10.2 Berger's classification of Riemannian holonomy
3 min read•august 9, 2024
Berger's classification of Riemannian holonomy groups is a game-changer in geometry. It narrows down the possible holonomy groups for simply connected, irreducible Riemannian manifolds, giving us a roadmap for understanding their structure.
This classification is crucial for grasping the link between a manifold's geometry and its holonomy group. It's not just theoretical - it has real-world applications in physics, especially in string theory and M-theory.
Berger's Theorem and Holonomy Types
Understanding Berger's Theorem
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Berger's theorem guides the search for manifolds with specific geometric properties
Helps in understanding the relationship between holonomy and manifold geometry
Plays a crucial role in theoretical physics, particularly in string theory and M-theory
Facilitates the construction of manifolds with desired holonomy groups
Classical Holonomy Groups
SO(n) and Its Properties
SO(n) represents the special orthogonal group in n dimensions
Consists of all n x n orthogonal matrices with determinant 1
Generic holonomy group for orientable Riemannian manifolds
Preserves orientation and metric on the tangent space
Dimension of SO(n) equals 2n(n−1)
U(n) and Complex Structures
U(n) denotes the unitary group in n complex dimensions
Holonomy group for Kähler manifolds
Preserves both the metric and complex structure
Dimension of U(n) equals n2
Examples include complex projective spaces and complex tori
SU(n) and Calabi-Yau Manifolds
SU(n) represents the special unitary group in n complex dimensions
Holonomy group for Calabi-Yau manifolds
Preserves metric, complex structure, and a holomorphic volume form
Dimension of SU(n) equals n2−1
Plays a crucial role in string theory and mirror symmetry
Sp(n) and Hyperkähler Geometry
Sp(n) denotes the symplectic group in n quaternionic dimensions
Holonomy group for hyperkähler manifolds
Preserves metric and three complex structures (I, J, K)
Dimension of Sp(n) equals 2n2+n
Examples include K3 surfaces and hyper-Kähler quotients
Exceptional Holonomy Groups
Sp(n)Sp(1) and Quaternionic Kähler Manifolds
Sp(n)Sp(1) results from the product of symplectic groups
Holonomy group for quaternionic Kähler manifolds
Preserves quaternionic structure but not individual complex structures
Dimension of Sp(n)Sp(1) equals 2n2+4n+3
Examples include quaternionic projective spaces and Wolf spaces
G2 Holonomy and 7-Dimensional Manifolds
G2 represents the smallest exceptional Lie group
Holonomy group for 7-dimensional G2 manifolds
Preserves a special 3-form and its Hodge dual 4-form
Dimension of G2 equals 14
G2 manifolds have applications in M-theory compactifications
Spin(7) and 8-Dimensional Geometry
Spin(7) denotes the double cover of SO(7)
Holonomy group for 8-dimensional Spin(7) manifolds
Preserves a self-dual 4-form called the Cayley form
Dimension of Spin(7) equals 21
Spin(7) manifolds appear in string theory and M-theory constructions
Key Terms to Review (19)
Affine connection: An affine connection is a mathematical structure that allows for the comparison of vectors in tangent spaces of a manifold, enabling the definition of parallel transport and covariant derivatives. It provides a way to differentiate vector fields along curves on the manifold and plays a crucial role in understanding the geometric properties of Riemannian manifolds.
Berger's Theorem: Berger's Theorem states that a Riemannian manifold can be characterized by its holonomy group and its sectional curvature. Specifically, it establishes a connection between the curvature properties of a manifold and the types of parallel transport that can occur within it. This theorem plays a crucial role in understanding how different geometric structures are tied to their intrinsic curvatures and holonomy groups.
Calabi-Yau manifold: A Calabi-Yau manifold is a special type of compact, complex Kähler manifold that has a vanishing first Chern class and admits a Ricci-flat metric. These manifolds are significant in both mathematics and theoretical physics, particularly in string theory, as they provide the necessary geometric structures for compactifying extra dimensions.
Curvature: Curvature is a measure of how much a geometric object deviates from being flat or straight. In the context of Riemannian geometry, it describes how the geometry of a manifold bends and can be quantified through different types such as sectional curvature, Ricci curvature, and scalar curvature, affecting the behavior of geodesics and the manifold's overall structure.
G2: g2 is a specific type of Riemannian holonomy group associated with a seven-dimensional manifold that admits a special geometric structure. This holonomy group is particularly significant as it reflects the symmetries and geometric properties of the underlying manifold, specifically indicating the presence of a G2 structure, which has implications for both topology and differential geometry.
Hyperkähler manifold: A hyperkähler manifold is a special type of Riemannian manifold that possesses a metric with holonomy group contained in $SU(2) \times SU(2) \times SU(2)$ and admits three compatible symplectic structures. This unique structure allows for rich geometric properties, including the existence of a nontrivial holomorphic structure and the ability to support quaternionic structures. Hyperkähler manifolds are important in the study of both mathematics and theoretical physics, particularly in string theory and the theory of special holonomy.
