This is MAGIC core course on differential geometry on manifolds, aimed at both pure and applied PhD students.
We begin with the basic ideas of differential geometry: manifolds, vectors and tensors, maps of manifolds, the Lie derivative, connections. Our choice of further topics includes curvature, Riemannian geometry, differential forms and integration, cohomology, and symplectic geometry. This could be adjusted a bit depending on your comments.
We plan to have a complete set of typed notes by the end of the course, but these will be written as we go along and put online in pieces.
Most of the material in the course is covered by both Aubin, Lee/Lee or Schutz. Pure mathematicians may prefer Aubin or Lee/Lee and applied mathematicians may prefer Schutz. You may want to own one of these three books. Warner is another good pure text, except that it does not cover connections. None of these books cover symplectic geometry, for which we use Berndt. Abraham/Marsden/Ratiu (does not cover connections) and Choquet-Bruhat/DeWitt-Morette/Dillard-Bleick are classic texts that can serve as background reading.
For lectures, we will use the electronic whiteboard. Screen shots will be saved on the website.
At the end of each lecture, we will suggest an exercise or two to do immediately. Other exercises can be found in the notes or the three recommended books.
Calculus of several variables (integration, implicit function theorem). Linear algebra (axioms of a vector space, linear operators, bases).
The differential geometry of curves and surfaces in 3-dimensional Euclidean space is neither a prerequisite nor part of the syllabus, but if you know some you will see how it fits in as a special case.
Manifolds, charts, partitions of unity.
Vector fields as tangents to curves and derivative operators. Lie bracket.
Covectors, tensors, bases. Abstract index and index-free notation.
Maps of manifolds, pull-back and push-forward.
Lie derivatives of scalar, vector, tensor fields.
Submanifolds. Statement of Frobenius theorem (for vector fields).
Connection as another way of taking a derivative of tensor
fields. Geometric meaning: geodesics. Torsion.
Curvature of a connection. Geometric meaning: geodesic deviation.
Brief overview of principal fibre bundles, Lie groups, connections on fibre bundles.
Differential forms. Exterior product. Exterior derivative.
Exterior derivative and Lie derivative (Cartan's formula).
Integration of differential forms over (sub)manifolds. Metric volume
form and Jacobi determinant. Stokes's theorem.
Poincare Lemma. De Rham cohomology, basic examples with partial proof. Statement of
Symplectic geometry and Poisson brackets.