Differential Geometry
2026/2027 mastermath course

This is the web-site for the mastermath course "Differential Geometry" given in the Fall of 2026. Here we keep all the practical information about the course, updates/changes that take place during the year, lecture notes, etc.

♣ Aim: The aim of this course is to provide an introduction to vector bundles and principal bundles and, along the way, illustrate their relevance to the study of various geometric structures on manifolds (Riemannian, foliations, etc). The plan is to start with vector bundles and their geometry (connections, curvature, etc); when discussing metrics we will make a small digression into Riemannian geometry and the tubular neighborhood theorem. Then we move on to principal bundles, their geometry, describe their relationship with vector bundles and their relevance to the theory of G-structures. Along the way we will also improvise a crash course on Lie groups (only what we need).

♣ Prerequisites: The standard basic notions that are taught 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 vector field, the tangent space (and bundle), differential forms, manipulations with differential forms including de Rham operator and Cartan's magic formula, the definition of de Rham cohomology, integration and Stokes theorem. See e.g. the introductory "Manifolds" lecture notes of M. Crainic or the book "Introduction to smooth manifolds" by John Lee.

♣ Literature: LECTURE NOTES.

For exercises, here is a COMPILATION OF EXERCISES (AND SOME OF THE PREVIOUS EXAMS).

♣ The lectures: Thursdays, 11:00-12:45, in BBG - 001. Lecturer: Marius Crainic.

♣ The exercise classes: Thursdays, 10:00-10:45, in BBG - 001. Teaching assistants: Sven Holtrop and Yuri Sampieri.

Feedback exercises: we assign feedback exercises every exercise class, for which you have two weeks to hand them in. They are not mandatory and will not contribute to your final grade, but are a way to get feedback on your understanding of the material.

Exercises for the exam: There will be several lists of exercises to help you prepare for the exam. The intention is that a large part of the exam questions will come from these lists (like 3/4 or 4/5, though that may also mean that one exam exercise will have one new sub-question/item). But the idea is that if, next to the werkcollege exercises, if you go through all these lists, you should be pretty well prepared for the exam. See above for the lists,

The final mark: The final mark is the outcome of a written exam at the end of the course (Thursday, 21 January 2027, 10:00 in EDUC - MEGARON). Please be aware that a large part of the exam will be based on the lists of "Collection of exercises for the exam" that we will distribute throughout the semester. So far, we have posted:

List 1 of exercises for the exam.

PLEASE BE AWARE: Although I will be trying to add here all the essential information, it is impossible to add here everything that is discussed during the lectures (e.g. my answer to little questions asked by students during the lecture or in the pause, or simply hints of tricks that may be useful at the exam, or etc etc). So, if you miss one (or more) lecture, please keep in touch with one of your colleagues that is present to find out more about what was discussed in class.

THE SCHEDULE WEEK BY WEEK (to be added, after each lecture: a description of what was discussed in the lectures + the exercises for the werkcollege):

♣ September 10: Lecture 1. The concept of fiber bundle and the main concepts around it (fibers, (locl) sections, morphisms, etc), local triviality, vector bundles, the module of sections, (local frames). Then some first examples, including the tautological line bundle and tangent bundles of manifolds.

Blackboard photos of lecture 1 (and a bit of the werkcollege)..

Exercises related to the background material (submersions). and here is also a feedback exercise .

Exercises for the next werkcollege: 1.1, 1.3, 1.6, 1.7 and 1.14. Review the standard smooth structure on RP^n and CP^n (bachelor notes 2.2.2 and 2.2.3).

♣ September 17: Lecture 2. Revisited the concept of vector bundles and morphism, pointing out that the functor from vector bundle to $C^{\infty}(M)$-morphisms is fully faithful, and that one can use local frames of $E$ to prove not only local triviality but even to exhibit the smooth structure on $E$, then looked at $TM$ in more detail, pointing out that for $T^*M$ (and similar ones) the story is the same, while deriving a general procedure/method for getting a smooth vector bundle from a family of vector spaces together with local frames whose change of coordinates is smooth. Then we moved to the local point of view on vector bundle- the fact that they can be described via transition data.

Blackboard photos of lecture 2..

Exercises for the next werkcollege: 1.24-1.29 and 1.34. The feedback exercise is 1.32.

♣ September 24: Lecture 3. Revisited the local story, with the two main points: the general procedure to construct of vector bundle, as a smooth (!!!) vector bundle once we have the fibers, and some local frames that changes smoothly, and then the transition data view-point on vector bundles. Then we moved to examples of operations, that provide constructions of new vector bundles. We arranged them in a table, where we looked at the various aspects: fibers, sections, the $C^{\infty}(M)$-module of sections, local frames, transition functions (though the transition functions we discussed them orally/hand waving, we did not write things down on the blackboard). The constructions we looked at were: sums, tensor products, duals, Hom, the bilinear symmetric construction ($S^2E^*$), exterior powers $\Lambda^kE^*$. Also, we spent a bit of time explaining why $C^{\infty}(M)$-linear maps $\Gamma(E)\to \Gamma(F)$ correspond (bijectively) to sections $u$ of $Hom(E, F)$ or,. equivalently, to morphisms of vector bundles $u: E\to F$. In the last minutes I introduced the concept of inner product on vector bundles.

