PORTUGALIAE MATHEMATICA Vol. 55, No. 4, pp. 485504 (1998) 

Some Embeddings of the Space of Partially Complex StructuresCecília Ferreira and Armando MachadoCMAF da Universidade de Lisboa,Av. Prof. Gama Pinto 2, 1699 Lisboa Codex  PORTUGAL Email: cecilia@ptmat.lmc.fc.ul.pt, armac@ptmat.lmc.fc.ul.pt 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 $O(n)/(U(k)\times O(n2k))$. 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 oneparameter 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; semiKähler manifold; minimal submanifold; $(2,0)$geodesic. Classification (MSC2000): 53C40; 53C15, 53C30, 53C55 Full text of the article:
Electronic version published on: 29 Mar 2001. This page was last modified: 27 Nov 2007.
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