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Umbilical Properties of Spacelike Co-dimension Two Submanifolds

Authors:
Nastassja Cipriani at University of Turin
Nastassja Cipriani
José M. M. Senovilla at University of the Basque Country
José M. M. Senovilla
Joeri Van der Veken at KU Leuven
Joeri Van der Veken
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Abstract

For Riemannian submanifolds of a semi-Riemannian manifold, we introduce the concepts of \emph{total shear tensor} and \emph{shear operators} as the trace-free part of the corresponding second fundamental form and shape operators. The relationship between these quantities and the umbilical properties of the submanifold is shown. Several novel notions of umbilical submanifolds are then considered along with the classical concepts of totally umbilical and pseudo-umbilical submanifolds. Then we focus on the case of co-dimension 2, and we present necessary and sufficient conditions for the submanifold to be umbilical with respect to a normal direction. Moreover, we prove that the umbilical direction, if it exists, is unique ---unless the submanifold is totally umbilical--- and we give a formula to compute it explicitly. When the ambient manifold is Lorentzian we also provide a way of determining its causal character. We end the paper by illustrating our results on the Lorentzian geometry of the Kerr black hole.
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Results Math 72 (2017), 25–46
c
2017 Springer International Publishing
1422-6383/17/010025-22
published online January 6, 2017
DOI 10.1007/s00025-016-0640-x Results in Mathematics
Umbilical Properties of Spacelike
Co-dimension Two Submanifolds
Nastassja Cipriani, Jos´e M. M. Senovilla, and Joeri Van der Veken
Abstract. For Riemannian submanifolds of a semi-Riemannian manifold,
we introduce the concepts of total shear tensor and shear operators as the
trace-free part of the corresponding second fundamental form and shape
operators. The relationship between these quantities and the umbilical
properties of the submanifold is shown. Several novel notions of umbilical
submanifolds are then considered along with the classical concepts of to-
tally umbilical and pseudo-umbilical submanifolds. Then we focus on the
case of co-dimension 2, and we present necessary and sufficient conditions
for the submanifold to be umbilical with respect to a normal direction.
Moreover, we prove that the umbilical direction, if it exists, is unique
—unless the submanifold is totally umbilical— and we give a formula to
compute it explicitly. When the ambient manifold is Lorentzian we also
provide a way of determining its causal character. We end the paper by
illustrating our results on the Lorentzian geometry of the Kerr black hole.
Mathematics Subject Classification. 53B25, 53B30, 53B50.
Keywords. Umbilical submanifolds, shear, pseudo-umbilical.
This work is partially supported by the Belgian Interuniversity Attraction Pole P07/18
(Dygest), by the KU Leuven Research Fund project 3E160361 “Lagrangian and calibrated
submanifolds”, and was initiated during a visit of the first author to the UPV/EHU sup-
ported by a travel grant of the Research Foundation–Flanders (FWO). 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 under project UFI 11/55
(UPV/EHU).
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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... Here ⃗ Σ AB is the total shear tensor [13] and ⃗ H is the mean curvature vector of S in M [23,34,44]. By definition, ...
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