This is a reading seminar aimed at master students interested in Differential Geometry.
Talks will be prepared and given by the participants, in close coordination with the organisers. Along with their talk, speakers will also provide a handout with homework exercises for all participants, to be approved by the organisers. The speaker is responsible for grading this homework.
The aim of the course is twofold: students will learn about principal bundles and the general theory of geometric structures on manifolds on the one hand, and improve their ability to present mathematics on the other.
Francesco Cattafi, Marius Crainic and Maarten Mol
Day, time and place
Mondays 13.15-15.00 in room 610 of the Hans Freudenthal building, weekly from 11 February 2019 until June 2019.
Students should be familiar with the contents of a first course in differentiable manifolds, covering differential forms and vector bundles. Knowledge of connections on vector bundles is useful but is not necessary. No prerequisite on Lie groups and principal bundles is required.
Plan of the talks (with main and secondary references)
• 11 February - Lecture 0 (Francesco): Introduction to geometric structures
• 18 February - Lecture 1 (Wilmer): Lie groups; examples, actions and representations, Wilmer's exercises, Crainic 47-50 [also Kobayashi 38-44]
• 25 February - Lecture 2 (Jan): Lie algebras; examples, correspondence with Lie groups; infinitesimal actions and representations, Jan's exercises, Crainic 51-60 [also Kobayashi 38-44]
• 4 March - Lecture 3 (Pascal): Principal bundles, Pascal's notes and exercises, Crainic 61-65 [also Kobayashi 50-54, Sternberg 294-297]
• 11 March - Lecture 4 (Aaron): Correspondence between vector and principal bundles, Aaron's exercises, Crainic 66-72 [also Kobayashi 54-58]
• 18 March - Lecture 5 (Nina): G-atlases, G-structures and integrability; first examples, Crainic 83-85, 98-104 [also Kobayashi 287-290, Sternberg 309-315]
• 25 March - Lecture 6 (Kevin): Symplectic structures, Crainic 91-94, 111-114; Foliations, Crainic 88, 105-107
• 1 April - Lecture 7 (Santiago): Principal connections; correspondence between connections on principal and vector bundles, Crainic 73-81 [also Kobayashi 63-67, Sternberg 298-300]
• 8 April - Lecture 8 : Connections compatible with G-structures, Crainic 117-124
• 15 April - Lecture 9: Intrinsic torsion of a G-structure, Crainic 125-128
• 22 April: no lecture (Easter Monday)
• 29 April - Lecture 10: Intrinsic curvature and higher torsions of G-structures; examples, Crainic 128-135
• 6 May - Lecture 11: Structure function of a G-structure and local equivalence, Sternberg 315-320, 331-335 [also Singer-Sternberg 41-45]
• 13 May - Lecture 12: Prolongation of Lie algebras, of Lie groups and of G-structures, Sternberg 335-338 [also Singer-Sternberg 45-52]
• 20 May - Lecture 13: Integrability theorem for G-structures of finite type, Guillemin [also Sternberg 339]
• 27 May - Lecture 14:
• 3 June - Lecture 15:
• 10 June: no lecture (Pentecost)
• 17 June - Lecture 16:
Bonus topics (depending on the number of participants)
• Holonomy and Ambrose-Singer theorem, Kobayashi 71-73, 83-91 [also Sternberg 300-308, Joyce 26-33]
• G-structures and holonomy, Joyce 34-40
• Berger classification of Riemannian holonomy, Joyce 42-59
• Transitive G-structures and Exterior Differential Systems, Singer-Sternberg 59-67
References (in debetable order of importance for the purpose of this course)
• Marius Crainic, Lecture notes for the Mastermath course Differential Geometry 2015/2016, see here
• Shlomo Sternberg, Lectures on differential geometry, Chelsea Publishing Co., New York, 1983
• Shoshich Kobayashi, Katsumi Nomizu, Foundations of differential geometry, Vol. I and II, John Wiley & Sons, Inc., New York, 1996
• Victor Guillemin, The integrability problem for G-structures, Trans. Amer. Math. Soc. 116, 1965
• Isadore Singer, Shlomo Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15, 1965.
• Dominic Joyce, Compact manifolds with special holonomy, Oxford University Press, 2000
Participants are expected to give two seminar talks (each of which is a 2x45 min presentation), possibly more or less depending on the number of participants. They will study the material beforehand, hold a blackboard presentation about it, and distribute a hand-in exercise to the other seminar participants (to be approved beforehand by the seminar organizers and to be handed in by the students at the next lecture). The speaker is responsible for grading this hand-in exercise. In case of discussion about the solutions, the seminar organisers decide. Participants are expected to attend every seminar meeting. The final grade for the seminar is based on your talks and handouts (40%) and on your homework formulation and grades (60%).