Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Remote Access
Green Open Access
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. Bejancu, A., CR-submanifolds of a Kaehler manifold I. Proc. Amer. Math. Soc. 69 (1978), 134-142. MR 57:7486
  • 2. Bejancu, A., Geometry of CR-Submanifolds. D. Reidel, Dordrecht, 1986. MR 87k:53126
  • 3. Chen, B. Y., Geometry of Submanifolds and its Applications. Sci. Univ. Tokyo, 1981. MR 82m:53051
  • 4. Chen, B. Y., Slant immersions. Bull. Austral. Math. Soc. 41 (1990), 135-147. MR 91h:53051
  • 5. Chen, B. Y., Tazawa, Y., Slant submanifolds in complex Euclidean spaces. Tokyo J. Math. 14 (1991), 101-120. MR 92e:53075
  • 6. Etayo, F., On quasi-slant submanifolds of an almost Hermitian manifold. Publ. Math. Debrecen 53 (1998), 217-223. MR 99j:53073
  • 7. Etayo, F., Fioravanti, M., Trías, U. R., The dimension function of holomorphic spaces of a real submanifold of an almost complex manifold. Czechoslovak Math. J. 51 (126) (2001), 139-141. MR 2002a:53037
  • 8. Hodge, W. V. D., Pedoe, D., Methods of Algebraic Geometry, I. Cambridge Univ. Press, Cambridge, 1994 (first published in 1947 MR 10:396b). MR 95d:14002a
  • 9. Oguie, K., Differential Geometry of Kähler Submanifolds. Advances in Math. 13 (1974), 73-114. MR 49:11444
  • 10. Wells, R. O. Jr., Differential Analysis on Complex Manifolds. Springer, New York, 1980. MR 83f:58001
  • 11. Yano, K., Kon, M., Anti-invariant Submanifolds. M. Dekker Inc., New York, 1976. MR 54:13799

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C40, 53C55

Retrieve articles in all journals with MSC (2000): 53C40, 53C55

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

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

Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia