Umbilical properties of spacelike co-dimension two submanifolds

Cipriani, Nastassja; Senovilla, José M. M.; Veken, Joeri Van der

doi:10.48550/arxiv.1604.06375

Cited by 3 publications

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“…In the present paper we generalize what has been done in [6] and [1]. We consider spacelike submanifolds of arbitrary co-dimension and give a characterization of those which are umbilical with respect to some normal directions.…”

Section: Introductionmentioning

confidence: 84%

“…They have mainly been studied in the Riemannian setting and, apart from very few exceptions, they have been applied to submanifolds of co-dimension one (hypersurfaces). In [1] the authors studied these concepts in a slightly more general framework, namely, allowing the ambient manifold to have arbitrary signature while requiring the submanifold to be spacelike (i.e. endowed with a Riemannian induced metric).…”

Section: Introductionmentioning

confidence: 99%

“…endowed with a Riemannian induced metric). Motivated by the applications to gravitation and general relativity, in [1] the authors focused on spacelike submanifolds of co-dimension two. Notice that the results presented in [1] generalized those presented in [6].…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the applications to gravitation and general relativity, in [1] the authors focused on spacelike submanifolds of co-dimension two. Notice that the results presented in [1] generalized those presented in [6]. In the latter, the study of umbilical surfaces (2-dimensional) in 4-dimensional Lorentzian manifolds was carried out.…”

Section: Introductionmentioning

confidence: 99%

“…This is defined as the trace-free part of the second fundamental form and it appears often in the mathematical literature, especially in conformal geometry. Nevertheless, it had never been given a name prior to [1] and, surprisingly, its relationship with the umbilical properties of submanifolds seemed to be almost unknown or is not explicitly mentioned at least.…”

Section: Introductionmentioning

confidence: 99%

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Umbilical Spacelike Submanifolds of Arbitrary Co-dimension

Cipriani

Senovilla

2017

Lorentzian Geometry and Related Topics

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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.2010 Mathematics Subject Classification. 53B25, 53B30, 53B50.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Umbilical Spacelike Submanifolds of Arbitrary Co-dimension

Cipriani

Senovilla

2017

Lorentzian Geometry and Related Topics

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Geometric horizons

2017

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We discuss black hole spacetimes with a geometrically defined quasilocal horizon on which the curvature tensor is algebraically special relative to the alignment classification. Based on many examples and analytical results, we conjecture that a spacetime horizon is always more algebraically special (in all of the orders of specialization) than other regions of spacetime. Using recent results in invariant theory, such geometric black hole horizons can be identified by the alignment type II or D discriminant conditions in terms of scalar curvature invariants, which are not dependent on spacetime foliations. The above conjecture is, in fact, a suite of conjectures (isolated vs dynamical horizon; four vs higher dimensions; zeroth order invariants vs higher order differential invariants). However, we are particularly interested in applications in four dimensions and especially the location of a black hole in numerical computations.

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Identification of black hole horizons using scalar curvature invariants

Coley¹,

McNutt²

2017

Class. Quantum Grav.

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We introduce the concept of a geometric horizon, which is a surface distinguished by the vanishing of certain curvature invariants which characterize its special algebraic character. We motivate its use for the detection of the event horizon of a stationary black hole by providing a set of appropriate scalar polynomial curvature invariants that vanish on this surface. We extend this result by proving that a non-expanding horizon, which generalizes a Killing horizon, coincides with the geometric horizon. Finally, we consider the imploding spherically symmetric metrics and show that the geometric horizon identifies a unique quasi-local surface corresponding to the unique spherically symmetric marginally trapped tube, implying that the spherically symmetric dynamical black holes admit a geometric horizon. Based on these results, we propose a suite of conjectures concerning the application of geometric horizons to more general dynamical black hole scenarios.

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Umbilical properties of spacelike co-dimension two submanifolds

Cited by 3 publications

References 0 publications

Umbilical Spacelike Submanifolds of Arbitrary Co-dimension

Umbilical Spacelike Submanifolds of Arbitrary Co-dimension

Geometric horizons

Identification of black hole horizons using scalar curvature invariants

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