Riemannian Geometry Unit 10 Review: Holonomy Groups and Symmetric Spaces

Key Concepts and Definitions

• Riemannian manifold: A smooth manifold equipped with a Riemannian metric, which is a positive definite inner product on each tangent space
• Holonomy group: The group of linear transformations of the tangent space obtained by parallel transport around loops based at a point
• Parallel transport: A way to transport vectors along curves on a manifold while preserving the Riemannian metric and the connection
  • Allows for the comparison of vectors at different points on the manifold
• Levi-Civita connection: The unique torsion-free metric connection on a Riemannian manifold
• Symmetric space: A Riemannian manifold with a symmetry at each point, i.e., an involutive isometry that fixes the point and reverses geodesics through it
• Lie group: A smooth manifold that is also a group, where the group operations (multiplication and inversion) are smooth maps
• Lie algebra: The tangent space at the identity element of a Lie group, equipped with a bilinear operation called the Lie bracket

Historical Context and Development

• Early work on holonomy groups traces back to Élie Cartan's research on symmetric spaces in the 1920s and 1930s
• Cartan introduced the notion of a symmetric space and studied their classification
• Further developments in the theory of holonomy groups were made by Marcel Berger in the 1950s
  • Berger classified the possible holonomy groups of irreducible Riemannian manifolds
• Significant contributions to the study of symmetric spaces were made by Sigurdur Helgason in the 1960s and 1970s
• The Ambrose-Singer theorem, proved by Warren Ambrose and Isadore Singer in 1953, relates the holonomy group to the curvature of the manifold
• The concept of holonomy has been generalized to other geometric structures, such as complex manifolds and manifolds with special holonomy (Calabi-Yau manifolds, $G_2$ manifolds)

Holonomy Groups: Fundamentals

• The holonomy group at a point $p$ on a Riemannian manifold $M$ is denoted by $Hol_p(M)$
• $Hol_p(M)$ is a subgroup of the orthogonal group $O(T_pM)$, where $T_pM$ is the tangent space at $p$
• The restricted holonomy group, denoted by $Hol_p^0(M)$, is the connected component of the identity in $Hol_p(M)$
  • $Hol_p^0(M)$ is a Lie subgroup of $SO(T_pM)$, the special orthogonal group
• The holonomy group is independent of the base point $p$ up to conjugation in $O(T_pM)$
• The holonomy group measures the extent to which parallel transport around loops fails to preserve the geometry of the manifold
• The Ambrose-Singer theorem states that the Lie algebra of the holonomy group is generated by the curvature tensors at all points of the manifold

Classification of Holonomy Groups

• Berger's classification theorem lists the possible connected holonomy groups of irreducible, non-symmetric Riemannian manifolds
• The list includes:
  • $SO(n)$: Generic Riemannian manifolds
  • $U(n)$: Kähler manifolds
  • $SU(n)$: Calabi-Yau manifolds
  • $Sp(n)$: Hyperkähler manifolds
  • $Sp(n) \cdot Sp(1)$: Quaternion-Kähler manifolds
  • $G_2$: $G_2$ manifolds (7-dimensional)
  • $Spin(7)$: $Spin(7)$ manifolds (8-dimensional)
• Each of these holonomy groups corresponds to a specific type of geometric structure on the manifold
  • For example, Kähler manifolds have a complex structure compatible with the Riemannian metric
• Manifolds with special holonomy (Calabi-Yau, $G_2$, $Spin(7)$) have applications in string theory and M-theory

Symmetric Spaces: Introduction

• A Riemannian manifold $M$ is a symmetric space if for each point $p \in M$, there exists an involutive isometry $s_p$ (called a symmetry at $p$) such that:
  • $s_p(p) = p$
  • $d(s_p)_p = -Id_{T_pM}$ (the differential of $s_p$ at $p$ is minus the identity on the tangent space)
• The symmetry $s_p$ reverses geodesics through $p$, i.e., if $\gamma(t)$ is a geodesic with $\gamma(0) = p$, then $s_p(\gamma(t)) = \gamma(-t)$
• Examples of symmetric spaces include:
  • Euclidean spaces
  • Spheres
  • Hyperbolic spaces
  • Grassmann manifolds
• Symmetric spaces have a rich algebraic structure and are closely related to Lie groups and Lie algebras
• The isometry group of a symmetric space acts transitively on the manifold, making it a homogeneous space

Types of Symmetric Spaces

• Symmetric spaces can be classified into three main types:
  • Euclidean type: The isometry group contains translations, and the space is flat (zero curvature)
  • Compact type: The isometry group is compact, and the space has positive sectional curvature
  • Non-compact type: The isometry group is non-compact, and the space has negative sectional curvature
• Symmetric spaces of compact and non-compact type are dual to each other
  • They share the same local geometry but have different global properties
• Irreducible symmetric spaces are those that cannot be decomposed into a product of smaller symmetric spaces
• The classification of irreducible symmetric spaces is related to the classification of simple Lie algebras
  • Each simple Lie algebra corresponds to a unique compact type and a unique non-compact type symmetric space

Connections to Lie Groups and Algebras

• Every symmetric space $M$ can be written as a quotient of Lie groups, $M = G/K$, where:
  • $G$ is the identity component of the isometry group of $M$
  • $K$ is the stabilizer subgroup of a point in $M$
• The Lie algebra $\mathfrak{g}$ of $G$ admits a Cartan decomposition, $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$, where:
  • $\mathfrak{k}$ is the Lie algebra of $K$
  • $\mathfrak{p}$ is a complementary subspace
  • $[\mathfrak{k}, \mathfrak{k}] \subseteq \mathfrak{k}$, $[\mathfrak{k}, \mathfrak{p}] \subseteq \mathfrak{p}$, and $[\mathfrak{p}, \mathfrak{p}] \subseteq \mathfrak{k}$
• The Cartan decomposition satisfies the following properties:
  • $\mathfrak{p}$ can be identified with the tangent space of $M$ at the base point
  • The Riemannian metric on $M$ is induced by the Killing form on $\mathfrak{g}$ restricted to $\mathfrak{p}$
• The curvature tensor of a symmetric space can be expressed in terms of the Lie bracket of $\mathfrak{g}$

Applications in Physics and Geometry

• Symmetric spaces appear in various areas of physics and geometry:
  • General relativity: Spacetimes with a high degree of symmetry (maximally symmetric spaces) are modeled by symmetric spaces
    • Examples include de Sitter space and anti-de Sitter space
  • Supergravity and string theory: Symmetric spaces are used to construct supergravity theories and to describe the moduli spaces of string compactifications
  • Harmonic analysis: Symmetric spaces provide a natural setting for the study of harmonic analysis and representation theory of Lie groups
  • Integrable systems: Many integrable systems, such as the Toda lattice and the Calogero-Moser system, are related to symmetric spaces
• The geometry of symmetric spaces has applications in:
  • Optimization and machine learning: Symmetric spaces (Grassmann manifolds, Stiefel manifolds) are used in problems involving subspace tracking, principal component analysis, and matrix completion
  • Computer vision: Symmetric spaces arise in the study of shape spaces and the analysis of image data
  • Coding theory: Symmetric spaces, particularly the Grassmann manifolds, are used in the construction of error-correcting codes (space-time codes, network codes)