On closed six-manifolds admitting Riemannian metrics with positive sectional curvature and non-abelian symmetry (Q6661070)

Historically, the search for manifolds admitting Riemannian metrics with positive sectional curvature becomes tractable only when one settles on a subclass, such as the simply connected normal homogeneous spaces selected for study by \textit{M. Berger} [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 15, 179--246 (1961; Zbl 0101.14201)] or compact even-dimensional homogeneous spaces [\textit{N. R. Wallach}, Ann. Math. (2) 96, 277--295 (1972; Zbl 0261.53033)], or resulting from isometric actions on Lie groups such as in [\textit{J.-H. Eschenburg}, Differ. Geom. Appl. 2, No. 2, 123--132 (1992; Zbl 0778.53033)].\N\NOver the various subclasses that have been considered, but in a handful of dimensions, the known simply connected manifolds admitting Riemannian metrics with positive sectional curvature amount to ones already found by Berger in his initial landmark study, namely, the compact rank one symmetric spaces. The first dimension where one needs to go outside the realm of compact rank one symmetric spaces is dimension six, where the complex flag manifold of Wallach \(\mathrm{SU}(3)/T^2\) and twisted flag manifold of Eschenburg \(\mathrm{SU}(3)//T^2\) (where \(T^2=S^1 \times S^1\) acts on \(\mathrm{SU}(3)\) by \((z,w) \cdot A = \mathrm{diag}(z,w,zw)A\mathrm{diag}(1,1,(zw)^{-2})\)) feature.\N\NIn this article the author concerns himself with those six-manifolds admitting Riemannian metrics with positive sectional curvature and an effective isometric action of the special unitary group \(\mathrm{SU}(2)\) or special orthogonal group \(\mathrm{SO}(3)\). He first shows that under this assumption the Euler characteristic of the six-manifold must be 2, 4, or 6. Moreover, he shows that in the event the principal isotropy is trivial, the six-manifold is equivariantly diffeomorphic to the six-sphere with a standard linear action, or in the event the principal isotropy is non-trivial, the exceptional isotropy groups are restricted to be cyclic or dihedral.\N\NSpecializing to the special unitary group case, he shows that if the effective isometric action has fixed points, then the six-manifold is equivariantly diffeomorphic to a six-sphere or complex projective three-space with a linear action; if the action is free, he shows the six-manifold is diffeomorphic to one of \(S^6\), \(S^2 \times S^4\), or Wallach's complex flag manifold \(\mathrm{SU}(3)/T^2\) (where \(T^2\) is the maximal torus in \(\mathrm{SU}(3)\)).\N\NFinally, in the special orthogonal group case, the author shows that if the orbit space of the action is a closed three-ball and its boundary two-sphere contains more than one orbit type and has no exceptional orbits or interior singular orbits, then the six-manifold is equivariantly homeomorphic to the six-sphere with a linear action.\N\NThe author attacks proofs of the above results by dividing his analysis according to the topological type of the orbit space of the effective isometric action. Namely, by whether it is homeomorphic to the four-ball, three-ball, or three-sphere.