Generalities. The goal of this Ph.D. course is to present some aspects of classical results in Riemannian geometry relying on the representation theory of some orthogonal group. Most of these results build on work of Weyl [12] on the representation theory and the invariant theory of classical Lie groups. Since the general theory is too wide for such a course, some emphasis will be given to concrete examples, particularly in low dimension. Explicit calculations will be carried out where necessary to illustrate relevant techniques.
The list of topics to be covered includes the following:
- Adams [1], Bröcker–Dieck [4], Fulton–Harris [7], Salamon [11]: quick introduction to Riemannian geometry, special geometric structures, basics of representation theory.
- Salamon [11]: Kahler manifolds, decomposition of exterior algebra under the action of the unitary group, real structures.
- Berger–Gauduchon–Mazet [2]: the Riemannian curvature tensor, Ricci curvature, scalar curvature.
- Besse [3], Tricerri–Vanhecke [9]: decomposition of the Ricci tensor under the action of orthogonal and unitary groups, applications.
- Gray–Hervella [8], Fernnádez–Gray [6]: decomposition of the intrinsic torsion of almost Hermitian manifolds and manifolds with structure group G2.
- Tricerri–Vanhecke [10]: classification of homogeneous structures on smooth manifolds.
- Bryant [5]: Riemannian manifolds with exceptional holonomy.
According to the interests of the audience we will reserve more time to selected topics. If time permits, we will expand on the above topics by adding extra material.
Prerequisites. General knowledge of linear algebra, matrix groups, Lie groups and algebras, and group actions, is essential. Some basic knowledge of differential and Riemannian geometry is required (smooth manifolds, fibre bundles, connections, curvature, etc.), but essential concepts will be recalled to make lectures as self-contained as possible. Familiarity with quaternions and octonions would also be ideal.
Time. The course will last 20 hours and will be run in the Fall 2024. We propose to have two lectures per week of 2 hours each.
Materials. Here below there is a list of useful references, far from exhaustive. Notes on the material covered will be provided.
References.
[1] J. F. Adams, “Lectures on Lie Groups”. W. A. Benjamin, New York, 1969.
[2] M. Berger, P. Gauduchon, and E. Mazet, “Le Spectre d’une Vari´et´e Riemannienne”. Lecture Notes in Mathematics 194, Springer-Verlag, 1971.
[3] A. L. Besse, “Einstein manifolds”. Classics in Mathematics. Springer-Verlag, Berlin, 2008.
[4] T. Bröcker, T. Dieck, “Representations of Compact Lie Groups”. Graduate Texts in Mathematics 98, Springer, Berlin-Heidelberg-New York, 1985.
[5] R. L. Bryant, “Metrics with exceptional holonomy”. In: Annals of Mathematics 126, no. 3 (1987), pp. 525–576.
[6] M. Fernández, A. Gray, “Riemannian Manifolds with Structure Group G2”. In: Annali di Matematica Pura ed Applicata 132 (1982), pp. 19–45.
[7] W. Fulton, J. Harris, “Representation Theory: A First Course”. Graduate Texts in Mathematics, Springer New York, 2004.
[8] A. Gray, L. M. Hervella, “The Sixteen Classes of Almost Hermitian Manifolds and Their Linear Invariants”. In: Annali di Matematica Pura ed Applicata 123, no. 1 (1980), pp. 35–58.
[9] F. Tricerri, L. Vanhecke, “Curvature Tensors on Almost Hermitian Manifolds”. In: Transactions of the American Mathematical Society 267, n. 2 (1981), pp. 365–398.
[10] F. Tricerri, L. Vanhecke, “Homogeneous Structures on Riemannian Manifolds”. Cambridge University Press, 1983.
[11] S. M. Salamon, “Riemannian Geometry and Holonomy Groups”. Longman Scientific & Technical, 1989.
[12] H. Weyl, “The Classical Groups, their Invariants and Representations”. Princeton University Press, 1939.