Blackboard photos of lecture 3..

Exercises for the next werkcollege:

- ordinary exercises:1.39, 1.44, 1.54, 1.55, 1.57, 1.62.

- feedback exercise: Make 1.56 (c) and 1.62 for the case of Sym^k. Transform these two exercises into a small explanatory piece of writing. How many perspectives are there on on Sym^k(E, F) and how do you efficiently show that they are all the same?

- recap exercises: Lie brackets, integral curves and flows, see this file.

♣ October 1: Lecture 4. Recap of constructions from previous time to which I added pull-backs, and of inner products on vector bundles to which I added the concept/existence of (local) orthonormal frames. Then we moved to the concept of vector-sub-bundle, with two characterisations, and some interesting examples; along the way, as "applications", I pointed out/explained a couple of fundamental facts on vector bundles such as: the dual of a vector bundle $E$ is always (noncanonically) isomorphic to $E$, every vector bundle can be embedded inside a trivial one, every sub-bundle admits a complementary sub-bundle, for any vector bundle $E$ there exists another one $E'$ such that $E\oplus E'$ is trivializable (isomorphic to the trivial one), the space of sections $\Gamma(E)$ of any vector bundle is a finitely generated projective $^{\infty}(M)$-module, and pointed out Serre-Swan's theorem. Then I moved to complex vector bundle, very fast pointing out that all the basic definitions on vector bundles work in the complex context, replacing all the real vector spaces by complex ones (hence $M$ is still a real manifold, but the fibers of $E$ are complex vector spaces, etc), with the warning that inner products are replaced by Hermitian products, and then the argument that the dual of a vector bundle $F$ is isomorphic to $F$ itself is no longer true in the complex context. Instead, $F^*$ is isomorphic to the conjugate of $F$. To be recaped and continued next time, when we will also move to complex structures on manifolds.

Exercises for the next werkcollege:

- ordinary exercises: 1.20, 1.41, 1.64 - 1.66, 1.70.

- feedback exercise: 1.46.

- recap: the definition of SMOOTH structures via SMOOTH atlases, the complex version of smoothness (holomorphicity), and the CR-equations (as a condition for a smooth function to be holomorphic). .

♣ October 8: Lecture 5. Parallelisms, then foliations and Frobenius theorem. Next time I will mention a couple of examples, then we move to complex structures and the Newlander-Nirenberg theorem (now without a proof).

Blackboard photos of lecture 5..

Exercises for the next werkcollege: 2.1, 2.2, 2.7, 2.8, 2.11, 2.13.

Feedback exercise: 2.9.

Please be aware that List 1 of exercises to prepare for the exam has been uploaded (see also above).

♣ October 15: Lecture 6. Reminder on Frobenius/foliations, then the notion of leaf, then various examples, including the Reeb foliation on $S^3$ and a bit on the history. Then moved to complex structures $J$ ... first on vector bundles (and their characterisation via the $i$-eigenspace as a subspace. of trhe complexification of the vector bundle) ... then moved to manifolds and almost complex structures on them, relationship with complex structures in sense of holomorphic atlases ... that allowed us to make sense of the integrability of an almost complex structure (by requiring that it comes from a holomorphic atlas) and that stated Newlander-Nirenberg in the form: $J$ is integrable if and only if the corresponding $i$-eigenspace is involutive (closed under taking the Lie bracket of vector fields). TBC next time.

Blackboard photos of lecture 6..

Exercises for the next werkcollege: 2.3-2.5 plus this one.

Feedback exercise: 2.14.

Recap exercises on differential forms: 4.54, 4.57.4.70 and 4.71 from the lecture notes of the bachelor.

♣ October 22: Lecture 7. Newlander-Nirenberg, Nijenhuis tensor, the similar tensor for foliations, a table overviewing various geometric structures, mentioned also symplectic structures, then, also for the later use, an overview on differential forms with coefficients in vector bundles.

Blackboard photos of lecture 7..

Exercises for the next werkcollege: to be added.

♣ October 29: Lecture 8. Connections, connection matrices, examples, paths and their $\nabla$-derivative and a bit about parallel transport.

Werkcollege photo .

Blackboard photos of lecture 8 ..

Exercises for the next werkcollege: to be added.

♣ November 5: Lecture 9. To be added.

Blackboard photos of lecture 9 (to be added)..

Exercises for the next werkcollege: to be added.

♣ November 12: Lecture 10. To be added.

Blackboard photos of lecture 10 (to be added)..

Exercises for the next werkcollege: to be added.