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This book offers a detailed exploration of the intrinsic geometrical properties of warped product spaces through the lens of mathematical analysis and global differential geometry. It touches upon key topics such as uniqueness results, height estimates, Riemannian immersions, and the geometrical behavior of submanifolds, while addressing complex phenomena that challenge traditional Euclidean assumptions. Divided into five comprehensive parts, the text provides clear refinements of recent findings, with connections to General Relativity and semi-Riemannian geometry.
Accessible yet thorough, this monograph is ideal for post-graduate students, researchers, and specialists across mathematics, geometry, and theoretical physics.
Inhaltsverzeichnis
Part I Uniqueness Results, Height Estimates and Half-Space Theorems for Hypersurfaces in Semi-Riemannian Warped Products. - 1 Riemannian Immersions. - 2 Uniqueness of complete hypersurfaces. - 3 Height estimates and half-space theorems. - 4 Spacelike hypersurfaces in standard static spacetimes. - Part II Riemannian Immersions in Weighted Semi-Riemannian Warped Products. - 5 Basic facts concerning weighted manifolds. - 6 Two-sided hypersurfaces in weighted Riemannian warped products. - 7 Spacelike hypersurfaces in weighted GRW spacetimes. - 8 Spacelike hypersurfaces in weighted standard static spacetimes. - Part III Submanifolds Immersed in Semi-Riemannian Warped Products. - 9 Submanifolds in a Riemannian warped product. - 10 Submanifolds immersed in a Killing warped product. - 11 Weakly trapped submanifolds immersed in a generalized Robertson-Walker spacetime. - 12 Studying the geometry of weakly trapped submanifolds in standard static spacetimes. - Part IV Stability of Riemannian Immersions in Semi-Riemannian Warped Products. - 13 A notion of stability to closed hypersurfaces in the hyperbolic space. - 14 Stable closed spacelike hypersurfaces in the de Sitter space. - 15 Stability in certain semi-Riemannian manifolds with density. - 16 -stability of zero -mean curvature hypersurfaces. - Part V Local Rigidity and Bifurcation of Riemannian Immersions in Semi-Riemannian Warped Products. - 17 Bifurcation of 2-hypersurfaces in Riemannian warped products. - 18 Bifurcation of spacelike hypersurfaces with constant mean curvature in spacetimes. - 19 Bifurcation of -minimal hypersurfaces in a weighted Killing warped product. - 20 Bifurcation of hypersurfaces with constant -mean curvature in × R. - References.
This monograph is a substantial and carefully crafted contribution to the theory of submanifolds and hypersurfaces in warped products. It successfully integrates classical submanifold geometry, Lorentzian geometry, weighted manifolds, stability theory, and bifurcation analysis into a single coherent framework. . . . Researchers working in Riemannian or Lorentzian geometry, geometric analysis, and mathematical physics will find this monograph to be a valuable reference. It also offers a rich source of ideas and techniques for advanced graduate students entering the field. (Gabriel Eduard Vilcu, zbMATH 1577. 53004, 2026)
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