**Announcements**:

**Teachers:**
Marius Crainic (UU)
Federica Pasquotto (VU)

**Location and time: **
Time: Wednesdays, from 10:15 to 13:00 (the first two hours are for the lectures, the last one is for the exercise classes, sometime the order is reversed)
Location: room C-121, W&N building, VU University - Faculty of Sciences
De Boelelaan 1081a, Amsterdam

**Organization: **
Lectures: 2 times 45 minutes
Assisted exercise session: 1 hour per week (immediately after the lectures). Assistants:
Jagna Wiśniewska (VU), j.j.wisniewska_at_vu.nl
Ori Yudilevich (UU), oriyudilevich_at_gmail.com
Kirsten Wang (UU), K.J.L.Wang_at_uu.nl

**Exam: **

There will be hand-in problems. They will count for 50 percent of your
grade. At the end of the course there will be a take home exam.

So far these were the homework problems:
First homework:
some "good" solutions to the first hand-in homework can be found here. Please discuss with the assistants any question you may still have about them

Second homework: the second exercise previously contained a typo which has now been corrected. The term *symmetric* here corresponds to the notion of *torsion-free* connection in the lecture notes.
Third homework with solutions.

And this was the final exam.

**Prerequisites: **

A good knowledge of multi-variable calculus.
Some basic knowledge of topology (such as compactness).
The standard basic notion that are tought in the first course on Differential Geometry, such as: the notion of manifold, smooth maps, immersions and submersions, tangent vectors, Lie derivatives along vector fields, the flow of a tangent vector, the tangent space (and bundle), the definition of differential forms, DeRham operator (and hopefully the definition of DeRham cohomology).

**Homework: **

Please do all the exercises in the lecture notes. The hand-in problems will be posted on this page, as a separate sheet

**Lecture notes: ** Lecture notes might be made available during the course, but only when the lecturer's treatment of the subject substantially differs from the treatment in the literature.

Lectures notes, last update (17-12-2013)
This reminder covers some of the basic notions of differential geometry that are necessary as background for the theory of G-structures on manifolds

**The schedule week by week **(here we will try to add, after each lecture, a description of what was discussed in the
lectures + the exercises):

** Week 1 **: Introduction, examples of linear G-structures: inner products, orientations, volume forms
** Week 2 **: More examples of linear G-structures: p-directions, integral affine structures, complex structures, symplectic forms, Hermitian structures. General definition of a linear G-structure (all topics still belong to Part 1 of the lecture notes)
** Week 3 **: G-structures on manifolds by examples (Part 2 of the lecture notes): general theory, frames and coframes, orientations, volume forms
** Week 4 **: more examples of G-structures on manifolds: distributions and foliations, (almost) complex structures
** Week 5 **: (almost) symplectic structures, Darboux's theorem, existence of Riemannian structures
** Week 6 **: affine structures, integrability of Riemannian structures
** Week 7 **: the exponential map for matrices, closed subgroups of GL(n) and their Lie algebra
** Week 8 **: smooth manifold structure on the bundle of frames, properties of the action of GL(n) on the bundle of frames, definition of principal G-bundle and homomorphism of principal bundles
** Week 9 **: vector bundles, sections, differential forms with values in a vector bundle
** Week 10 **: vector bundles associated to principal bundles. Please read: reduction and infinitesimal action
** Week 11**: the tautological form of a G-structure, connections on vector bundles, locality, derivatives along paths. Please read: De Rham-like operators and curvature of a connection (5.7 and 5.12 in the notes)
** Week 12**: parallel transport on vector bundles, principal bundles connections and connection 1-forms, parallel transport in principal bundles, from vector bundle connections to principal ones.
** Week 13**: Curvature of a connection on principal bundles, VB connections versus PB connections, G-compatible connections on TM
** Week 14**: equivalent definition of G-compatibility in some examples, curvature of G-compatible connections on TM. Please read: the torsion of connections on G-structures
** Week 15**: intrinsic torsion, integrability results for G-structures, examples (Riemannian metrics and symplectic forms). Please read: complex structures

**Aim/content of the course:**

The aim of this course is to provide an introduction to the general concept of a G-structure, which includes several significant geometric structures on differentiable manifolds (for instance, Riemannian or symplectic structures).

The course will start with a discussion of "geometric structures" on vector spaces and on manifolds.
Some of the key-words are: Riemannian metrics, distributions, foliations, symplectic structures, almost complex and complex structures.

We will then introduce the concept of a G-structure on a manifold and concentrate on the general framework that allows us to take this more general (abstract) point of view: Lie groups and Lie algebras, principal bundles, and connections.

The last part of the course will focus on topics such as equivalence and integrability of G-structures and discuss their interpretation in the some specific examples.

**Literature: **

We will occasionally use lecture notes (see above).

The main reference for this course is:

S. Sternberg, "Lectures on differential geometry", Prentice-Hall, First (1964) or Second (1983) edition.

Last update: 30-1-2014