Abstract
Given a semi-Riemannian manifold, we give necessary and sufficient conditions for a Riemannian submanifold of arbitrary co-dimension to be umbilical along normal directions. We do that by using the so-called total shear tensor, i.e., the trace-free part of the second fundamental form. We define the shear space and the umbilical space as the spaces generated by the total shear tensor and by the umbilical vector fields, respectively. We show that the sum of their dimensions must equal the co-dimension.
This work is partially supported by the Belgian Interuniversity Attraction Pole P07/18 (Dygest) and by the KU Leuven Research Fund project 3E160361 “Lagrangian and calibrated submanifolds”. NC and JMMS are supported under grant FIS2014-57956-P (Spanish MINECO–Fondos FEDER) and project IT956-16 of the Basque Government. JMMS is also supported by EU COST action No. CA15117 “CANTATA”.
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We would like to thank the referee for comments and improvements.
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Cipriani, N., Senovilla, J.M.M. (2017). Umbilical Spacelike Submanifolds of Arbitrary Co-dimension. In: Cañadas-Pinedo, M., Flores, J., Palomo, F. (eds) Lorentzian Geometry and Related Topics. GELOMA 2016. Springer Proceedings in Mathematics & Statistics, vol 211. Springer, Cham. https://doi.org/10.1007/978-3-319-66290-9_4
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DOI: https://doi.org/10.1007/978-3-319-66290-9_4
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