Some Embeddings of the Space of Partially Complex Structures

Cecília Ferreira and Armando Machado

Abstract: Let $E$ be a Euclidean $n$-dimensional vector space. A partially complex structure with dimension $k$ in $E$ is a couple $(F,J)$, where $F\subset E$ is a real vector subspace, with dimension $2k$, and $J: F\rightarrow F$ is a complex structure in $F$, 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 $(\mathcal{G}_{kt}(E))_{t\in[0,\pi[}$ 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.

Keywords: partially complex structure; semi-Kähler manifold; minimal submanifold; $(2,0)$-geodesic.

Classification (MSC2000): 53C40; 53C15, 53C30, 53C55

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