TAILIEUCHUNG - Đề tài " Positively curved manifolds with symmetry "

There are very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions above 24 all known examples are diffeomorphic to locally rank one symmetric spaces. We give a partial explanation of this phenomenon by showing that a positively curved, simply connected, compact manifold (M, g) is up to homotopy given by a rank one symmetric space, provided that its isometry group Iso(M, g) is large. More precisely we prove first that if dim(Iso(M, g)) ≥ 2 dim(M ) − 6, then M is tangentially homotopically equivalent to a rank one symmetric space or M. | Annals of Mathematics Positively curved manifolds with symmetry By Burkhard Wilking Annals of Mathematics 163 2006 607 668 Positively curved manifolds with symmetry By Burkhard Wilking Abstract There are very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions above 24 all known examples are diffeomorphic to locally rank one symmetric spaces. We give a partial explanation of this phenomenon by showing that a positively curved simply connected compact manifold M g is up to homotopy given by a rank one symmetric space provided that its isometry group Iso M g is large. More precisely we prove first that if dim Iso M g 2dim 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 . Introduction Studying positively curved manifolds is a classical theme in differential geometry. So far there are very few constraints known. For example there is not a single obstruction known that distinguishes the class of simply connected compact manifolds that admit positively curved metrics from the class of simply connected compact manifolds that admit nonnegatively curved metrics. On the other hand the list of known examples is rather short as well. In particular in dimensions other than 6 7 12 13 and 24 all known simply connected positively curved examples are diffeomorphic to rank one symmetric spaces. To advance the theory Grove 1991 proposed to classify positively curved manifolds with a large amount of symmetry. This program may also be viewed as part of a philosophy of . Hsiang that in each category one should pay particular attention to those objects with a large amount of symmetry. Another possible motivation is that once one understands the obstructions to positive curvature under symmetry assumptions

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