Bang-Yen Chen Knihy

The book offers a comprehensive introduction to pseudo-Riemannian manifolds and their non-degenerate submanifolds, requiring only basic knowledge of manifold theory. It includes recent findings on these submanifolds. The second part delves into -invariants, a concept introduced by the author in the 1990s to address challenges in relating extrinsic and intrinsic invariants. It also surveys significant developments in this area over the past two decades, highlighting the evolution and impact of -invariants in the field of geometry.

Warpped product manifolds are a key concept in both geometry and mathematical physics, particularly in general relativity. These manifolds feature a metric tensor that can be expressed as a Cartesian product, with the x-part being rescaled by a scalar function dependent on the y coordinates. This framework is crucial for understanding various solutions to the Einstein field equations, such as the Schwarzschild solution and Robertson-Walker models, highlighting their significance in the study of the universe's structure.

The first two chapters of this frequently cited reference provide background material in Riemannian geometry and the theory of submanifolds. Subsequent chapters explore minimal submanifolds, submanifolds with parallel mean curvature vector, conformally flat manifolds, and umbilical manifolds. The final chapter discusses geometric inequalities of submanifolds, results in Morse theory and their applications, and total mean curvature of a submanifold. Suitable for graduate students and mathematicians in the area of classical and modern differential geometries, the treatment is largely self-contained. Problems sets conclude each chapter, and an extensive bibliography provides background for students wishing to conduct further research in this area. This new edition includes the author's corrections.