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”.


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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