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Intro; Preface; Contents; Spacelike Hypersurfaces in the Lorentz-Minkowski Space with the Same Riemannian and Lorentzian Mean Curvature; 1 Introduction and Background; 2 Preliminaries; 3 Spacelike Hypersurfaces with HR=HL; 4 A Quasi-linear PDE Related to Spacelike Hypersurfaces with HR=HL; 4.1 On Rotationally Invariant Spacelike Graphs; References; Space-Time Convex Functions and Sectional Curvature; 1 Introduction; 1.1 Outline of the Paper; 2 Space-Time Convex Functions; 3 Comparisons for Lorentzian Distance; 4 Curvature Bounds mathcalRleK, mathcalRgeK; 4.1 Geometric Meaning
4.2 GRW Spaces4.3 Comparisons Based at a Point; 4.4 Geometric Characterization of mathcalRleK, mathcalRgeK; 4.5 Comparisons for Parallel Families of Hypersurfaces; 5 Results; 6 Conclusion; References; Recent Results on Oscillator Spacetimes; 1 Introduction; 2 The Oscillator Group; 3 Symmetries of the Oscillator Group; 4 Ricci Solitons of the Oscillator Group; References; Umbilical Spacelike Submanifolds of Arbitrary Co-dimension; 1 Introduction; 2 Preliminaries; 2.1 Basic Concepts of Submanifold Theory; 2.2 The Total Shear Tensor and the Shear Operators
2.3 Umbilical Points and Umbilical Submanifolds3 Results; 3.1 The Relationship Between mathscrU and Imwidetildeh; 3.2 Characterization of Umbilical Spacelike Submanifolds; 3.3 Special Cases; References; A Study in Stationary: Geometric Properties of Stationary Regions and Regularity of Their Horizons; 1 Introduction; 2 Preliminaries; 3 Global Structure of Stationary Spacetimes I: Topological Aspects; 4 Global Structure of Stationary Spacetimes I: Metrical Aspects; 5 Static Splitting and Komar Energy of Stationary Spacetimes; 6 General Properties of Stationary Regions
7 Regularity of S-Horizons8 Killing Horizons; References; Anti-De Sitter Spacetimes and Isoparametric Hypersurfaces in Complex Hyperbolic Spaces; 1 Introduction; 2 Anti-De Sitter Spacetime and Complex Hyperbolic Space; 3 Isoparametric Hypersurfaces; References; Future Completion of a Spacetime and Standard Causal Constructions; 1 Introduction; 2 Definitions and Properties; 3 Results; References; Some Criteria for Wind Riemannian Completeness and Existence of Cauchy Hypersurfaces; 1 Introduction; 2 A Lorentzian Overview on Wind Riemannian Structures; 2.1 Revisiting SSTK Spacetimes.
4.2 GRW Spaces4.3 Comparisons Based at a Point; 4.4 Geometric Characterization of mathcalRleK, mathcalRgeK; 4.5 Comparisons for Parallel Families of Hypersurfaces; 5 Results; 6 Conclusion; References; Recent Results on Oscillator Spacetimes; 1 Introduction; 2 The Oscillator Group; 3 Symmetries of the Oscillator Group; 4 Ricci Solitons of the Oscillator Group; References; Umbilical Spacelike Submanifolds of Arbitrary Co-dimension; 1 Introduction; 2 Preliminaries; 2.1 Basic Concepts of Submanifold Theory; 2.2 The Total Shear Tensor and the Shear Operators
2.3 Umbilical Points and Umbilical Submanifolds3 Results; 3.1 The Relationship Between mathscrU and Imwidetildeh; 3.2 Characterization of Umbilical Spacelike Submanifolds; 3.3 Special Cases; References; A Study in Stationary: Geometric Properties of Stationary Regions and Regularity of Their Horizons; 1 Introduction; 2 Preliminaries; 3 Global Structure of Stationary Spacetimes I: Topological Aspects; 4 Global Structure of Stationary Spacetimes I: Metrical Aspects; 5 Static Splitting and Komar Energy of Stationary Spacetimes; 6 General Properties of Stationary Regions
7 Regularity of S-Horizons8 Killing Horizons; References; Anti-De Sitter Spacetimes and Isoparametric Hypersurfaces in Complex Hyperbolic Spaces; 1 Introduction; 2 Anti-De Sitter Spacetime and Complex Hyperbolic Space; 3 Isoparametric Hypersurfaces; References; Future Completion of a Spacetime and Standard Causal Constructions; 1 Introduction; 2 Definitions and Properties; 3 Results; References; Some Criteria for Wind Riemannian Completeness and Existence of Cauchy Hypersurfaces; 1 Introduction; 2 A Lorentzian Overview on Wind Riemannian Structures; 2.1 Revisiting SSTK Spacetimes.