Positively curved manifolds with symmetry

• Autores: Burkhard Wilking
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 163, Nº 2, 2006, págs. 607-668
• Idioma: inglés
• DOI: 10.4007/annals.2006.163.607
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• Resumen

  • There are very few examples of Riemannian manifolds with positive sectionalcurvature known. In fact in dimensions above 24 all known examplesare diffeomorphic to locally rank one symmetric spaces. We give a partialexplanation of this phenomenon by showing that a positively curved, simplyconnected, compact manifold (M,g) is up to homotopy given by a rank onesymmetric space, provided that its isometry group Iso(M,g) is large. Moreprecisely we prove first that if dim(Iso(M,g)) ¡Ý 2 dim(M) . 6, then M is tangentially homotopically equivalent to a rank one symmetric space or M is homogeneous. Secondly, we show that in dimensions above 18(k +1)2 each M is tangentially homotopically equivalent to a rank one symmetric space, where k > 0 denotes the cohomogeneity, k = dim(M/Iso(M,g)).