LMS/EPSRC Short Instructional Course
DIFFERENTIAL GEOMETRY, HOMOGENEOUS SPACES AND INTEGRABLE
SYSTEMS
University of Durham, 16-20 September 2002
MARIO MICALLEF
INTRODUCTION TO DIFFERENTIAL GEOMETRY
The notion of continuity of real valued functions of a real
variable generalizes in a straightforward way to continuity of functions
between metric spaces. Differentiability is much harder to
generalize because it makes essential use of the underlying
linear structure of Euclidean space. Indeed, one must first introduce
a class of topological spaces (called manifolds) which,
near every point, look like a ball in Euclidean space.
A manifold structure not only allows us to define
the notion of a differentiable function
but also the geometric notions of a tangent space,
vector field and integral curves. These topics will form
the content of the first lecture.
In the second lecture we will introduce differential forms
which are the objects that can be integrated over manifolds.
The key results here are Stokes's Theorem and de Rham's Theorem
which provide a fundamental link between the homology of a
manifold (an algebraic topological property) and the calculus
of differential forms. There will be no time for full
proofs.
The remaining lectures will be devoted to the introduction of
basic geometric ideas like parallel transport and the
curvature associated to it, Riemannian manifolds in which
one can measure lengths of curves and angles
and the geometry of submanifolds.
Basic notions of symplectic, complex and Kahler manifolds will
also be discussed.
Particular attention will
be paid to examples including the sphere, hyperbolic
ball and complex projective space which will provide a link with
the course on Lie Groups and Homogeneous Spaces.
Differential Geometry is a highly computational subject and
students will be encouraged to compute curvature
tensors and other geometric quantities.
The material in this course is essential for anyone working in
the many areas of geometry and topology, including algebraic
geometry. It is also essential for anyone working in any
branch of theoretical physics (e.g. general relativity,
string theory, gauge theory) which is concerned with
a geometric formulation of the fundamental forces of
nature.
Students will be expected to have some knowledge of the following
topics, although an opportunity for their review will be provided
on the first day of the Short Course:
implicit and inverse function theorems;
dual vector spaces and theory of symmetric bilinear forms;
basic point-set topology, including quotient topology;
fundamental group;
existence and uniqueness of solutions to
ordinary differential equations,
smooth dependence on initial conditions.
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Last update: 15 March 2002