Abstract
We study Grassmannian bundles Gk(M) of analytical 2k-planes over an almost Hermitian manifold M2n, from the point of view of the generalized twistor spaces of [13], and with the method of the moving frame [9]. G1(M4) is the classical twistor space. We find four distinguished almost Hermitian structures, one of them being that of [13], and discuss their integrability and Kählerianity. For n=2, we compute the corresponding Hermitian connections, and derive consequences about the corresponding first Chern classes.
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Vaisman, I. On the Twistor Spaces of Almost Hermitian Manifolds. Annals of Global Analysis and Geometry 16, 335–356 (1998). https://doi.org/10.1023/A:1006551225776
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DOI: https://doi.org/10.1023/A:1006551225776