Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

The title immediately brings to mind a standard reference of almost the same title [1]. The authors are quick to point out the relationship between these two works: they are complementary. The purpose of this work is to explain what is known about a selection of exact solutions. As the authors state, it is often much easier to find a new solution of Einstein's equations than it is to understand it. Even at first glance it is very clear that great effort went into the production of this reference. The book is replete with beautifully detailed diagrams that reflect deep geometric intuition. In many parts of the text there are detailed calculations that are not readily available elsewhere. The book begins with a review of basic tools that allows the authors to set the notation. Then follows a discussion of Minkowski space with an emphasis on the conformal structure and applications such as simple cosmic strings. The next two chapters give an in-depth review of de Sitter space and then anti-de Sitter space. Both chapters contain a remarkable collection of useful diagrams. The standard model in cosmology these days is the ICDM model and whereas the chapter on the Friedmann-Lema?tre?Robertson?Walker space-times contains much useful information, I found the discussion of the currently popular a representation rather too brief. After a brief but interesting excursion into electrovacuum, the authors consider the Schwarzschild space-time. This chapter does mention the Swiss cheese model but the discussion is too brief and certainly dated. Space-times related to Schwarzschild are covered in some detail and include not only the addition of charge and the cosmological constant but also the addition of radiation (the Vaidya solution). Just prior to a discussion of the Kerr space-time, static axially symmetric space-times are reviewed. Here one can find a very interesting discussion of the Curzon?Chazy space-time. The chapter on rotating black holes is rather brief and, for example, does not contain reference to the insights found by Pretorius and Israel [2]. This is perhaps justifiable in view of the many specialized texts devoted to the Kerr space-time (e.g. [3]). The large clear diagrams that one becomes accustomed to in this book show off the Taub-NUT (and related) space-times in the next chapter. After perhaps a somewhat standard discussion of stationary axially symmetric space-times, there is a very informative discussion of accelerating black holes. For example, the global structure of the C-metric is considered in detail. This is followed by a brief discussion of solutions for uniformly accelerating particles. The discussion of the Pleba?ski-Demia?ski solutions contains two very useful flow charts that help to systematize two rather complex families of solutions. After a somewhat brief discussion of plane and pp-waves, the authors give an extensive discussion of the Kunt solutions. I note here that after this text was in production the importance of the Kunt space-times as regards the characterization of space-times by scalar curvature invariants was made clear [4]. The discussion of the Robinson-Trautman solutions that follows is extensive, containing, for example, details of the singularity structure and of the global structure. The final formal chapter in this text covers colliding plane waves. This contains, for example, discussions of the Khan?Penrose, Ferrari?Iba?ez and Chandrasekhar?Xanthopoulos solutions. The text ends with a `final miscellany'. This covers a number of interesting topics, but I found the discussion of the Lema?tre?Tolman solutions rather weak (compare e.g. [5]). The book has two quite useful appendices covering 2-spaces and 3-spaces of constant curvature. To conclude, I will quote from the dust jacket: `The book is an invaluable resource for both graduate students and academic researchers working in gravitational physics'. I highly recommend it. References [1] Stephani H, Kramer D, MacCallum M, Hoenselaers C and Herlt E 2003 Exact Solutions of Einstein's Field Equations (Second Edition) (Cambridge: Cambridge University Press) [2] Pretorius F and Israel W 1998 Class. Quantum Grav.15 2289 [3] Wiltshire D, Visser M and Scott S (ed) 2008 The Kerr Spacetime: Rotating Black Holes in General Relativity (Cambridge: Cambridge University Press) [4] Coley A, Hervik S and Pelavas N 2009 Class. Quantum Grav. 26 025013 [5] Pleba?ski J and Krasi?ski A 2006 An Introduction to General Relativity and Cosmology (Cambridge: Cambridge University Press)

Exact Space-Times in Einstein's General Relativity

In this paper we give some examples of a quasi Einstein manifold (QE)n. Next we prove the existence of (QE)n manifolds. Then we study some properties of a quasi Einstein manifold. Finally the hypersurfaces of a Euclidean space have been studied.

On quasi Einstein manifolds

The geometry and, especially, the geodesics of a class of spacetimes generalizing Robertson-Walker ones (without any assumption on the fiber) is studied, under a global point of view. Our study covers geodesic connectedness, geodesic completeness and stability of completeness.

