Vector fields on manifolds play a major role in mathematics and other sciences. In particular, the Poincaré-Hopf index theorem gives rise to the theory of Chern classes, key manifold-invariants in geometry and topology. It is natural to ask what is the ‘good’ notion of the index of a vector field, and of Chern classes, if the underlying space becomes singular. The question has been explored by several authors resulting in various answers, starting with the pioneering work of M.-H. Schwartz and R. MacPherson. We present these notions in the framework of the obstruction theory and the Chern-Weil theory. The interplay between these two methods is one of the main features of the monograph.

An application of the techniques of dynamical systems and bifurcation theories to the study of nonlinear oscillations. Taking their cue from Poincare, the authors stress the geometrical and topological properties of solutions of differential equations and iterated maps. Numerous exercises, some of which require nontrivial algebraic manipulations and computer work, convey the important analytical underpinnings of problems in dynamical systems and help readers develop an intuitive feel for the properties involved.

Presenting a classical approach to the foundations and development of the geometry of vector fields, this volume space, three orthogonal systems, and applications in mechanics. Other topics, including vector fields, Pfaff forms and systems in n-dimensional space, foliations and Godbillon-Vey invariant, are also considered. There is much interest in the study of geometrical objects in n-dimensional Euclidean space, and this volume provides a useful and comprehensive presentation.

This book provides an up-to-date introduction to the theory of manifolds, submanifolds, semi-Riemannian geometry and warped product geometry, and their applications in geometry and physics. It then explores the properties of conformal vector fields and conformal transformations, including their fixed points, essentiality and the Lichnerowicz conjecture. Later chapters focus on the study of conformal vector fields on special Riemannian and Lorentzian manifolds, with a special emphasis on general relativistic spacetimes and the evolution of conformal vector fields in terms of initial data. The book also delves into the realm of Ricci flow and Ricci solitons, starting with motivations and basic results and moving on to more advanced topics within the framework of Riemannian geometry. The main emphasis of the book is on the interplay between conformal vector fields and Ricci solitons, and their applications in contact geometry. The book highlights the fact that Nil-solitons and Sol-solitons naturally arise in the study of Ricci solitons in contact geometry. Finally, the book gives a comprehensive overview of generalized quasi-Einstein structures and Yamabe solitons and their roles in contact geometry. It would serve as a valuable resource for graduate students and researchers in mathematics and physics as well as those interested in the intersection of geometry and physics.

This textbook serves as an introduction to modern differential geometry at a level accessible to advanced undergraduate and master's students. It places special emphasis on motivation and understanding, while developing a solid intuition for the more abstract concepts. In contrast to graduate level references, the text relies on a minimal set of prerequisites: a solid grounding in linear algebra and multivariable calculus, and ideally a course on ordinary differential equations. Manifolds are introduced intrinsically in terms of coordinate patches glued by transition functions. The theory is presented as a natural continuation of multivariable calculus; the role of point-set topology is kept to a minimum. Questions sprinkled throughout the text engage students in active learning, and encourage classroom participation. Answers to these questions are provided at the end of the book, thus making it ideal for independent study. Material is further reinforced with homework problems ranging from straightforward to challenging. The book contains more material than can be covered in a single semester, and detailed suggestions for instructors are provided in the Preface.

This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings: