Wu, H., ‘Lipschitz Convergence of Riemannian Manifolds’,Pacific J. Math. 131 (1988), 119–141.

    Article  MathSciNet  MATH  Google Scholar 

  11. Gromov, M., Lafontaine, J. and Pansu, P.,Structures metriques pour les Varietes Riemanniennes, textes math. 1, Cedic/Fernand Nathan, Paris, 1981.

    Google Scholar 

  12. Grove, K. and Karcher, H., ‘On Pinched Manifolds with Fundamental Group Z2’.Compositio Math. 27 (1973), 49–61.

    MathSciNet  MATH  Google Scholar 

  13. Grove, K., Karcher, H., and Ruh, E. A., ‘Group Actions and Curvature’,Invent. Math. 23 (1974), 21–48.

    Article  MathSciNet  MATH  Google Scholar 

  14. Grove, K., Karcher, H. and Ruh, E. A., ‘Jacobi Fields and Finsler Metrics on Compact Lie Groups with an Application to Differentiable Pinching Problems’,Math. Ann. 211 (1974), 7–21.

    Article  MathSciNet  MATH  Google Scholar 

  15. Heintze, E., ‘The Curvature of SU(5)/(Sp(2)×S 1)’,Invent. Math. 13 (1971), 205–212.

    Article  MathSciNet  MATH  Google Scholar 

  16. Huang, H.-M., ‘Some Remarks on Pinching Problems’,Bull. Inst. Math. Acad. Sinica. 9 (1981), 321–340.

    MathSciNet  MATH  Google Scholar 

  17. Peters, S., ‘Cheeger's Finiteness Theorem for Diffeomorphism Classes of Riemannian Manifolds’,J. reine angew. Math. 394 (1984), 77–82.

    MATH  Google Scholar 

  18. Peters, S., ‘Convergence of Riemannian Manifolds’,Compositio Math. 62 (1987), 3–16.

    MathSciNet  MATH  Google Scholar 

  19. Shiohama, K., ‘Pinching Theorem for the Real Projective Space’,J. Math. Soc. Japan 26 (1974), 161–167.

    Article  MathSciNet  MATH  Google Scholar 

  20. Tsagas, G. and Xenos, P. J., ‘On the Cohomology Ring of a Homogeneous Manifold’,Tensor, N.S. 43 (1986), 248–254.

    MathSciNet  MATH  Google Scholar 

  21. Wallach, N. L., ‘Compact Homogeneous Riemannian Manifolds with Strictly Positive Curvature’,Ann. Math. 96 (1972), 277–295.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Karsten Grove

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  1. Karsten Grove
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Grove, K. The even dimensional pinching problem and SU(3)/T . Geom Dedicata 29, 327–334 (1989). https://doi.org/10.1007/BF00572449

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