LMS/EPSRC Short Instructional Course
DIFFERENTIAL GEOMETRY, HOMOGENEOUS SPACES AND INTEGRABLE
SYSTEMS
University of Durham, 16-20 September 2002
DMITRI ALEKSEEVSKY
LIE GROUPS AND HOMOGENEOUS SPACES
According to the Erlanger Programm of Felix Klein,
the main object of geometry is a G-space M, that is,
a set M with a given group G of transformations.
If the group G acts transitively on M, that is,
for any two points x and y in M there exists an
element in G which transforms x into y, then the G-space
is said to be homogeneous, and after picking a
point m in M we can identify M with the set G/H of
left cosets, where H is the subgroup of G consisting
of those elements which map m to itself. The homogeneous
geometry of such a space M = G/H is the study of
those geometrical properties of M and its subsets
which are invariant under G.
Some of the more famous examples of homogeneous
geometries are provided by Euclidean geometry,
affine geometry, conformal geometry, projective
geometry, hyperbolic geometry, and
Minkowski geometry of special relativity.
Using the identification of a homogeneous space M with
the quotient G/H, many hard problems in homogeneous
geometry can be reformulated in terms of the group G
and the
subgroup H, and in the case when G and H are Lie groups,
in terms of the corresponding infinitesimal objects:
the Lie algebra of G and its Lie subalgebra
associated with H. Such an infinitesimal approach
enables one to use linear algebra to tackle non-linear
problems and provides a very useful and powerful method
for solving problems in geometry, analysis, and the
theory of differential equations.
For example, the equations satisfied by an
Einstein metric (which, in particular, according
to general relativity describe the evolution of our
universe) is a very
complicated non-linear system of partial differential
equations. However, for invariant metrics on a
homogeneous space, this system reduces to a
system of algebraic equations, which can be solved in many cases.
One of the most significant advances of 20th century
mathematics is Cartan's classification of semisimple
Lie groups. This leads to the classification of
two important classes of homogeneous spaces, namely
symmetric spaces and flag manifolds, which have many
applications in real and complex analysis,
topology, geometry, dynamical systems, probability and
physics.
The aim of the lecture course is to give a short
introduction into the theory of Lie groups and Lie
algebras, including Cartan's structure theory of
semisimple Lie
algebras, and into the geometry of homogeneous spaces,
especially symmetric spaces and flag manifolds.
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Last update: 8 February 2002