On positively curved 4-manifolds with S¹-symmetry

2008/10/07 by Jin Hong Kim, Kim, Jin Hong · 1 citation
Mathematics · #53C20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.0810.1232

openalex publication_date 2008/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well-known by the work of Hsiang and Kleiner that every closed oriented positively curved 4-dimensional manifold with an effective isometric S¹-action is homeomorphic to S⁴ or CP². As stated, it is a topological classification. The primary goal of this paper is to show that it is indeed a diffeomorphism classification for such 4-dimensional manifolds. The proof of this diffeomorphism classification also shows an even stronger statement that every positively curved simply connected 4-manifold with an isometric circle action admits another smooth circle action which extends to a 2-dimensional torus action and is equivariantly diffeomorphic to a linear action on S⁴ or CP². The main strategy is to analyze all possible topological configurations of effective circle actions on simply connected 4-manifolds by using the so-called replacement trick of Pao.

Cited by

• Topology of positively curved 8-dimensional manifolds with symmetry

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