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Classification of quaternionic spaces with transitive solvable group of motions. (Russian) Zbl 0309.53045
Summary: A complete classification of quaternionic Riemannian spaces (that is, spaces \( \mathcal{V}^n\) with the holonomy group \( \Gamma\subset\text{Sp}(1)\cdot\text{Sp}(m)\), \( n=4m\)) which admit a transitive solvable group of motions is given. It turns out that the rank of these spaces does not exceed four and that all spaces \( \mathcal{V}^n\) whose rank is less than four are symmetric. The spaces \( \mathcal{V}^n\) of rank four are in natural one-to-one correspondence with the Clifford modules of Atiyah, Bott and Shapiro. In this correspondence, the simplest Clifford modules, which are connected with division algebras, are mapped to symmetric spaces of exceptional Lie groups. Other Clifford modules, which are obtained from the simplest with help of tensor products, direct sums and restrictions, correspond to nonsymmetric spaces.

MSC:
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C35 Differential geometry of symmetric spaces
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