Topological Geometry

Provides a route from first principles through standard linear and quadratic algebra to geometric algebra, with Clifford's geometric algebras taking pride of place.

Ian R. Porteous (Author)

9780521298391, Cambridge University Press

Paperback, published 5 February 1981

500 pages
22.8 x 15.2 x 2.7 cm, 0.73 kg

Review of the hardback: '… a remarkable book, which is likely to remain very useful, both for teaching and research, during many years to come.' J.Dieudonné in Zentralblatt Für Mathematik

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.

Foreword
Guide
1. Maps
2. Real and complex numbers
3. Linear spaces
4. Affine spaces
5. Quotient structures
6. Finite-dimensional spaces
7. Determinants
8. Direct sum
9. Orthogonal spaces
10. Quaternions
11. Correlations
12. Quadric Grassmannians
13. Clifford Algebras
14. The Cayley algebra
15. Normed linear spaces
16. Topological spaces
17. Topological groups and manifolds
18. Affine approximation
19. The inverse function theorem
20. Smooth manifolds
21. Triality
Bibliography
List of symbols
Index.

Subject Areas: Geometry [PBM]