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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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