Some embeddings of the space of partially complex structures.(English)Zbl 0951.53040

Authors’ abstract: “Let $$E$$ be an $$n$$-dimensional Euclidean vector space. A partially complex structure of dimension $$k$$ in $$E$$ is a couple $$(F,J)$$, where $$F\subset E$$ is a real vector subspace, of dimension $$2k$$, and $$J:F\to F$$ is a complex structure on $$F$$, compatible with the induced inner product. The space of all such structures can be identified with the holomorphic homogeneous non-symmetric space $$O(n)/(U(k)\times O(n- 2k))$$. We study a family of $$({\mathcal G}_{kt}(E))_{t\in [0,\pi[}$$ of equivariant models of this homogeneous space inside the orthogonal group $$O(E)$$, from the viewpoint of its extrinsic geometry. The metrics induced by the biinvariant metric of $$O(E)$$ correspond to an interval of the one-parameter family of invariant compatible metrics of this homogeneous space, including the Kähler and the naturally reductive ones. The manifolds $${\mathcal G}_{kt}(E)$$ are $$(2,0)$$-geodesic inside $$O(E)$$; some of them are minimal inside $$O(E)$$ and others are minimal inside a suitable sphere. We show also that the model $${\mathcal F}_k(E)$$ inside the Lie algebra $$o(E)$$, corresponding to the compatible $$f$$-structures of Yano, is $$(2,0)$$-geodesic and minimal inside a sphere”.

MSC:

 53C40 Global submanifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C30 Differential geometry of homogeneous manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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