The book under review studies methods and techniques of contemporary differential geometry as well as several different physical and non-physical applications. One may consider this book as a natural continuation of [V. G. Ivancevic and T. T. Ivancevic, Natural biodynamics, Hackensack, NJ: World Scientific (2005; Zbl 1120.92002)] where the same authors focus on applications of biodynamic type whereas in the current book, they mainly focus on applications in the area of theoretical physics such as Einstein-Feynman-Witten theories. In addition to that, they also consider applications of control theory, robotics, neurodynamics, psychodynamics, and socio-economical dynamics.
The book consists of six chapters and each chapter has both theoretical and related applied parts. The first chapter is devoted to fundamentals in manifold theory and related geometric structures on manifolds. In the second chapter, the authors study both classical and modern differential objects such as tensors, functors and also modern physical objects such as actions. In the third chapter, modern objects in manifold geometry (i.e., in Riemannian geometry and complex geometry) are discussed as well as their applications. Bundle geometry and its applications are considered in Chapter 4.
In Chapter 5 that is regarded as the second part of Chapter 4, the authors develop jet bundle geometry and its applications in the areas of non-autonomous mechanics and field physics.
The main topics of the last chapter include Feynman’s path integrals and their applications.
The book covers almost the full spectrum of topics in classical and modern applications of differential geometry so that it can be used in many branches of science ranging from theoretical disciplines (e.g., mathematics, physics, chemistry and biology) to more applied disciplines (e.g., control, robotic, mechatronic and computer engineering and neurodynamics) and even social disciplines (e.g., psychology, sociology and economy).