Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set M ~G. We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of M ~G. Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify nonnegatively curved fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.
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