Differential Geometry and Global Analysis

✍ Scribed by Bang-Yen Chen, Nicholas D. Brubaker, Takashi Sakai, Bogdan D. Suceavă, Makiko Sumi Tanaka, Hiroshi, Tamaru Mihaela B. Vajiac

• Publisher: American Mathematical Society
• Year: 2022
• Tongue: English
• Leaves: 242
• Series: Contemporary Mathematics, 777
• Category: Library

✦ Synopsis

This volume contains the proceedings of the AMS Special Session on Differential Geometry and Global Analysis, Honoring the Memory of Tadashi Nagano (1930–2017), held January 16, 2020, in Denver, Colorado. Tadashi Nagano was one of the great Japanese differential geometers, whose fundamental and seminal work still attracts much interest today. This volume is inspired by his work and his legacy and, while recalling historical results, presents recent developments in the geometry of symmetric spaces as well as generalizations of symmetric spaces; minimal surfaces and minimal submanifolds; totally geodesic submanifolds and their classification; Riemannian, affine, projective, and conformal connections; the $(M_{+}, M_{-})$ method and its applications; and maximal antipodal subsets. Additionally, the volume features recent achievements related to biharmonic and biconservative hypersurfaces in space forms, the geometry of Laplace operator on Riemannian manifolds, and Chen-Ricci inequalities for Riemannian maps, among other topics that could attract the interest of any scholar working in differential geometry and global analysis on manifolds.

✦ Table of Contents

Cover
Title page
Contents
Preface
To the memory of Professor Tadashi Nagano
1. Period I (1930–1967) by Takushiro Ochiai
2. Period II (1967–1986) by Bang-Yen Chen
3. Period III (1986–2000) by Makiko Sumi Tanaka
List of Professor Nagano’s publications Reseach papers
Books
Bifurcations of minimal surfaces via index theory
1. Introduction
2. Catenoid
3. Dihedral Enneper surfaces
4. Discussion
References
The (𝑀₊,𝑀₋)-method on compact symmetric spaces and its applications
1. Basics of compact symmetric spaces
2. (𝑀₊,𝑀₋)-theory
3. Applications to totally geodesic embeddings
4. Application to homotopy
5. An algorithm for stability and its applications
6. 2-numbers and maximal antipodal sets
7. Links between 2-number and topology
8. Applications to compact Lie groups
9. Applications to algebraic geometry
10. Applications of (𝑀₊,𝑀₋)-theory via real forms
11. Index numbers and flag manifolds
12. Index numbers and CW complex structures
Acknowledgment
References
Biharmonic and biconservative hypersurfaces in space forms
1. Introduction
2. Definitions and general properties
3. Surfaces in three dimensional space forms
4. Hypersurfaces in space forms
5. Open problems
Acknowledgments
References
Recent progress of biharmonic hypersurfaces in space forms
1. The original Chen’s conjecture
2. The generalized Chen’s conjecture
3. Biharmonic submanifolds in Euclidean sphere
4. Biharmonic hypersurfaces in ℝ⁵
References
A commutativity condition for subsets in quandles —a generalization of antipodal subsets
1. Introduction
2. Preliminaries
3. Poles and antipodal subsets
4. 𝑠-commutative subsets
5. Subsets in direct products and interaction-free unions of quandles
6. Examples
References
Spectral gaps of the Laplacian on differential forms
1. Introduction
2. The spectrum of the Laplacian on 𝑘-forms
3. Basic construction
4. Proof of the main theorems
References
Chen-Ricci inequalities for Riemannian maps and their applications
1. Introduction
2. Preliminaries
3. Chen-Ricci inequality for Riemannian maps with a real space form
4. Improved Chen-Ricci inequalities for Lagrangian Riemannian maps
References
Totally geodesic surfaces in the complex quadric
1. Introduction
2. Preliminaries
3. Explicit descriptions of totally geodesic surfaces in 𝑄ⁿ
4. An alternative description in dimension two
5. Totally geodesic surfaces in the hyperbolic complex quadric
References
Parallel Kähler submanifolds and 𝑅-spaces
1. Introduction
2. 𝑅-spaces and Olmos-Sánchez’s characterization
3. Homogeneous structure on the inverse images of parallel Kähler submanifolds under the Hopf fibration
4. Classification of parallel Kähler submanifolds
Acknowledgments
References
A survey on natural Γ-symmetric structures on 𝑅-spaces
1. Introduction
2. Γ-symmetric spaces
3. 𝑅-spaces and symmetric 𝑅-spaces
4. Natural Γ-symmetric structures on 𝑅-spaces
5. Extrinsically Γ-symmetric spaces
6. Maximal antipodal sets of Γ-symmetric 𝑅-spaces
References
On the first eigenvalue of the 𝑝-Laplacian on Riemannian manifolds
1. Introduction
2. Estimates for the first eigenvalue of the 𝑝-Laplacian
3. Extension to differential forms
References
Polars of disconnected compact Lie groups
1. Introduction
2. Compact Lie groups
3. Disconnected compact Lie groups
4. Semidirect products
5. Classical compact Lie groups
References
Back Cover