9.3 Symmetric spaces and their classification

Symmetric spaces are special Riemannian manifolds with involutive isometries at each point. They include familiar spaces like spheres and hyperbolic spaces, and have applications in geometry, analysis, and physics.

The classification of symmetric spaces is tied to semisimple Lie algebras and their involutive automorphisms. They're categorized by rank and type, with curvature properties playing a key role in their structure and behavior.

Symmetric spaces: Definition and Applications

Definition and examples

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• A symmetric space is a Riemannian manifold M such that for each point p in M, there exists an involutive isometry sp that fixes p and reverses geodesics through p
• Symmetric spaces include Euclidean spaces, spheres, hyperbolic spaces, Grassmannians, and certain matrix groups (special orthogonal group SO(n), special unitary group SU(n))

Applications in geometry, analysis, and physics

• In geometry, symmetric spaces are used to study isometries, geodesics, and the structure of Riemannian manifolds
• Symmetric spaces have applications in analysis, including harmonic analysis, representation theory, and the study of differential operators on manifolds
• Physics applications of symmetric spaces include gauge theories, string theory, and integrable systems, where the symmetries and geometry of these spaces play a crucial role

Classifying symmetric spaces

Rank, type, and associated Lie algebras

• The rank of a symmetric space is the maximum dimension of a flat totally geodesic submanifold, which is connected to the structure of its associated Lie algebra
• Symmetric spaces are classified into compact type (non-negative sectional curvature), noncompact type (non-positive sectional curvature), and Euclidean type (zero sectional curvature) based on their curvature properties
• Compact type symmetric spaces are diffeomorphic to compact Lie groups modulo maximal tori, while noncompact type spaces are diffeomorphic to noncompact semisimple Lie groups modulo maximal compact subgroups

Relationship to semisimple Lie algebras and involutive automorphisms

• The classification of symmetric spaces is closely tied to the classification of semisimple Lie algebras and their involutive automorphisms
• The Killing form of the associated Lie algebra is essential in determining the type and structure of the symmetric space
• Involutive automorphisms of semisimple Lie algebras give rise to symmetric spaces through the construction of Lie triple systems and the exponential map

Curvature properties of symmetric spaces

Constant sectional curvature and its implications

• Symmetric spaces have constant sectional curvature, which can be positive, negative, or zero depending on the type of the space
• The Ricci curvature and scalar curvature of a symmetric space are fully determined by its sectional curvature and dimension
• The constant curvature properties affect the behavior of geodesics, conjugate points, and Jacobi fields on symmetric spaces

Spectral theory and harmonic analysis

• The Laplacian operator on a symmetric space has a rich spectral theory that is closely related to the curvature properties and the representation theory of the associated Lie group
• Harmonic analysis on symmetric spaces, including the study of spherical functions and the Helgason-Fourier transform, relies on the curvature properties and the underlying structure of the space
• The curvature properties also influence the heat kernel, Green's functions, and other fundamental solutions of differential equations on symmetric spaces

Fundamental theorems for symmetric spaces

Existence and uniqueness of geodesic symmetries

• The existence and uniqueness of geodesic symmetries at each point of a symmetric space are proven using the properties of the Levi-Civita connection and the exponential map
• The Cartan-Ambrose-Hicks theorem characterizes symmetric spaces as connected Riemannian manifolds where the geodesic symmetries are isometries, providing an alternative definition

Structure theorems and rigidity results

• The de Rham decomposition theorem states that every simply connected, complete Riemannian manifold is isometric to a product of irreducible Riemannian manifolds, which has implications for the structure of symmetric spaces
• The Mostow rigidity theorem asserts that the geometry of compact symmetric spaces of rank at least 2 is determined by their fundamental group, leading to consequences for the classification of locally symmetric spaces
• The Margulis superrigidity theorem connects the representation theory of lattices in semisimple Lie groups to the geometry of symmetric spaces, with applications in number theory and ergodic theory