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.