# Differential Geometry (Mastermath course, Fall 2013)

Announcements:
• You can find the final results for this course here.
• the marking for exercise 1 was: 2p (question 1)+ 4p (question 2)+ 3p (question 3)+ 1p (question 4)
• the marking for exercise 2 was: 0.5p (question 0)+ 0.5p (question 1)+ 0.5p (question 2)+ 0.5p (question 3)+ 0.5 p (question 4) + 1p (question 5)+ 1p (question 6)+ 0.5p (question 7)+ 0.5p (question 8)+ 0.5p (question 9)+ 0.5p (question 10)+ 0.5p (question 11)+ 1p (question 12)+ 0.5p (question 13)+ 0.5p (question 14)+ 1p (question 15)
• the exam mark was the weighted average (Ex1+ 2 Ex2)/3.
• The average of homework and exam mark were rounded (to give the final mark); the way the final rounding was done (whenever it was not clear or not fair) was influenced by the exam mark.
• If you did not pass the exam and are interested in a retake (in the form of an oral exam) please contact one of the instructors

• 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