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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
DOI: https://doi.org/10.1090/proc/12987
Published electronically: December 22, 2015
MathSciNet review: 3451244
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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.


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Antonio J. Di Scala
Affiliation: Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000, Córdoba, Argentina
Email: antonio.discala@polito.it

Carlos Olmos
Affiliation: Dipartimento di Scienze Matematiche ‘G.L. Lagrange’, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy
Email: olmos@famaf.unc.edu.ar

DOI: https://doi.org/10.1090/proc/12987
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