On the Geometry of Generalized Robertson-Walker Spacetimes: Geodesics

The monograph deals with the theory of conformal, geodesic,
holomorphically projective, F-planar and others mappings and
transformations of manifolds with affine connection,
Riemannian, Kahler and Riemann-Finsler manifolds. Concretely,
the monograph treats the following: basic concepts of
topological spaces, the theory of manifolds with affine
connection (particularly, the problem of semigeodesic
coordinates), Riemannian and Kahler manifolds (reconstruction
of a metric, equidistant spaces, variational problems in
Riemannian spaces, SO(3)-structure as a model of statistical
manifolds, decomposition of tensors), the theory of
differentiable mappings and transformations of manifolds (the
problem of metrization of affine connection, harmonic
diffeomorphisms), conformal mappings and transformations
(especially conformal mappings onto Einstein spaces, conformal
transformations of Riemannian manifolds), geodesic mappings
(GM; especially geodesic equivalence of a manifold with affine
connection to an equiaffine manifold), GM onto Riemannian
manifolds, GM between Riemannian manifolds (GM of equidistant
spaces, GM of Vn(B) spaces, its field of symmetric linear
endomorphisms), GM of special spaces, particularly Einstein,
Kahler, pseudosymmetric manifolds and their generalizations,
global geodesic mappings and deformations, GM between
Riemannian manifolds of different dimensions, global GM,
geodesic deformations of hypersurfaces in Riemannian spaces,
some applications of GM to general relativity, namely three
invariant classes of the Einstein equations and geodesic
mappings, F-planar mappings of spaces with affine connection,
holomorphically projective mappings (HPM) of Kahler manifolds
(fundamental equations of HPM, HPM of special Kahler manifolds,
HPM of parabolic Kahler manifolds, almost geodesic mappings,
which generalize geodesic mappings, Riemann-Finsler spaces and
their geodesic mappings, geodesic mappings of Berwald spaces
onto Riemannian spaces.

Differential Geometry of Special Mappings

On curvature properties of quasi-einstein hypersurfaces in semi-euclidean spaces

The symmetry of warped product space-times

We investigate Lagrangian submanifolds of the nearly Kahler 6-sphere. In particular we investigate Lagrangian quasi-Einstein submanifolds of the 6-sphere. We relate this class of submanifolds to certain tubes around almost complex curves in the 6-sphere. 1. Introduction. In this paper, we investigate 3-dimensional totally real submani- folds M3 of the nearly Kahler 6-sphere S6. A submanifold M3 of S6 is called totally real if the almost complex structure / on S6 interchanges the tangent and the normal space. It has been proven by Ejiri ((El)) that such submanifolds are always minimal and orientable. In the same paper, he also classified those totally real submanifolds with constant sectional curva- ture. Note that 3-dimensional Einstein manifolds have constant sectional curvature. Here, we will investigate the totally real submanifolds of S6 for which the Ricci tensor has an eigenvalue with multiplicity at least 2. In general, a manifold Mn whose Ricci tensor has an eigenvalue of multiplicity at least n — 1 is called quasi-Einstein. The paper is organized as follows. In Section 2, we recall the basic formulas about the vector cross product on RΊ and the almost complex structure on S6. We also relate the standard Sasakian structure on S5 with the almost complex structure on S6. Then, in Section 3, we derive a necessary and sufficient condition for a totally real submanifold of S6 to be quasi-Einstein. Using this condition, we deduce from (C), see also (CDVV1) and (DV), that totally real submanifolds M with 8M = 2 are quasi-Einstein. Here, 8M is the Riemannian invariant defined by

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Quasi-einstein totally real submanifolds of the nearly kahler 6-sphere

On some class of warped product manifolds

coordinate neighbourhoods { u j x r } . We denote by |B-|;}» H v r g t t R v t and R the Chr is tof f .e l symbols, the operator of covar iant d i f f e r e n t i a t i o n , the curvature t e n s o r , the Rioci t ensor and the s ca l a r curvature of N. The ind ices r , s , t , u,v,w run over the range | l , 2 , . . . , n | , n ^ 4 . A tensor f i e l d i r s t u i s ca l l ed a general ized curvature tensor on K ( [ 1 3 ] ) , i f

Ryszard Deszcz

Papers

On curvature properties of quasi-einstein hypersurfaces in semi-euclidean spaces

The symmetry of warped product space-times

Quasi-einstein totally real submanifolds of the nearly kahler 6-sphere

On some class of warped product manifolds

On totally umbilical submanifolds of some class riemannian manifolds