PhD in Mathematics

Topics on Lie groups, nilmanifolds and solvmanifolds

Topics on Lie groups, nilmanifolds and solvmanifolds

Academic year 2023/2024

Sommario del corso

• Course objectives
• Program
• Course delivery
• Learning assessment methods
• Suggested readings and bibliography
• Further information
• Teaching tools
• Course card

Course objectives

The aim of the course is to present the main results concerning the geometry of Lie groups and their quotients, in particular nilmanifolds and solvmanifolds.

Program

25 hours.

The topics covered will be the following. Review of elementary theory of Lie groups and Lie algebras. Lie's Theorem and Engel's Theorem. Cartan's criteria for solvability and semisimplicity. The exponential map on solvable and nilpotent Lie groups. Left-invariant and biinvariant metrics. Some results on the curvature. Lattices and uniform subgroups. Unimodularity. Existence of cocompact lattices, Mal'cev's criterion. Mostow structure theorem for solvmanifolds and its consequences. Cohomology, Hattori's and Nomizu's Theorem, minimal models. Relation with Kähler structures, Benson-Gordon's Theorem, formality and Hasegawa's Theorem.
Time permitting: complex structures on nilmanifolds and solvmanifolds. Basic facts and results on the canonical bundle.

The topics may be subjected to variations according to the interests of the audience.

In detail:

Lecture 1 (05/04): Review of elementary theory of Lie groups and Lie algebras. Basic definitions and properties. Lie groups-Lie algebras correspondence. Normal Lie subgroups and ideals. Exponential map and its properties. Characterization of connected abelian Lie groups as products of a torus and an Euclidean factor.

Lecture 2 (12/04): Solvability and nilpotency. Definitions and essential facts. Examples. Statement of Lie's and Engel's Theorems and first consequences.

Lecture 3 (15/04): Proofs of Lie's and Engel's Theorems. Semidirect products. Simplicity and semisimplicity. The exponential map on 1-connected nilpotent Lie groups is a diffeomorphism.

Lecture 4 (17/04): The exponential map on solvable Lie groups. Curvatures of left-invariant metrics: flatness (Hano), Ricci-flatness (Alekseevskii-Kimelfeld), Scalar curvature (Jensen).

Lecture 5 (19/04): Biinvariant metrics, their properties and the constraints they impose. The Killing form. Statement of Cartan's criteria. A connected Lie group admits a biinvariant metric if and only if it is isomorphic to a product of a compact Lie group and an Euclidean factor. A compact connected solvable Lie group is isomorphic to a torus.

Lecture 6 (22/04): Left and right Haar measures on Lie groups. Modular function. Unimodularity. Left-invariant measures on homogeneous spaces. Characterization of the existence of left-invariant measures on homogeneous spaces.

Lecture 7 (24/04): Proof of Weil's theorem on the characterization of the existence of left-invariant measures on homogeneous spaces. Finiteness of left-invariant measures. Lattices and uniform subgroups. Nilmanifolds. A nilmanifold is diffeomorphic to the quotient of a connected simply-connected nilpotent Lie group by a discrete subgroup. A discrete subgroup in a connected simply-connected nilpotent Lie group is a lattice if and only if it is uniform.

Lecture 8 (29/04): Given a discrete subgroup of a Lie group G there exists a unique closed connected Lie subgroup of G in which the discrete subgroup is uniform. Rigidity: an isomorphism of lattices in connected simply-connected nilpotent Lie groups admits a unique extension to an isomorphism of the groups. Mal'cev's criterion: A connected simply-connected nilpotent Lie group admits a lattice if and only if there exists a basis of the Lie algebra with respect to which the structure constants are rational.

Course delivery

Schedule:

Fri 5/04, 15.30-17.30, aula 1
Fri 12/04, 12.30-14.30, aula 5
Mon 15/04, 14.30-16.30, aula lodi
Wed 17/04, 14.30-16.30, aula lodi
Fri 19/04, 10.30-12.30, aula 1
Mon 22/04, 14.30-16.30, aula 2
Wed 24/04, 10.30-12.30, aula 3
Mon 29/04, 14.30-16.30, aula 2
Fri 3/05, 10.30-12.30, aula 1
Tue 7/05, 16.30-18.30, aula lodi
Fri 10/05, 10.30-12.30, aula 1
Tue 14/05, 16.30-18.30, aula 2
Fri 17/05, 10.30-12.30, aula 1

Learning assessment methods

Students in need to take the exam will be required to prepare a seminar on an argument related to those presented in the course.

The lecture notes of the course will be available at the link below.

Further information

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