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On a theorem of Grove and Searle

A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with effective circular symmetry group of dimension 8 or less have positive Euler characteristic X(M): the fixed point set N consists of even dimensional positive curvature manifolds and has the Euler characteristic X(N)=X(M). It is not empty by Berger. If N has a co-dimension 2 component, Grove-Searle forces M to be in { RP^2d,S^2d,CP^d }. By Frankel, there can be not two codimension 2 cases. In the remaining cases, Gauss-Bonnet-Chern forces all to have positive Euler characteristic. This simple proof does not quite reach the record 10 or less which uses methods of Wilking but it motivates to analyze the structure of fixed point components N and in particular to look at positive curvature manifolds which admit a U(1) or SU(2) symmetry with connected or almost connected fixed point set N. They have amazing geodesic properties: the fixed point manifold N agrees with the caustic of each of its points and the geodesic flow is integrable. In full generality, the Lefschetz fixed point property X(N)=X(M) and Frankel's dimension theorem dim(M) is less than dim(A) + dim(B) for two different connectivity components A,B of N produce already heavy constraints in building up M from smaller components. It is possible that S^2d, RP^2d, CP^d, HP^d, OP^2, W^6,E^6,W^12,W^24 are actually a complete list of even-dimensional positive curvature manifolds admitting a continuum symmetry. Aside from the projective spaces, the Euler characteristic of the known cases is always 1,2 or 6, where the jump from 2 to 6 happened with the Wallach or Eschenburg manifolds W^6,E^6 which have four fixed point components N=S^2 + S^2 + S^0, the only known case which are not of the Grove-Searle form N=N_1 or N=N_1 + p} with connected N_1.