Holonomy groups measure how parallel transport changes vectors along closed loops in a manifold. They provide crucial insights into a manifold's geometry and topology, helping us understand its structure and properties.

In this section, we'll explore the definition and classification of holonomy groups. We'll look at reducible and irreducible holonomy, their significance, and how they relate to curvature through the Ambrose-Singer theorem.

Holonomy Groups and Parallel Transport

Defining Holonomy and Parallel Transport

• Holonomy group measures the failure of parallel transport to preserve vectors along closed loops in a manifold
• Parallel transport moves vectors along curves while maintaining their angle and length
• Restricted holonomy group consists of transformations arising from parallel transport along contractible loops
• Full holonomy group includes transformations from all loops, contractible or not

Mathematical Representation of Holonomy

• Holonomy group denoted as $Hol(M,g)$ for a Riemannian manifold $(M,g)$
• Parallel transport along a curve $\gamma$ represented by a linear map $P_\gamma: T_pM \rightarrow T_qM$
• Restricted holonomy group written as $Hol_0(M,g)$, a connected Lie subgroup of $O(n)$
• Full holonomy group may have multiple connected components, with $Hol_0(M,g)$ as the identity component

Properties and Applications of Holonomy

• Holonomy groups provide information about the global geometry and topology of manifolds
• Parallel transport preserves inner products, resulting in holonomy groups being subgroups of orthogonal groups
• Restricted holonomy group used to study local geometric properties (curvature)
• Full holonomy group reveals global topological information (fundamental group)

Types of Holonomy

Reducible Holonomy

• Reducible holonomy occurs when the holonomy group preserves a proper non-trivial subspace of the tangent space
• Manifolds with reducible holonomy can be decomposed into simpler geometric components
• De Rham decomposition theorem states that a simply connected complete Riemannian manifold with reducible holonomy splits as a Riemannian product
• Examples include product manifolds (torus $T^2 = S^1 \times S^1$) and Kähler manifolds with reducible holonomy

Irreducible Holonomy

• Irreducible holonomy means the holonomy group acts irreducibly on the tangent space
• Manifolds with irreducible holonomy often have rich geometric structures
• Berger's classification theorem lists all possible irreducible holonomy groups for simply connected Riemannian manifolds
• Examples include special holonomy groups such as $G_2$ (7-dimensional manifolds) and $Spin(7)$ (8-dimensional manifolds)

Significance of Holonomy Classification

• Classification of holonomy groups provides a framework for understanding geometric structures
• Reducible holonomy indicates simpler geometric components or product structures
• Irreducible holonomy often corresponds to special geometric structures (Kähler, hyper-Kähler, $G_2$, etc.)
• Understanding holonomy aids in the study of Einstein manifolds and Ricci-flat metrics

Holonomy and Lie Algebra

Ambrose-Singer Theorem and Curvature

• Ambrose-Singer theorem relates holonomy groups to the curvature tensor of a Riemannian manifold
• States that the Lie algebra of the holonomy group equals the space spanned by curvature operators
• Provides a local characterization of holonomy in terms of curvature
• Curvature operator $R(X,Y): T_pM \rightarrow T_pM$ defined by $R(X,Y)Z = \nabla_X\nabla_YZ - \nabla_Y\nabla_XZ - \nabla_{[X,Y]}Z$

Lie Algebra of Holonomy

• Lie algebra of holonomy, denoted $\mathfrak{hol}(M,g)$, is the infinitesimal version of the holonomy group
• Dimension of $\mathfrak{hol}(M,g)$ provides information about the complexity of the manifold's geometry
• Holonomy principle states that parallel tensors correspond to fixed points of the holonomy representation
• Lie algebra of holonomy used to study killing vector fields and isometries of Riemannian manifolds

Applications in Differential Geometry

• Holonomy and its Lie algebra play crucial roles in the study of special geometric structures
• Kähler manifolds characterized by $U(n)$ holonomy, with Lie algebra $\mathfrak{u}(n)$
• Calabi-Yau manifolds have $SU(n)$ holonomy, important in string theory and algebraic geometry
• Holonomy Lie algebra used to construct parallel spinors and study Ricci-flat metrics