preprint
Inserted: 22 apr 2025
Last Updated: 18 may 2026
Year: 2025
Accepted by Routledge and CRC Press, Taylor & Francis Group
Abstract:
These are notes for a Ph.D. course I held at SISSA, Trieste, in the Winter 2025. We review well-known topics in Riemannian geometry where Lie groups play a fundamental role. Part of the theory of compact connected Lie groups, their invariants, and representations is discussed, with particular emphasis on low dimensional examples. We go through a number of applications in Riemannian geometry, in particular the classification of Riemannian holonomy groups, and the first construction of exceptional holonomy metrics. Some more recent advances in the field of Riemannian geometry with symmetries are mentioned.