It is well known that submanifolds \(M\) of a semi-Riemannian (in another terminology, pseudo-Riemannian) manifold \(\overline {M}\) can be either nondegenerate (spacelike or timelike) or degenerate (lightlike). The type depends on the location of the tangent subspace \(T_x(M)\) with respect to the light cone \(C_x\). For a lightlike submanifold, \(T_x (M)\) is tangent to \(C_x\). The lightlike submanifolds, especially the lightlike hypersurfaces, are extensively used in relativity. The main difference between the lightlike (degenerate) submanifolds and the spacelike and timelike (nondegenerate) submanifolds is that for the former the normal vector bundle intersects the tangent bundle of the submanifold.
The book under review is the first book on the theory of lightlike submanifolds. It is based mostly on a series of the authors’ papers. The authors provide an up-to-date information on null curves, lightlike hypersurfaces and submanifolds consistent with the theory of nondegenerate submanifolds and give considerable amount of geometric and physical results on two- and three-dimensional lightlike surfaces and hypersurfaces.
Chapters 1 and 2 review most of the prerequisites for reading the book. The main formulas and results are presented in both the invariant form and the index form. In Chapter 3 the authors present the general theory of null curves in Lorentz manifolds. They find the Frenet frame and the Frenet equations and prove the existence and uniqueness theorem for null curves. In addition, some properties of null curves of four- and three-dimensional Lorentz manifolds and \(N\)-dimensional Minkowski space are considered.
In Chapter 4 (which is the core of the book) the authors introduce the general techniques for studying lightlike hypersurfaces and develop the differential geometry of a lightlike hypersurface of a proper semi-Riemannian manifold. Namely, they define a nondegenerate screen distribution, construct the corresponding lightlike transversal vector bundle and use them to define the projection of the bundle \(T \overline{M}|_M\) onto the tangent bundle \(TM\), a linear connection, the second fundamental form, etc., and to prove the fundamental theorem for lightlike hypersurfaces.
In addition, for physical applications in the following chapters the authors provide information on the general theory of lightlike submanifolds and Cauchy-Riemann (CR-) submanifolds, respectively. Namely, in Chapter 5, they show how by using the techniques developed in Chapter 4 one can solve the basic problem of the theory of lightlike submanifolds \(M\): to replace the part of the normal vector bundle lying in the tangent bundle \(TM\) by a vector subbundle whose sections are not tangent to \(M\). In addition, in Chapter 5 some results on differential geometry of lightlike surfaces of Lorentz manifolds (in particular, the manifold \(\mathbb{R}^4_1\)) are presented. In Chapter 6 a complete proof of the existence of CR-lightlike products is presented. In Chapters 7, 8, and 9 the authors apply the theory of lightlike hypersurfaces developed in Chapter 4, to relativity. They restrict the scope of applications to results on Killing horizon, electromagnetic and radiation fields and asymptotically flat spacetimes.
The basic results are new and of impressive depth. They will strongly influence further research in the field. The book will be of interest to scientists working in differential geometry and mathematical physics.