On the twistor bundle of De Sitter space $\DS$

Eduardo Hulett

Abstract: We study the twistor bundle $\cal{Z}$ over De Sitter space $\DS$. Viewing $\cal{Z}$ as an $SO(1,1)$-principal bundle over the Grassmannian $G_2(\bb{L}^4)$ of oriented space-like planes in Lorentz-Minkowski $4$-space, the orthogonal complement of the fibers of $\pi':\cal{Z}\to G_2(\bb{L}^4)$ defines a 4-dimensional horizontal neutral (of signature $(++--)$) distribution $\cal{H} \subset T\cal{Z}$. Two $SO(3,1)$-invariant almost Cauchy-Riemann structures $ \cal{J}^I$ and $ \cal{J}^{II}$ on $\cal{H}$ are introduced. According to which structure is considered two classes of horizontal holomorphic maps arise. These maps are projected to $\DS$ onto space-like surfaces with different properties. We characterize both classes of horizontal maps in terms of the geometry of their projections to $\DS$.

Keywords: De Sitter space-time, twistor bundle, harmonic maps, holomorphic structures

Classification (MSC2000): 53C43, 53C42, 53C28

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