Unitarily Invariant Geometry On Grassmann Manifold

This work offers a contribution in the geometric form of the theory of several complex variables. Since complex Grassmann manifolds serve as classifying spaces of complex vector bundles, the cohomology structure of a complex Grassmann manifold is of importance for the construction of Chern classes of complex vector bundles. The cohomology ring of a Grassmannian is therefore of interest in topology, differential geometry, algebraic geometry, and complex analysis. Wilhelm Stoll treats certain aspects of the complex analysis point of view. This work originated with questions in value distribution theory. Here analytic sets and differential forms rather than the corresponding homology and cohomology classes are considered. On the Grassmann manifold, the cohomology ring is isomorphic to the ring of differential forms invariant under the unitary group, and each cohomology class is determined by a family of analytic sets.

This volume contains the papers presented at a symposium on differential geometry at Shinshu University in July of 1988. Carefully reviewed by a panel of experts, the papers pertain to the following areas of research: dynamical systems, geometry of submanifolds and tensor geometry, lie sphere geometry, Riemannian geometry, Yang-Mills Connections, and geometry of the Laplace operator.

Differential geometry is a very active field of research and has many applications to areas such as physics, in particular gravity. The chapters in this book cover a number of subjects that will be of interest to workers in these areas. It is hoped that these chapters will be able to provide a useful resource for researchers with regard to current fields of research in this important area.

Articles on classical convex geometry, geometric functional analysis, computational geometry, and related areas of harmonic analysis, first published in 1999.

Geometry of Manifolds

The goal of this book is to introduce the reader to some of the main techniques, ideas and concepts frequently used in modern geometry. It starts from scratch and it covers basic topics such as differential and integral calculus on manifolds, connections on vector bundles and their curvatures, basic Riemannian geometry, calculus of variations, DeRham cohomology, integral geometry (tube and Crofton formulas), characteristic classes, elliptic equations on manifolds and Dirac operators. The new edition contains a new chapter on spectral geometry presenting recent results which appear here for the first time in printed form.

This book is devoted to Killing vector fields and the one-parameter isometry groups of Riemannian manifolds generated by them. It also provides a detailed introduction to homogeneous geodesics, that is, geodesics that are integral curves of Killing vector fields, presenting both classical and modern results, some very recent, many of which are due to the authors. The main focus is on the class of Riemannian manifolds with homogeneous geodesics and on some of its important subclasses. To keep the exposition self-contained the book also includes useful general results not only on geodesic orbit manifolds, but also on smooth and Riemannian manifolds, Lie groups and Lie algebras, homogeneous Riemannian manifolds, and compact homogeneous Riemannian spaces. The intended audience is graduate students and researchers whose work involves differential geometry and transformation groups.