* A geometric approach to problems in physics, many of which cannot be solved by any other methods * Text is enriched with good examples and exercises at the end of every chapter * Fine for a course or seminar directed at grad and adv. undergrad students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics

Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data. Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods. Beyond medical image computing, the methods described in this book may also apply to other domains such as signal processing, computer vision, geometric deep learning, and other domains where statistics on geometric features appear. As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces. The foundational material and the advanced techniques presented in the later parts of the book can be useful in domains outside medical imaging and present important applications of geometric statistics methodology Content includes: - The foundations of Riemannian geometric methods for statistics on manifolds with emphasis on concepts rather than on proofs - Applications of statistics on manifolds and shape spaces in medical image computing - Diffeomorphic deformations and their applications As the methods described apply to domains such as signal processing (radar signal processing and brain computer interaction), computer vision (object and face recognition), and other domains where statistics of geometric features appear, this book is suitable for researchers and graduate students in medical imaging, engineering and computer science. - A complete reference covering both the foundations and state-of-the-art methods - Edited and authored by leading researchers in the field - Contains theory, examples, applications, and algorithms - Gives an overview of current research challenges and future applications

The present textbook is a somewhat expanded version of the material of a three-semester course I gave in Bochum. It attempts a synthesis of geometric and analytic methods in the study of Riemannian manifolds. In the first chapter, we introduce the basic geometric concepts, like dif ferentiable manifolds, tangent spaces, vector bundles, vector fields and one parameter groups of diffeomorphisms, Lie algebras and groups and in par ticular Riemannian metrics. We also derive some elementary results about geodesics. The second chapter introduces de Rham cohomology groups and the es sential tools from elliptic PDE for treating these groups. In later chapters, we shall encounter nonlinear versions of the methods presented here. The third chapter treats the general theory of connections and curvature. In the fourth chapter, we introduce Jacobi fields, prove the Rauch com parison theorems for Jacobi fields and apply these results to geodesics. These first four chapters treat the more elementary and basic aspects of the subject. Their results will be used in the remaining, more advanced chapters that are essentially independent of each other. In the fifth chapter, we develop Morse theory and apply it to the study of geodesics. The sixth chapter treats symmetric spaces as important examples of Rie mannian manifolds in detail.

This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.

This book presents a selection of the most recent algorithmic advances in Riemannian geometry in the context of machine learning, statistics, optimization, computer vision, and related fields. The unifying theme of the different chapters in the book is the exploitation of the geometry of data using the mathematical machinery of Riemannian geometry. As demonstrated by all the chapters in the book, when the data is intrinsically non-Euclidean, the utilization of this geometrical information can lead to better algorithms that can capture more accurately the structures inherent in the data, leading ultimately to better empirical performance. This book is not intended to be an encyclopedic compilation of the applications of Riemannian geometry. Instead, it focuses on several important research directions that are currently actively pursued by researchers in the field. These include statistical modeling and analysis on manifolds,optimization on manifolds, Riemannian manifolds and kernel methods, and dictionary learning and sparse coding on manifolds. Examples of applications include novel algorithms for Monte Carlo sampling and Gaussian Mixture Model fitting, 3D brain image analysis,image classification, action recognition, and motion tracking.

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

Hoping to make the text more accessible to readers not schooled in the probabalistic tradition, Stroock (affiliation unspecified) emphasizes the geometric over the stochastic analysis of differential manifolds. Chapters deconstruct Brownian paths, diffusions in Euclidean space, intrinsic and extrinsic Riemannian geometry, Bocher's identity, and the bundle of orthonormal frames. The volume humbly concludes with an "admission of defeat" in regard to recovering the Li-Yau basic differential inequality. Annotation copyrighted by Book News, Inc., Portland, OR.

This is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. Specifically, I have added three chapters on Riemannian and pseudo Riemannian geometry, that is, covariant derivatives, curvature, and some applications up to the Hopf-Rinow and Hadamard-Cartan theorems, as well as some calculus of variations and applications to volume forms. I have rewritten the sections on sprays, and I have given more examples of the use of Stokes' theorem. I have also given many more references to the literature, all of this to broaden the perspective of the book, which I hope can be used among things for a general course leading into many directions. The present book still meets the old needs, but fulfills new ones. At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differentiable maps in them (immersions, embeddings, isomorphisms, etc.).