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.
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
𝑳𝝋-stability of zero 𝝋-mean curvature hypersurfaces.-
Part V Local Rigidity and Bifurcation of Riemannian Immersions in
Semi-Riemannian Warped Products.- 17 Bifurcation of 𝑯2-hypersurfaces
in Riemannian warped products.- 18 Bifurcation of spacelike hypersurfaces
with constant mean curvature in spacetimes.- 19 Bifurcation of
𝝋-minimal hypersurfaces in a weighted Killing warped product.- 20
Bifurcation of hypersurfaces with constant 𝝋-mean curvature in
𝑴𝒏𝝋 ×𝝆 R.- References.
Henrique Fernandes de Lima was born in Fortaleza, Brazil, and graduated in Mathematics from the Federal University of Ceará (Brazil), completed his Master's degree in Mathematics and obtained his PhD in Mathematics also from the Federal University of Ceará (Brazil), and has been a professor in the Department of Mathematics at the Federal University of Campina Grande (Brazil) since 2004, where he regularly teaches postgraduate courses on Differential Geometry, Riemannian Geometry, Isometric Immersions and Submanifolds Theory. Author of several research articles, Henrique F. de Lima has been awarded a CNPq Research Grant in Differential Geometry since 2013. Since joining the Federal University of Campina Grande, Henrique F. de Lima has also dedicated himself to training master students and doctors in Mathematics.
Giovanni Molica Bisci is currently full professor of Mathematical Analysis at the San Raffaele University of Rome (Italy). He got his PhD in 2004 at the University of Messina. In his career, he has been an assistant professor at the Mediterranea University of Reggio Calabria (Italy) and in 2018 became an associate professor at the University of Urbino Carlo Bo (Italy). His research interests are nonlinear partial differential equations, variational and topological methods in Nonlinear Analysis, and Differential Geometry. He published more than 140 papers in high level international mathematical journals and two monographs. In 2019 he was a Highly Cited Researcher (Thomson Reuters). Since 2020 he is in the Worlds Top 2% Scientists List - Stanford University.
Marco Antonio Lázaro Velásquez was born in Trujillo, Peru, and graduated in Mathematics from the Universidad Nacional de Trujillo, completed his Master's degree in Mathematics at the Federal University of Campina Grande (Brazil), obtained his PhD in Mathematics from the Federal University of Ceará (Brazil), and has been a professor in the Department of Mathematics at the Federal University of Campina Grande since 2010, where he regularly teaches postgraduate courses on Differential Geometry, Smooth Manifolds, Semi-Riemannian Geometry, Isometric Immersions, and Minimal Submanifolds. Author of several research articles, Marco A.L. Velásquez has been awarded a CNPq Research Grant in Differential Geometry since 2015. Since joining the Federal University of Campina Grande, Marco A.L. Velásquez has also dedicated himself to training master students and doctors in Mathematics.
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