Modern Geometric Structures And Fields (Graduate Studies in Mathematics) by S. P. Novikov (PDF)

Description

The book presents the basics of Riemannian geometry in its modern form as geometry of differentiable manifolds and the most important structures on them. The authors’ approach is that the source of all constructions in Riemannian geometry is a manifold that allows one to compute scalar products of tangent vectors. With this approach, the authors show that Riemannian geometry has a great influence to several fundamental areas of modern mathematics and its applications. In particular, Geometry is a bridge between pure mathematics and natural sciences, first of all physics. Fundamental laws of nature are formulated as relations between geometric fields describing various physical quantities. The study of global properties of geometric objects leads to the far-reaching development of topology, including topology and geometry of fiber bundles. Geometric theory of Hamiltonian systems, which describe many physical phenomena, led to the development of symplectic and Poisson geometry. Field theory and the multidimensional calculus of variations, presented in the book, unify mathematics with theoretical physics. Geometry of complex and algebraic manifolds unifies Riemannian geometry with modern complex analysis, as well as with algebra and number theory. Prerequisites for using the book include several basic undergraduate courses, such as advanced calculus, linear algebra, ordinary differential equations, and elements of topology.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book conveys the idea that differential geometry in a broad sense is about smooth manifolds with a 2-tensor on it (that is a bilinear form on each tangent space). Examples of such tensors that have proved to be useful are (pseudo)-Riemannian metrics, hermitian metrics and symplectic forms. These structures are closely related and in some manifolds (Kähler manifolds) all of them exist in a compatible way. The symmetry of those structures is measured by the action of a group which in some important cases is a manifold itself, these groups are called Lie groups. When a manifold is regarded as a phase space or a configuration space its points are not as fundamental as the related quantities we can observe: so one considers the (Lie) algebra of real smooth functions on the manifold. Surprisingly one can learn a lot about the topology of a manifold by studying the critical points of generic real-valued smooth functions on it, this is Morse theory. When one considers instead real-valued ‘functionals’ on the space of curves on a manifold, the study of their critical points is still relevant since for instance the critical points of the energy functional on a Riemannian manifold (geodesics) describe the motion of a free particle according to classical mechanics (in a sense every kind of motion can be seen as geodesic motion but to prove it sometimes requires ad hoc tricks). Obtaining physical equations as conditions of being a critical point of a certain functional has been a successful method in Physics: the Einstein field equations and Yang-Mills equation are examples of this. Perhaps one of the most important aspects of this trick lies in the fact that one can apply for example Noether’s Theorem to the functional and obtain “conserved quantities” in the presence of symmetry.This is a sketch of the topics presented in this book, obviously it is not a comprehensive treatment but there are plenty of examples of each concept, the main strength of this book is the choice of material. If you want an elementary introduction to all the main branches of modern geometry and how this geometry has been used in physics (which explains its importance), this is the book for you.

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