Einstein submanifolds with flat normal bundle in space forms are holonomic
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- by Marcos Dajczer, Christos-Raent Onti and Theodoros Vlachos PDF
- Proc. Amer. Math. Soc. 146 (2018), 4035-4038 Request permission
Abstract:
A well-known result asserts that any isometric immersion with flat normal bundle of a Riemannian manifold with constant sectional curvature into a space form is (at least locally) holonomic. In this note, we show that this conclusion remains valid for the larger class of Einstein manifolds. As an application, when assuming that the index of relative nullity of the immersion is a positive constant we conclude that the submanifold has the structure of a generalized cylinder over a submanifold with flat normal bundle.References
- E. Cartan, Sur les variétés de courbure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 47 (1919), 125–160 (French). MR 1504786
- E. Cartan, Sur les variétés de courbure constante d’un espace euclidien ou non-euclidien, Bull. Soc. Math. France 48 (1920), 132–208 (French). MR 1504796
- Marcos Dajczer and Ruy Tojeiro, An extension of the classical Ribaucour transformation, Proc. London Math. Soc. (3) 85 (2002), no. 1, 211–232. MR 1901374, DOI 10.1112/S0024611502013552
- Marcos Dajczer, Luis A. Florit, and Ruy Tojeiro, Reducibility of Dupin submanifolds, Illinois J. Math. 49 (2005), no. 3, 759–791. MR 2210258
- M. Dajczer, L. A. Florit, and R. Tojeiro, The vectorial Ribaucour transformation for submanifolds and applications, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4977–4997. MR 2320656, DOI 10.1090/S0002-9947-07-04211-0
- John Douglas Moore, Submanifolds of constant positive curvature. I, Duke Math. J. 44 (1977), no. 2, 449–484. MR 438256
- Helmut Reckziegel, Krümmungsflächen von isometrischen Immersionen in Räume konstanter Krümmung, Math. Ann. 223 (1976), no. 2, 169–181 (German). MR 425846, DOI 10.1007/BF01360880
Additional Information
- Marcos Dajczer
- Affiliation: IMPA – Estrada Dona Castorina, 110, 22460–320, Rio de Janeiro, Brazil
- MR Author ID: 54140
- Email: marcos@impa.br
- Christos-Raent Onti
- Affiliation: Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
- Address at time of publication: IMPA - Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, Brazil
- Email: christos.onti@impa.br
- Theodoros Vlachos
- Affiliation: Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
- MR Author ID: 291296
- Email: tvlachos@uoi.gr
- Received by editor(s): September 26, 2017
- Received by editor(s) in revised form: December 4, 2017
- Published electronically: April 18, 2018
- Communicated by: Lei Ni
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 4035-4038
- MSC (2010): Primary 53B25; Secondary 53C40, 53C42
- DOI: https://doi.org/10.1090/proc/14057
- MathSciNet review: 3825856