Irreducible Holonomy: Irreducible holonomy refers to the situation in which the holonomy group of a Riemannian manifold cannot be reduced to a smaller subgroup of the general linear group acting on the tangent space. This concept is crucial in understanding how the geometry of the manifold relates to its curvature and structure. In essence, if the holonomy is irreducible, it implies that the manifold cannot be decomposed into simpler geometric pieces, leading to more complex and interesting geometrical properties.
Kähler manifold: A Kähler manifold is a special type of complex manifold equipped with a Riemannian metric that is both Hermitian and symplectic, meaning it has a compatible complex structure that allows for the integration of geometric and complex analysis. This unique combination allows Kähler manifolds to have rich geometric properties, making them significant in both Riemannian geometry and complex geometry. The interplay between the metric, the symplectic structure, and the complex structure is key to understanding their holonomy properties and behavior under deformation.
Levi-Civita connection: The Levi-Civita connection is a unique, compatible affine connection on a Riemannian manifold that preserves the metric and is torsion-free. This connection allows for the definition of parallel transport, covariant derivatives of tensor fields, and plays a crucial role in understanding the geometric structure of Riemannian spaces.
Non-positive curvature: Non-positive curvature refers to a geometric property of a space where, intuitively, triangles formed within the space have angles that sum to less than or equal to 180 degrees. This concept is crucial because it implies that geodesics can diverge, indicating the presence of flat or saddle-like geometry. This property connects to various significant results in geometry and topology, influencing how manifolds behave under certain curvature constraints and shaping the classification of their structures.
Parallel Transport: Parallel transport is a way of moving vectors along a curve in a manifold such that they remain parallel according to the connection. This process is crucial for understanding how vectors change as they move through curved spaces, linking concepts of geodesics, affine connections, and holonomy.
Positive Curvature: Positive curvature is a property of a geometric space where, intuitively, the surface bends outward, like the surface of a sphere. In such spaces, geodesics tend to converge, and triangles formed within them have angles that sum to more than 180 degrees. This concept is crucial for understanding various phenomena in Riemannian geometry, affecting properties of geodesics, curvature behavior, and geometric structures.
Quaternionic kähler manifold: A quaternionic kähler manifold is a special type of Riemannian manifold equipped with a metric that is not only Kähler but also allows for the existence of a hypercomplex structure. This means it has a holonomy group that is a subgroup of the unitary group, specifically related to quaternionic structures. These manifolds play a significant role in the study of special holonomy, which is important in various areas of mathematics and theoretical physics.
Reducible holonomy: Reducible holonomy refers to a situation where the holonomy group of a Riemannian manifold can be represented by a subgroup of the general linear group that is reducible, meaning it can be decomposed into smaller, invariant subspaces. This concept is essential in understanding how curvature and geometric properties can influence the structure of the manifold, linking it to the classification of holonomy groups and their properties.
So(n): The notation so(n) represents the Lie algebra of the special orthogonal group SO(n), which consists of all n x n skew-symmetric matrices. This algebra plays a crucial role in understanding the properties and structures of rotations in n-dimensional Euclidean space and is foundational in both the study of Lie groups and in analyzing geometric structures influenced by Riemannian holonomy.
Sp(n): The special unitary group sp(n) is a mathematical concept that describes symmetries in an even-dimensional space, specifically the symmetries of a symplectic vector space. It consists of linear transformations that preserve a symplectic form, which is crucial in areas like Riemannian geometry and physics, particularly in Hamiltonian mechanics. Understanding sp(n) is essential for classifying holonomy groups and analyzing the geometric structures associated with Riemannian manifolds.
Spin(7): Spin(7) is a special type of symmetry group that arises in the study of Riemannian geometry, specifically relating to the holonomy groups of 8-dimensional Riemannian manifolds. It is part of a larger classification system for holonomy groups that can be understood through their actions on various geometric structures, such as metrics and connections. Understanding Spin(7) helps in identifying specific geometrical properties of manifolds and their potential applications in theoretical physics.
Su(n): The Lie algebra su(n) is the algebra of skew-Hermitian matrices of trace zero, playing a crucial role in the study of Riemannian holonomy. It serves as the algebraic structure that corresponds to the special unitary group SU(n), which consists of n x n unitary matrices with determinant one. This concept is significant in understanding the holonomy groups of Riemannian manifolds, especially in classifications and applications related to curvature and geometric structures.
U(n): u(n) refers to the unitary group of degree n, which consists of all n x n unitary matrices. These matrices are important in various areas, particularly in the study of Riemannian geometry and holonomy groups, as they preserve the inner product in complex vector spaces and play a key role in understanding symmetries in geometric contexts.