Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? This invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms de Rham theory , and applications such as the Poincare-Hopf theorem relating the Euler number of a manifold and the index of a vector field.
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The basic objects of differential topology are manifolds, introduced by Riemann as "multiply-extended quantities'' to generalize surfaces to many dimensions. The appeal of manifolds is the richness of available structures that follow from the definition. Such neighborhoods may overlap and this gives rise to coordinate transformations. Restricting the class of coordinate transformations determines some of the particular properties of the manifold, for example, if the transformations are complex analytic a complex manifold , or have Jacobians of positive determinant an orientable manifold.
Furthermore, the neighborhoods may be taken as parameter spaces for other geometric data, glued together with the coordinate transformations into fibre bundles. In a locally Euclidean space, we can do calculus, and so manifolds admit differentiable functions, vector fields, a tangent space, all intrinsically defined via the coordinate transformations.
The main point of differential topology is to sort out the consequences of all this structure, and the interrelations between its various aspects.
The problem with a book about manifolds is that the basic definitions are so many, and you need them all to study their interactions with one another. The book of Barden and Thomas is based on courses taught at the University of Cambridge. The direction is topological, leaving the geometric Riemannian metrics for another course. With this decisive turn, the authors can strike deep into relevant structure with purpose.
After the basic structure is set out, they treat fibre bundles with the Submersion Theorem of Ehresman as goal, illustrating how manifolds with geometry might be classified by the local geometries. The next few chapters treat topological equipment — the exterior derivative and de Rham cohomology, which feature the relations between the calculus and the topology on a manifold. The next chapter gives the reader some real substance for all the structure. Degrees, indices, the Gauss map, and Morse theory employ the available structure in a dramatic way to give back geometric data.
A rich source of examples is considered next with a thumbnail introduction to Lie groups. There is a final chapter that provides the background in analysis and algebra to support the course. I am saddened by the death of Professor Thomas. I am glad that we have a record of his notes and the course or Professor Barden; they are rich with insight into how to think about manifolds, introducing students to a subject that is central to mathematics and to our framework for understanding physics.
Short and insightful, this is a good way to get started. Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R.
Ford Awards Merten M. Dennis Barden and Charles Thomas. Publication Date:. Number of Pages:. Differential Geometry. Log in to post comments.
C3.3 Differentiable Manifolds - Material for the year 2019-2020
Useful but not essential: B3. A manifold is a space such that small pieces of it look like small pieces of Euclidean space. Thus a smooth surface, the topic of the B3 course, is an example of a 2-dimensional manifold. Manifolds are the natural setting for parts of classical applied mathematics such as mechanics, as well as general relativity. They are also central to areas of pure mathematics such as topology and certain aspects of analysis. In this course we introduce the tools needed to do analysis on manifolds.
ISBN 13: 9781860943553
It does not only guide the reader gently into the depths of the theory of differential manifolds but also careful on giving advice how one can place the information in the right context. It is certainly written in the best traditions of great Cambridge mathematics. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.
An Introduction to Differential Manifolds