Ebook: Topological Geometry

Mathematicians frequently use geometrical examples as aids to the study of more abstract concepts and these examples can be of great interest in their own right. Yet at the present time little of this is to be found in undergraduate textbooks on mathematics. The main reason seems to be the standard division of the subject into several watertight compartments, for teaching purposes. The examples get excluded since their construction is normally algebraic while their greatest illustrative value is in analytic subjects such as advanced calculus or, at a slightly more sophisticated level, topology and differential topology. Experience gained at Liverpool University over the last few years, in teaching the theory of linear (or, more strictly, affine) approximation along the lines indicated by Prof. J. Dieudonne in his pioneering book Foundations of Modern Analysis [14], has shown that an effective course can be constructed which contains equal parts of linear algebra and analysis, with some of the more interesting geometrical examples included as illustrations. The way is then open to a more detailed treatment of the geometry as a Final Honours option in the following year. This book is the result. It aims to present a careful account, from first principles, of the main theorems on affine approximation and to treat at the same time, and from several points of view, the geometrical examples that so often get forgotten. The theory of affine approximation is presented as far as possible in a basis-free form to emphasize its geometrical flavour and its linear algebra content and, from a purely practical point of view, to keep notations and proofs simple. The geometrical examples include not only projective spaces and quadrics but also Grassmannians and the orthogonal and unitary groups. Their algebraic treatment is linked not only with a thorough treatment of quadratic and hermitian forms but also with an elementary constructive presentation of some little-known, but increasingly important, geometric algebras, the Clifford algebras. On the topological side they provide natural examples of manifolds and, particularly, smooth manifolds. The various strands of the book are brought together in a final section on Lie groups and Lie algebras.

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The earlier chapter of this self-contained text provide a route from first principles through standard linear and quadratic algebra to geometric algebra, with Clifford's geometric algebras taking pride of place. In parallel with this is an account, also from first principles, of the elementary theory of topological spaces and of continuous and differentiable maps that leads up to the definitions of smooth manifolds and their tangent spaces and of Lie groups and Lie algebras. The calculus is presented as far as possible in basis free form to emphasize its geometrical flavour and its linear algebra content. In this second edition Dr Porteous has taken the opportunity to add a chapter on triality which extends earlier work on the Spin groups in the chapter on Clifford algebras. The details include a number of important transitive group actions and a description of one of the exceptional Lie groups, the group G2. A number of corrections and improvements have also been made. There are many exercises throughout the book and senior undergraduates in mathematics as well as first-year graduate students will continue to find it stimulating and rewarding.