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In this chapter, we generalize the notion of a semi-Riemannian map to the notion of a semi-Riemannian map with respect to a semi-Riemannian foliation on the target manifold. If this semi-Riemannian foliation is taken to be the points of the target manifold, then the definition of a semi-Riemannian map with respect to such a foliation reduces to the definition of a semi-Riemannian map.

As we know from the semi-Riemannian regular interval theorem (Theorem 6.3.1), the existence of an affine solution f of a nonnull eikonal equation on a semi-Riemannian manifold (M, g) yields a splitting of (M, g) into a semi-Riemannian product manifold, provided that ∇ f is a complete vector field on (M, g). We also know from Proposition 6.2.5 that being affine for a solution f of a semi-Riemannian eikonal equation is related to the Ricci curvature of (M,g) and the Hessian tensor of f. Now, by combining these results, one can obtain splitting theorems for certain semi-Riemannian manifolds which admit solutions to nonnull eikonal equations. In fact, in this chapter we obtain such splitting theorems in a more general context for semi-Riemannian, Riemannian and Lorentzian manifolds. We devote sections 7.1, 7.2 and 7.3 to such splitting theorems for semi-Riemannian, Riemannian and Lorentzian manifolds, respectively. Section 7.3 may be considered of separate interest for applications of semi-Riemannian maps in General Relativity.

In this chapter we introduce semi-Riemannian eikonal equations and investigate the consequences of the existence of their solutions.

In this chapter, we introduce the second fundamental form of a map between semi-Riemannian manifolds. The second fundamental form of a map is used largely in the analysis of certain properties of maps between manifolds in the literature. However, in this chapter, we study the second fundamental form of a map only as much as we need to carry out our work on semi-Riemannian maps. We devote the first section to the definition and basic properties of the second fundamental form of a map between semi-Riemannian manifolds. In section 2, we define affine and harmonic maps between semi-Riemannian manifolds and derive certain properties of affine maps. Then in section 3, we define the divergence of a vector field along a map between semi-Riemannian manifolds and generalize the semi-Riemannian divergence theorem to such vector fields. Finally in section 4, we derive the Bochner identity of a map between semi-Riemannian manifolds which will be used in showing a Riemannian map to be affine under certain curvature conditions. Throughout this chapter, and in all following chapters, let M 1 M 2 and M be manifolds with dimensions n 1, n 2 and n, respectively. Also let (M l, g 1), (M 2, g 2 and (M, g) be semi-Riemannian manifolds of indices v l, v 2 and v with Levi-Civita connections \(\mathop \nabla \limits^1 ,\mathop \nabla \limits^2 \) and ∇, respectively, unless otherwise stated.

In this chapter, we introduce the concept of a semi-Riemannian manifold and develop the necessary machinery to analyze its geometric structure. In section 2.1, we define the metric tensor on a manifold. Then in section 2.2, we introduce connections on a semi-Riemannian manifold compatible with the metric tensor. We also define parallel translation and geodesics determined by the semi-Riemannian structure of a manifold. In section 2.3 we study the curvatures of the Levi-Civita connection of a semi-Riemannian manifold. In section 2.4, we construct some differential operators on a semi-Riemannian manifold such as, divergence, gradient, Hessian tensor and Laplacian. Finally in section 2.5 we state the semi-Riemannian divergence theorem.

In this chapter we collect the mechanisms behind the linear algebra of real inner product spaces. Some of these were developed for certain classes of geometric objects in other literature, yet their proofs involve only the linear algebraic properties of the geometric objects at hand.

A generalization of semi-Riemannian submersions allowing degenerate submanifolds as fibres is given by making an application of semi-Riemannian maps to submersions. Also a fundamental equation of a regular semi-Riemannian submersion is obtained.

In this chapter we introduce the notion of semi-Riemannian maps. Intuitively a semi-Riemannian map is a map between semi-Riemannian manifolds which is as “isometric as it can be.” Hence, as one expects, the existence of a semi-Riemannian map between semi-Riemannian manifolds enables us to compare some geometric properties of the semi-Riemannian manifolds in its domain and range. Clearly, this concept is a generalization of the notion of isometry.

In this chapter, we will investigate the structure of quaternionic Kahler manifolds with degenerate quaternionic metric tensors.

In this chapter, we will investigate the structure of Kahler manifolds with degenerate Hermitian metric tensors.

In this chapter we will collect the mechanisms behind the linear algebra of real inner product spaces. Some of these works were developed for certain classes of geometric objects, yet their proofs completely involve the linear algebraic properties of the geometric objects we have in hand.

In this chapter, we will review the linear algebra of Hermitian inner product spaces as parallel to Preliminaries I in Part I.

