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Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank

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Abstract.

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m≥5) of positive sectional curvature on which an (m−1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By [Wi], these results imply a homeomorphism classification for positively curved n-manifolds (n≥8) of almost maximal symmetry rank

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Authors and Affiliations

  1. Nankai Institute of Mathematics, Nankai University, Tianjing, 300071, P.R. China

    Fuquan Fang

  2. Mathematics Department, Rutgers University, New Brunswick, NJ, 08903, USA

    Xiaochun Rong

  3. Mathematics Department, Capital Normal University, Beijing, P.R. China

    Fuquan Fang

  4. Mathematics Department, Capital Normal University, Beijing, P.R. China

    Xiaochun Rong

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  1. Fuquan Fang
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  2. Xiaochun Rong
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Correspondence to Fuquan Fang.

Additional information

Supported by CNPq of Brazil, NSFC Grant 19741002, RFDP and Qiu-Shi Foundation of China.

Supported partially by NSF Grant DMS 0203164 and a research grant from Capital normal university.

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Fang, F., Rong, X. Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank. Math. Ann. 332, 81–101 (2005). https://doi.org/10.1007/s00208-004-0618-y

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  • DOI: https://doi.org/10.1007/s00208-004-0618-y

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