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2010 | Buch

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Differential Geometry of Lightlike Submanifolds

verfasst von: Krishan L. Duggal, Bayram Sahin

Verlag: Birkhäuser Basel

Buchreihe : Frontiers in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract

Denote by R the set of real numbers and Rn their n-fold Cartesian product R × … × R, the set of all ordered n-tuples (x¹, …, xⁿ). Define a function

$$ d:R^n \times R^n ,{\text{ where }}d(x,y) = {\text{ }}\parallel x - y\parallel $$

for every pair (x, y) of the points x, y ∈ Rⁿ. This function d is known as the Euclidean metric in Rⁿ. Then, we call Rⁿ with the metric d the n-dimensional Euclidean space. Consider V a real n-dimensional vector space with a symmetric bilinear mapping g: V × V → R. We say that g is positive (negative) definite on V if g(v, v) ≥ 0 (g(v, v) ≤ 0) for any non-zero v ∈ V. On the other hand, if g(v, v)=0 (g(v, v) ≤ 0) for any v ∈ V and there exists a non-zero u ∈ V with g(u, u)=0, we say that g is positive (negative) semi-definite on V.

Chapter 2. Lightlike hypersurfaces

Abstract

Since for any semi-Riemannian manifold $ \bar M $ there is a natural existence of null (lightlike) subspaces, their study is equally desirable. In particular, from the point of physics lightlike hypersurfaces are of importance as they are models of various types of horizons, such as Killing, dynamical and conformal horizons, studied in general relativity (see some details in Chapter 3). However, due to the degenerate metric of a lightlike submanifold M, one fails to use, in the usual way, the theory of non-degenerate geometry. The primary difference between the lightlike submanifolds and the non-degenerate submanifolds is that in the first case the normal vector bundle intersects the tangent bundle. In other words, a vector of a tangent space $ T_x \bar M $ cannot be decomposed uniquely into a component tangent to T_xM and a component of normal space T_xM^⊥. Therefore, the standard definition of the second fundamental form and the Gauss-Wiengarten formulas do not work, in the usual way, for the lightlike case.

Chapter 3. Applications of lightlike hypersurfaces

Abstract

In this chapter we present the latest work on applications of lightlike hypersurfaces in two active ongoing research areas in mathematical physics. First, we deal with black hole horizons. We prove a Global Null Splitting Theorem and relate it with physically significant works of Galloway [197] on null hypersurfaces in general relativity, Ashtekar and Krishnan’s work [16] on dynamical horizons and Sultana-Dyer’s work [378, 379] on conformal Killing horizons, with related references. Secondly, we present the latest work on Osserman lightlike hypersurfaces [20].

Chapter 4. Half-lightlike submanifolds

Abstract

There are two cases of codimension 2 lightlike submanifolds M since for this type the dimension of their radical distribution RadTM is either one or two. A codimension 2 lightlike submanifold is called half-lightlike [147] if dim(Rad TM)=1. The objective of this chapter is to present up-to-date results of this sub-case.

Chapter 5. Lightlike submanifolds

Abstract

The objective of this chapter is to present an up-to-date account of the works published on the general theory of lightlike submanifolds of semi-Riemannian manifolds. This includes unique existence theorems for screen distributions, geometry of totally umbilical, minimal and warped product lightlike submanifolds.

Chapter 6. Submanifolds of indefinite Kähler manifolds

Abstract

In the 1996 book [149] there is a brief discussion on Cauchy-Riemann(CR) lightlike submanifolds of an indefinite Kähler manifold. Contrary to the non-degenerate case [45, 133 373], CR-lightlike submanifolds are non-trivial (i.e., they do not include invariant (complex) and real parts). Since then considerable work has been done on new concepts to obtain a variety of classes of lightlike submanifolds. In this chapter we present up-to-date new results on all possible (complex, screen real and totally real) lightlike submanifolds of an indefinite Kähler manifold.

Chapter 7. Submanifolds of indefinite Sasakian manifolds

Abstract

In this chapter, we first give a review of indefinite Sasakian manifolds, contact CR-submanifolds and a variety of other submanifolds of Sasakian manifolds. Then, similar to the case of the previous chapter, we focus on up-to-date published results (with complete proofs) on the geometry of invariant, contact Cauchy-Riemann (CR) and contact screen Cauchy-Riemann (SCR) lightlike submanifolds of indefinite Sasakian manifolds.

Chapter 8. Submanifolds of indefinite quaternion Kähler manifolds

Abstract

In this chapter, we first recall the structure of indefinite quaternion Käahler manifolds. Then, we give a review of Riemannian submanifolds of quaternion Käahler manifolds. We study the geometry of real lightlike hypersurfaces, the structure of lightlike submanifolds, both, of indefinite quaternion Kähler manifolds and show that a quaternion lightlike submanifold is always totally geodesic. This result implies that the study of lightlike submanifolds, other than quaternion lightlike submanifolds, is interesting. Then, we deal with the geometry of screen real submanifolds in detail. As a generalization of real lightlike hypersurfaces of quaternion Kähler manifolds, we introduce QR-lightlike submanifolds. We show that the class of QR-lightlike submanifolds does not include quaternion lightlike submanifolds and screen real submanifolds. Then, we introduce and study the geometry of screen QR-lightlike and screen CR-lightlike submanifolds as generalizations of quaternion lightlike submanifolds and screen real submanifolds, and provide examples for each class of lightlike submanifolds of indefinite quaternion Kähler manifolds.

Chapter 9. Applications of lightlike geometry

Abstract

In this chapter we present applications of lightlike geometry in the study of null 2-surfaces in spacetimes, lightlike versions of harmonic maps and morphisms, CRstructures in general relativity and lightlike contact geometry in physics.

Backmatter

Titel: Differential Geometry of Lightlike Submanifolds
verfasst von: Krishan L. Duggal
Bayram Sahin
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-0346-0251-8
Print ISBN: 978-3-0346-0250-1
DOI: https://doi.org/10.1007/978-3-0346-0251-8

Springer Professional

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Chapter 2. Lightlike hypersurfaces

Chapter 3. Applications of lightlike hypersurfaces

Chapter 4. Half-lightlike submanifolds

Chapter 5. Lightlike submanifolds

Chapter 6. Submanifolds of indefinite Kähler manifolds

Chapter 7. Submanifolds of indefinite Sasakian manifolds

Chapter 8. Submanifolds of indefinite quaternion Kähler manifolds

Chapter 9. Applications of lightlike geometry

Backmatter

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