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Codimension reduction for real submanifolds of a complex hyperbolic space. (English) Zbl 0805.53014
The main result of the article is the following: If \(M\) is a real submanifold of the complex hyperbolic space \(\mathbb{C} H^ n\) and if there exists a \(J\)-invariant, parallel subbundle of rank \(2d\) in the normal bundle of \(M\) orthogonal to the first normal spaces of \(M\) everywhere, then \(M\) lies in a complex \((n - d)\)-dimensional hyperbolic subspace of \(\mathbb{C} H^ n\). For the proof the author lifts the problem to the anti-de Sitter space via the Hopf fibration, for which he derives an appropriate theorem on the reduction of the codimension. The analogous theorem for submanifolds of the complex projective space is due to M. Okumura [see Colloq. Math. Soc. János Bolyai 56, 573-585 (1992; Zbl 0789.53035)]. The results can also be derived from Theorem 1 of Th. Meumertzheim and H. Reckziegel [J. Reine Angew. Math. 407, 19-32 (1990; Zbl 0692.53006)].

MSC:
53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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