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 and revisions. AUTHOR: Bang-yen Chen is University Distinguished Professor Emeritus at Michigan State University. His other books include Pseudo-Riemannian Geometry, D-Invariants and Applications, and Differential Geometry of Warped Product Manifolds and Submanifolds.