Geometry of Submanifolds : AMS Special Session on Geometry of Submanifolds in Honor of Bang Yen Chen’s 75th Birthday October 20–21 2018 University of Michigan Ann Arbor Michigan Cover Title page Contents Foreword by the Editors Reflections on some research work of Bang Yen Chen 1. invariants and basic inequalities 2. A new approach to compact symmetric spaces 3. Two numbers and applications 4. Finite type theory for maps and immersions 5. Kaehler manifolds 6. Riemannian submersions 7. Production functions and constant elasticity of substitution property in economics 8. Some other topics studied by Bang Yen Chen References My education in differential geometry and my indebtedness 1. My education in differential geometry in Taiwan 2. My study at University of Notre Dame and my indebtedness to Professor Tadashi Nagano 3. My indebtedness to Professor Shiing Shen Chern 4. My indebtedness to Professor Tominosuke Otsuki 5. My indebtedness to Professor Kentaro Yano 6. In conclusion References Submanifold theory A contemplation of submanifolds Acknowledgments References Spectral gaps on complete Riemannian manifolds 1. Introduction 2. The New Weyl criterion 3. On a theorem of Sturm 4. Spectral continuity and a gap theorem References On stability and index of minimal submanifolds 1. Introduction 2. Minimal submanifolds of arbitrary codimension 3. Minimal hypersurfaces with lower index 4. Proof of Theorem 3.6 Acknowledgments References Maximally warped metrics with harmonic curvature Introduction 1. Preliminaries 2. Vector fields 3. Integrable complement Killing fields 4. Multiply warped metrics with =0 5. The local structure 6. The scalar curvature integral 7. Completeness Appendix: Warped products with harmonic curvature References CR submanifolds of Chen type two in non flat complex space forms 1. Introduction 2. The notation and relevant formulas. This volume contains the proceedings of the AMS Special Session on Geometry of Submanifolds in honor or Bang Yen Chens 75th birthday held from October 20 21 2018 at the University of Michigan Ann Arbor Michigan.The development of contemporary geometry of submanifolds benefited greatly from Bang Yen Chens contributions as several interesting questions actively pursued today originate in his work. Chen is known for several fundamental ideas in differential geometry including Chen inequalities Chen invariants Chens conjectures Chen surface Chen Ricci inequality Chen submanifolds Chen equality submanifolds of finite type and slant submanifolds.The papers in this volume represent a celebration of the geometry of submanifolds and its connections with other areas of mathematics and cover themes rooted in Chens work from investigations on the spectrum of the Laplacian on complete Riemannian manifolds to the geometry of symmetric spaces. These contributions are written with the hope to inform and inspire.