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Level sets of Lorentzian distance functions with respect to a point and with respect to an achronal spacelike hypersurface, are analyzed. Some bounds for the Laplacian of such Lorentzian distance functions are obtained and, in relation to them, some spacetime singularity theorems are given.

A local characterization of Schwarzschild metrics is made by showing the space-time is locally a 2 by 2 warped product and admitting a static reference frame on its certain open subsets under some assumptions on the global analytic structure and stress-energy tensor of the space-time, such as, assuming the existence of solutions to certain partial differential equations and the existence of a radiation stress-energy tensor consistent with these solutions on the space-time.

Some characterizations of certain rank-one symmetric Riemannian manifolds by the existence of nontrivial solutions to certain partial differential equations on Riemannian manifolds are surveyed.

A partial differential equation, the local Möbius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local Möbius equation and an additional partial differential equation.

A characterization of time functions on a spacetime is made by using theMobius equation. It is shown that a time function characterized in this wayyields past timelike geodesic incompleteness and local Lorentzian warpedproduct decomposition of spacetime, provided that the stress-energy tensoris a fluid. Also, by imposing additional assumptions on the stress-energytensor and global analytic structure of the spacetime, more restrictivedecompositions closer to Robertson–Walker spacetimes are obtained.

It is shown that a mixed Ricci-flat twisted product semi-Riemannian manifold can be expressed as a warped product semi-Riemannian manifold. As a consequence, any Einstein twisted product semi-Riemannian manifold is in fact, a warped product semi-Riemannian manifold.

Osserman property is studied for affine torsion-free connections with special attention to the 2-dimensional case. As an application, examples of nonsymmetric and even not locally homogeneous Osserman pseudo-Riemannian metrics are constructed on the cotangent bundle of a manifold equipped with a torsionfree connection by looking at their Riemann extensions. Also, timelike and spacelike Osserman conditions are analyzed for general pseudo-Riemannian manifolds showing that they are equivalent.

A rigid singularity theorem for spacetimes admitting irrotational reference frames is proven as an extension of a result (Theorem 3.5) of Petersen and Walschap [Observer fields and the strong energy condition, Class. Quantum Grav. 13 (1996) 1901–1908.].

The relations to curvatures of a Riemannian map to be affine are investigated. Also some splitting theorems related to Riemannian maps and curvatures are obtained.

We prove a splitting theorem for stably causal spacetimes and another splitting theorem for finitely compact spacetimes admitting a proper time synchronizable reference frame.

Splitting theorems for stable causal spacetimes admitting certain metric related reference frames are obtained in connection to timelike geodesic incompletenes.

Variation of the volume of semi-Riemannian submanifolds is studied. A possible error in the second variational formula in [4] is corrected and it is shown that a semi-Riemannian submanifold cannot be extremal unless it is both definite and codefinite.

The concept of a semi-Riemannian map is introduced and it is shown that such maps are solutions of the eikonal equation. Also the existence of solutions to the eikonal equation are discussed and their relation to the Laplace-Beltrami equation is investigated.

A local characterization of Schwarzschild and Reissner metrics is made by using the concepts of infinitesimal null anisotropy (or equivalently, infinitesimal isotropy) and weak affinity.

Null and infinitesimal isotropy are defined for semi-Riemannian manifolds in a more general context. A theorem of Karcher is extended to Semi-Riemannian manifolds in a more general setting. Also, by using this theorem, a characterization of static blackhole metrics can be made as well as a characterization of Robertson-Walker metrics as Karcher made.

5.1 Nonsymmetric Semi-Riemannian Osserman Manifolds of Arbitrary Signature 5.1.1 Indefinite Kahler Osserman Manifolds 5.1.2 Semi-Riemannian Osserman Metric Tensors on Tangent Bundles 5.2 Special Semi-Riemannian Osserman Manifolds 5.3 Classification of Semi-Riemannian Special Osserman Manifolds 5.3.1 Multiplicities of the Eigenvalues 5.3.2 Semi-Riemannian Special Osserman Manifolds of Dimensions Different from 16 and 32 5.3.3 Semi-Riemannian Special Osserman Manifolds of Dimensions 16 and 32

4.1 Nonsymmetric Semi-Riemannian Osserman Manifolds with Metric Tensors of Signature (2,2) 4.2 Semi-Riemannian Osserman and Jordan-Osserman Manifolds with Metric Tensors of Signature (2,2) 4.2.1 Osserman Algebraic Curvature Tensors on Four-Dimensional Vector Spaces of Neural Signature 4.2.2 Four-Dimensional Semi-Riemannian Pointwise Osserman Manifolds 4.2.3 Symmetric Semi-Riemannian Osserman Manifolds with Metric Tensors of Signature (2,2) 4.2.4 Classification of Semi-Riemannian Jordan-Osserman Manifolds with Metric Tensors of Signature (2,2)

3.1 The Complete Solution to the Osserman Problem 3.2 The Null Osserman Condition 3.2.1 The Null Osserman Condition at a Point 3.2.2 Decomposition Theorems

1.1 The Jacobi Operator 1.2 The Timelike and Spacelike Osserman Conditions at a Point 1.3 Semi-Riemannian Pointwise and Global Osserman Manifolds

2.1 Partial Solution to the Osserman Conjecture 2.1.1 Odd-Dimensional Riemannian Pointwise Osserman Manifolds 2.1.2 (4m + 2)-Dimensional Riemannian Pointwise Osserman Manifolds 2.1.3 Four-Dimensional Riemannian Osserman Manifolds 2.2 A Solution to the Osserman Conjecture with Additional Assumptions 2.2.1 Kahler and Quaternionic Kahler Manifolds 2.2.2 Riemannian Osserman Manifolds with Two Distinct Eigenvalues 2.2.3 Homogeneous Manifolds of Negative Sectional Curvature

6.1 Semi-Riemannian Generalized Osserman Manifolds 6.2 The Osserman Condition in Affine Differential Geometry 6.2.1 Affine Osserman Manifolds 6.2.2 Semi-Riemannian Osserman Metric Tensors on Cotangent Bundles 6.3 Riemannian Manifolds with Isoparametric Geodesic Sheres 6.4 Riemannian \(\frak{C}\)-manifolds 6.5 Skew-Symmetric Curvature Operators 6.5.1 Riemannian Manifolds Whose Skew-Symmetric Curvature Operators Have Constant Eigenvalues along each Circle

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.