Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The relative nullity of complex submanifolds and the Gauss map

Authors: Antonio J. Di Scala and Carlos Olmos
Journal: Proc. Amer. Math. Soc. 144 (2016), 1689-1695
MSC (2010): Primary 53C29, 53C40
Published electronically: December 22, 2015
MathSciNet review: 3451244
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a short and geometric proof, based on Jacobi fields, of a theorem of K. Abe that asserts that the relative index of nullity is trivial for complete non-totally geodesic complex projective submanifolds. Using this idea we prove a splitting theorem for complex Euclidean submanifolds with a non-trivial relative index of nullity.

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

  • [1] Dmitri V. Alekseevsky and Antonio J. Di Scala, The normal holonomy group of Kähler submanifolds, Proc. London Math. Soc. (3) 89 (2004), no. 1, 193-216. MR 2063664 (2005m:53075),
  • [2] Kinetsu Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions, Tôhoku Math. J. (2) 25 (1973), 425-444. MR 0350671 (50 #3163)
  • [3] Jürgen Berndt, Sergio Console, and Carlos Olmos, Submanifolds and holonomy, Chapman & Hall/CRC Research Notes in Mathematics, vol. 434, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1990032 (2004e:53073)
  • [4] Eugenio Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1-23. MR 0057000 (15,160c)
  • [5] Marcos Dajczer and Pedro Morais, Isometric rigidity in codimension 2, Michigan Math. J. 58 (2009), no. 3, 759-770. MR 2595563 (2011c:53086),
  • [6] Dirk Ferus, Totally geodesic foliations, Math. Ann. 188 (1970), 313-316. MR 0271872 (42 #6753)
  • [7] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725 (80b:14001)
  • [8] Phillip Griffiths and Joseph Harris, Algebraic geometry and local differential geometry, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 3, 355-452. MR 559347 (81k:53004)
  • [9] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1969 original; A Wiley-Interscience Publication. MR 1393941 (97c:53001b)
  • [10] Joseph Lewittes, Differentials and matrices on Riemann surfaces, Trans. Amer. Math. Soc. 139 (1969), 311-318. MR 0237769 (38 #6050)
  • [11] Carlos Olmos and Francisco Vittone, On completeness of integral manifolds of nullity distributions, Rev. Un. Mat. Argentina 53 (2012), no. 2, 89-90. MR 3086806
  • [12] F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, vol. 127, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by the author. MR 1234494 (94i:14053)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C29, 53C40

Retrieve articles in all journals with MSC (2010): 53C29, 53C40

Additional Information

Antonio J. Di Scala
Affiliation: Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000, Córdoba, Argentina

Carlos Olmos
Affiliation: Dipartimento di Scienze Matematiche ‘G.L. Lagrange’, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

Keywords: Gauss map, relative nullity index, Jacobi vector fields
Received by editor(s): February 20, 2015
Published electronically: December 22, 2015
Additional Notes: The first author is a member of PRIN 2010-2011 “Varieta’ reali e complesse: geometria, topologia e analisi armonica” and a member of GNSAGA of INdAM.
The second author was supported by Famaf-UNC, CIEM-Conicet and Project Erasmus Mundus Action 2 Eurotango II
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society