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holomorphic space; submanifold; almost complex
Let $ M $ be a real submanifold of an almost complex manifold $ (\overline{M},\overline{J}) $ and let $ H_{x}=T_{x}M\cap \overline{J}(T_{x}M) $ be the maximal holomorphic subspace, for each $ x\in M $. We prove that $ c\:M\rightarrow \mathbb{N} $, $ c(x)=\dim _{\mathbb{R}} H_{x} $ is upper-semicontinuous.
[C] B. Y. Chen: Geometry of Submanifolds and its Applications. Sci. Univ. Tokyo, 1981. MR 0627323 | Zbl 0474.53050
[K-N] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, II. Interscience, New York, 1969.
[W] R. O. Wells, Jr.: Differential Analysis on Complex Manifolds. Springer, New York, 1980. MR 0608414 | Zbl 0435.32004
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