250A. Differentiable Manifolds
Math 250A.
Differentiable Manifolds - Fall 04 - Hans Lindblad
250A is roughly vector calculus on manifolds.
An example of a manifold is a surface in space
when you ignore the rigid structure of space.
Many things in geometry and physics,
such as Einstein's equations of general relativity,
are best stated independent of
a particular coordinate system.
A manifold together with invariantly defined differential operators
is the natural setting for many nonlinear equations of physics.
Applications include general relativity, fluid mechanics,
electromagnetism, Hamiltonian mechanics, dynamical systems and control theory.
The core topics in the course are:
- Differentiability. Inverse and implicit function theorem.
Submanifolds of Euclidean space.
- Topological and smooth manifolds.
Tangent space and bundle. Vector bundles.
- Vector fields, existence for ODE, Lie bracket and derivative,
Frobenius theorem.
- Tensors and differential forms.
Tensor bundles and cotangent bundle.
- Integration, partition of unity, orientation,
Poincare lemma, Stokes formula, De Rham
Additional topics that we might cover:
- Applications to Hamiltonian Mechanics and Electromagnetism
- Connections and covariant derivatives. Parallel transport.
- Transversality and tubular neighborhood.
Sards theorem and embedding theorems.
- Lie groups.
The course continues in the Winter quarter
with Riemannian geometry, surfaces and relativity
see
250B
The main text for the course is:
- Abraham, Marsden, Ratiu
Manifolds, Tensor Analysis, and Applications
Springer 2005, 3ed
We will cover ch. 2-4, 6-8 (skipping the supplements),
and if time, 9.1, 9.3 and connections.
I choose the book because it is directed towards applications
but there are several shorter introductions:
- Lee
Introduction to Smooth Manifolds
Springer 2003
- Boothby
An introduction to differentiable manifolds and Riemannian geometry
Acad. Press 2002
- Spivak
A Comprehensive Introduction to Differential Geometry Vol I
Publish & Perish 1999
I plan to put my lecture notes on the web
and the
references will
be on reserve in the library.
If you have had no prior encounter with the material you can
look at any
introduction or undergraduate text.
There will be no exams but some homework problems.
Lectures MWF 3-4 in APM6218. Office hour W 4-5 in APM7220.
Problem session F 4-5 in APM5829.
Lecture Notes:
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Homeworks