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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The measure of holomorphicness of a real submanifold of an almost Hermitian manifold


Author: Fernando Etayo
Journal: Proc. Amer. Math. Soc. 131 (2003), 2911-2920
MSC (2000): Primary 53C40; Secondary 53C55
Published electronically: April 9, 2003
MathSciNet review: 1974349
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Abstract: In this note we define the measure of holomorphicness $\mu (M)$ of a compact real submanifold $M$ of an almost Hermitian manifold $(\overline{M},\overline{J},\overline{g})$. The number $\mu (M)\in [0,1]$verifies the following properties: $M$ is a complex submanifold iff $\mu (M)=1$; if $\dim M$ is odd, then $\mu (M)=0$. Explicit examples of surfaces in ${\mathbb C}^{2}$ are obtained, showing that $\mu (S^{2})=\frac{1}{5}$ and that $0\leq \mu (T)\leq \frac{3}{8}$, $T$ being the Clifford torus.


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Additional Information

Fernando Etayo
Affiliation: Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda. de los Castros, s.n., E-39071 Santander, Spain
Email: etayof@unican.es

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07112-0
PII: S 0002-9939(03)07112-0
Keywords: Measure of holomorphicness, almost Hermitian manifold, submanifold, Pl\"{u}cker
Received by editor(s): May 25, 2001
Published electronically: April 9, 2003
Additional Notes: The author’s research was partially supported by the Spanish Ministerio de Ciencia y Technología (BFM 2002-00141)
Communicated by: Mohan Ramachandran
Article copyright: © Copyright 2003 American Mathematical Society