## 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

### Keywords:

G-symmetry; homogeneous manifold; complete classification
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