Let M be a semi-Riemannian manifold with boundary ∂M (possibly ∂M = O). If the metric on M is indefinite then the Laplace-Beltrami equation Δf= 0 on M is an ultra-hyperbolic type equation. Hence even if M is compact and ∂M = O, unlike the Riemannian case, Δf = 0 may have nonconstant solutions. In Partial Differential Equations, a common method of solving this equation is the method of separation of variables. But if one examines this method, it can be seen that the domain of solution is given as a product manifold and then the spectral theory for Laplacian is applied. The purpose of this note is to generalize this method to double-warped semi-Riemannian manifolds, which is the maximal extension of this method of finding solutions. As in Partial Differential Equations, when we choose the factor manifolds Riemannian and impose the Dirichlet or Neumann boundary conditions on M, the Laplace-Beltrami equation decomposes to elliptic, self-adjoint operators on factor manifolds. Then by making use of the spectral theory of such operators, we will obtain some solutions of the Laplace-Beltrami equation. We will also define the generalized wave equation and show that there exists a unique solution for that equation for the usual wave boundary conditions. Intuitively, the generalized wave equation may be considered that “drum” is changing it’s shape in time.

Recall that a nondegenerate Kahler manifold (M, J, Η) of complex type (v, η) is a Hermitian manifold. Hence, if H is a submanifold of M, then TH is invariant under J if and only if H is a complex submanifold of M, since the vanishing torsion tensor N of J on M implies that N is also vanishing on H. (see Theorem 6.2.2 for the details). Now, let H be a complex submanifold of a nondegenerate Kahler manifold (M, J, Η) of complex type (v, η) and let Η H be the restriction of Η to (TH, J). If Η H is a Hermitian metric tensor on (H, J) of complex type (μ′, v′, η′), then (H, J, Η H ) is called a Hermitian submanifold of complex type (μ′, v′, η′) in (M, J, Η). Also, if Η = g + iΩ, let Η H = gH + iΩ H , Η TH ┴= g TH ┴ + iΩ TH ┴ and \({H_{N_{gH}^ \bot }} = {g_{N_{gH}^ \bot }} + i{\Omega _{N_{gH}^ \bot }}\) be the restrictions of Η to (TH, J), (TH┴, J) and (N gH ┴ , J), respectively. Also note that, since TH and N gH are invariant under J, TH┴ and N gH ┴ are invariant under J and, therefore, we will use the same notation J for the restrictions of the canonical almost complex structure J of M to TH, TH┴ and N gH ┴ .

In this chapter, we will investigate the structure of quaternionic semi-Riemannian submanifolds of nondegenerate quaternionic Kahler manifolds. In fact, we will show that such manifolds are totally geodesic.

Let M be an n-dimensional manifold and let g be a metric tensor of type in (μ, v, η) in TM. Then we will call (M, g) a semi-Riemannian manifold of type (μ, v, η). If g is of type (0, v, η) we will call (M, g) a nondegenerate semi-Riemannian manifold of type (v, η).

A problem of Osserman related to the constancy of the eigenvalues of the Jacobi operator is studied in Lorentzian Geometry. It is proven that pointwise timelike Osserman Lorentzian manifolds are of constant curvature as well as those of pointwise spacelike Osserman Lorentzian spaces of dimension 3 and 4. It is shown that a 4-dimensional globally null Osserman Lorentzian manifold is either of constant curvature or is locally a Robertson-Walker spacetime.

In this chapter, we will give a brief review of covariant derivative operators in real vector bundles. Also, in this way, we will establish some notation and terminology to be used in later chapters. A full discussion of the contents of this chapter may be found in [21]. In this chapter, we will assume that every object in hand is smooth, unless otherwise stated. Also, we will assume that the manifolds are paracompact for sake of using the partition of unity.

In this chapter, we will give a review of linear algebra of quaternionic inner product spaces parallel to Preliminaries II in Part II.

In this chapter, we will analyze the structure of submanifolds in nondegenerate semi-Riemannian manifolds, which have an induced metric tensor of type (μ′, v′, η′).

Preface. I: Singular Semi-Riemannian Manifolds. 1. Preliminaries I: The Linear Algebra of Real Inner Product Spaces. 2. A Review of Covariant Derivative Operators in Real Vector Bundels. 3. Singular Semi-Riemannian Manifolds. 4. Semi-Riemannian Submanifolds in Nondegenerate Semi-Riemannian Manifolds. II: Singular Kahler Manifolds. 5. Preliminaries II: Linear Algebra of Hermitian Inner Product Spaces. 6. A Review of Covariant Derivative Operators on Complex Vector Bundles. 7. Singular Kahler Manifolds. 8. Hermitian Submanifolds of Nondegenerate Kahler Manifolds. III: Singular Quaternionic Kahler Manifolds. 9. Preliminaries III: Linear Algebra of Quaternionic Inner Product Spaces. 10. Singular Quaternionic Kahler Manifolds. 11. Quaternionic Semi-Riemannian Submanifolds of Nondegenerate Quaternionic Kahler Manifolds. References. Index.