## Hypercomplex structures on Stiefel manifolds.(English)Zbl 0843.53030

This paper proves the existence of uncountably many distinct hypercomplex structures $$\{{\mathcal I}^a ({\mathbf p})\}$$ on Stiefel manifolds $$V$$ of complex 2-planes in complex $$n$$-space, and studies their properties. A hypercomplex structure on a smooth manifold $$M$$ is a $$\text{GL} (n, \mathbb{H})$$-structure preserved by a torsion-free connection. In particular every such $$M$$ has three complex structures $$I$$, $$J$$ and $$K$$ satisfying algebraic-relations of imaginary quaternions.
If $$n>2$$ and $${\mathbf p}= (p_1, p_2, \dots, p_n)\in (\mathbb{R}^*)^n$$, the first result asserts that for each such $${\mathbf p}$$ there is a compact hypercomplex manifold $$({\mathcal N} ({\mathbf p}), {\mathcal I} ({\mathbf p}))$$, where $${\mathcal N} ({\mathbf p})$$ is diffeomorphic to the Stiefel manifold $$V$$. To study these hypercomplex structures let $$C_n= \{{\mathbf p}\in \mathbb{R}^n\mid 0< p_1\leq p_2\leq \dots \leq p_n\}$$ be the positive cone and $${\mathbf p}$$ be commensurable if each of the ratios $$p_i/ p_j$$ is a rational number. From the property that permuting the coordinates of $${\mathbf p}$$ or changing the signs does not change the hypercomplex structure represented by $${\mathbf p}$$, one assumes that $${\mathbf p}\in C_n$$.
The second result asserts that if $${\mathbf p}$$ and $${\mathbf q}$$ are both commensurable sequences in $$C_n$$, then $${\mathcal N} ({\mathbf p})$$ and $${\mathcal N} ({\mathbf q})$$ are hypercomplex equivalent if and only if $${\mathbf p}= {\mathbf q}$$. Furthermore $${\mathcal N} ({\mathbf p})$$ is hypercomplex homogeneous if and only if $${\mathbf p}= \lambda (1, 1, \dots, 1)$$ for some $$\lambda\in \mathbb{R}^*$$. The discrete quotients are also studied. A commensurable $${\mathbf p}$$ is called basic if all $$p_i$$ are integers and the greatest common divisor of all $$p_i$$ is one. A basic $${\mathbf p}$$ is called coprime if the $$p_i$$ are pairwise relatively prime, and $$k$$-coprime if $${\mathbf p}$$ is an integer multiple of a basic sequence such that $$(p_i, p_j, k)$$ have no common factor for $$1\leq i< j\leq n$$. Let $${\mathbf p}$$ be a $$k$$-coprime. Then there is a compact hypercomplex manifold $${\mathcal H} ({\mathbf p}, k)$$ with universal cover $$\rho_k: {\mathcal N} ({\mathbf p})\to {\mathcal H} ({\mathbf p}, k)$$ such that $$\pi_1 ({\mathcal H} ({\mathbf p}, k))\cong \mathbb{Z}_k$$ and $$\rho_k$$ is a hypercomplex map.
Finally, this paper determines the connected component of the group of hypercomplex automorphisms of $${\mathcal N} ({\mathbf p})$$, and shows the existence of a natural hyperhermitian metric $$h({\mathbf p})$$ such that every infinitesimal automorphism is an infinitesimal isometry with respect to $$h({\mathbf p})$$.

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 32Q99 Complex manifolds 53C56 Other complex differential geometry 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 32M05 Complex Lie groups, group actions on complex spaces
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### References:

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