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A diffeomorphism classification of 7-dimensional homogeneous Einstein manifolds with \(SU(3)\times SU(2)\times U(1)\)-symmetry. (English) Zbl 0649.53029

Let G be a compact Lie group. A manifold M has G-symmetry if there exists a Riemannian metric such that the full group of isometries has the same Lie algebra as G. If G acts transitively, M is a homogeneous space and is called a homogeneous manifold with G-symmetry. Let \(M_{k,l}\), \(k\neq 0\), \(l\neq 0\), be the orbit space of \(S^ 1\)-actions on \(S^ 5\times S^ 3\) parametrized by coprime integers (k,l) where \(z\in S^ 1\) acts by (x,y)\(\to (z^ kx,z^ ly)\). For different k, l with the restrictions above, one obtains all 7-dimensional 1-connected homogeneous manifolds with SU(3)\(\times SU(2)\times U(1)\) symmetry [see M. J. Duff, B. E. W. Nilsson and C. N. Pope, Kaluza-Klein supergravity, Phys. Rep. 130, 1-142 (1986)].
The main results, Theorems A and B give a complete classification of the above manifolds. In Theorem A, the homeomorphism and diffeomorphism type is determined by analytic invariants, described in section 2 of the paper. Theorem B classifies the manifolds in terms of k, l and shows that for some of the homogeneous spaces one obtains a negative answer to the question of W.-c and W.-y. Hsiang: Is it true that two homeomorphic homogeneous spaces are necessarily diffeomorphic?
Reviewer: I.Dotti Miatello

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds
57R55 Differentiable structures in differential topology
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