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-Nomizu 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-Nomizu 38-44]
• 4 March - Lecture 3 (Pascal): Principal bundles, Pascal's notes and exercises, Crainic 61-65 [also Kobayashi-Nomizu 50-54, Sternberg 294-297]
• 11 March - Lecture 4 (Aaron): Correspondence between vector and principal bundles, Aaron's exercises, Crainic 66-72 [also Kobayashi-Nomizu 54-58]
• 18 March: lecture cancelled (building closed for safety measures again the shooting)
• 25 March - Lecture 5 (Nina): G-atlases, G-structures and integrability; first examples, Nina's exercises, Crainic 83-85, 98-104 [also Kobayashi-Nomizu 287-290, Sternberg 309-315]
• 1 April - Lecture 6 (Kevin): Symplectic structures, Crainic 91-94, 111-114; Foliations, Crainic 88, 105-107, Kevin's notes and exercises
• 8 April - Lecture 7 (Santiago): Principal connections; correspondence between connections on principal and vector bundles, Santiago's notes and exercises, Crainic 73-81 [also Kobayashi-Nomizu 63-67, Sternberg 298-300]
• 15 April - Lecture 8 (Pascal) : Connections compatible with G-structures, Pascal's notes and exercises, Crainic 117-124
• 22 April: no lecture (Easter Monday)
• 29 April - Lecture 9 (Nina): Intrinsic torsions of G-structures and examples, Nina's exercises, Crainic 125, 127, 132-134; ROOM CHANGE: BBG005
• 6 May - Lecture 10 (Aaron): Intrinsic curvatures of G-structures and examples, Aaron's exercises, Crainic 126, 128-131
• 13 May: no lecture (organisers away)
• 20 May - Lecture 11 (Kevin): Tautological form and structure function of a G-structure, Kevin's notes and exercises, Struchiner 22-23, 25-26 [also Crainic 135-136, Sternberg 315-320, 331-335, Singer-Sternberg 41-45]
• 27 May - Lecture 12 (Santiago): Prolongations of G-structures, Struchiner 26-29 [also Sternberg 331-338, Singer-Sternberg 45-52]
• 3 June - Lecture 13 (Jan): Integrability theorem for G-structures of finite type, Stermberg 220-221, 339 [also Guillemin 550-554]
• 10 June: no lecture (Pentecost)
• 17 June - Lecture 14 (Wilmer): Automorphism group of a G-structure, Kobayashi 15-19, 22 [also Sternberg 347-351]
• 24 June - Bonus lecture?
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
• Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry, Vol. I and II, John Wiley & Sons, Inc., New York, 1996
• Ivan Struchiner, The Classifying Lie Algebroid of a Geometric Structure, PhD thesis, 2009, see here
• Shoshichi Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin, 1995
• 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%).