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Topics in representation theory and Riemannian geometry

Giovanni Russo

TL;DR

These notes explore how Lie groups and their representations shape Riemannian geometry, emphasizing compact connected groups and the decomposition of geometric tensors into irreducible modules. The text develops a representation-theoretic framework for curvature and torsion, leading to Berger’s holonomy classification and the construction of exceptional holonomy metrics via $\mathrm{G}_2$ and $\mathrm{Spin}(7)$. It interleaves algebraic topics (weights, roots, Weyl groups, Clebsch–Gordan) with geometric structures (principal bundles, $G$-structures, holonomy) to connect differential geometry with physics applications in gravity and string theory. Concrete treatments of $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ representations, invariant theory, and the Gray–Hervella/nearby geometries illustrate the broad reach of representation theory in geometry. The notes are intended as foundational material for researchers entering the intersection of Lie theory and differential geometry.

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.

Topics in representation theory and Riemannian geometry

TL;DR

These notes explore how Lie groups and their representations shape Riemannian geometry, emphasizing compact connected groups and the decomposition of geometric tensors into irreducible modules. The text develops a representation-theoretic framework for curvature and torsion, leading to Berger’s holonomy classification and the construction of exceptional holonomy metrics via and . It interleaves algebraic topics (weights, roots, Weyl groups, Clebsch–Gordan) with geometric structures (principal bundles, -structures, holonomy) to connect differential geometry with physics applications in gravity and string theory. Concrete treatments of and representations, invariant theory, and the Gray–Hervella/nearby geometries illustrate the broad reach of representation theory in geometry. The notes are intended as foundational material for researchers entering the intersection of Lie theory and differential geometry.

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.

Paper Structure

This paper contains 30 sections, 63 theorems, 291 equations, 4 figures.

Table of Contents

  1. Representation theory
  2. Lie groups and representations
  3. Irreducible representations of SU(2) and SO(3)
  4. The Clebsch--Gordan formula
  5. Real and quaternionic representations
  6. Invariant theory
  7. Linear algebra
  8. Orthogonal groups
  9. Unitary groups
  10. On G2 and Spin7
  11. Fibre bundles
  12. Principal and associated bundles
  13. Connections and holonomy
  14. On vector bundles
  15. On G-structures
  16. ...and 15 more sections

Key Result

Theorem 1.1

Any compact connected Lie group can be realised as a linear connected reductive group.

Figures (4)

  • Figure 1: Roots of $\mathrm{SU}(2)$ (left) and $\mathrm{SU}(3)$ (right).
  • Figure 2: Dominant weights of $\mathrm{SU}(2)$ (left) and $\mathrm{SU}(3)$ (right). The red dots correspond to dominant roots. The shaded green region is the fundamental dual Weyl chamber for the positive simple roots.
  • Figure 3: Dominant weights of $\mathrm{SU}(2)$ (left) and $\mathrm{SU}(3)$ (right). The blue dots correspond to the fundamental weights.
  • Figure 4: Roots and weights of $\mathrm G_2$. The red roots correspond to a choice of positive simple roots. The black dots correspond to some dominant weights, the blue ones are the fundamental weights. The shaded green region is the fundamental dual Weyl chamber for the positive simple roots chosen.

Theorems & Definitions (289)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.1
  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Example 1.5
  • ...and 279 more