6.5 Symmetric spaces

Symmetric spaces are a special class of Riemannian manifolds with high symmetry. They generalize Euclidean spaces and spheres, exhibiting large groups of isometries that preserve their geometry. This makes them important objects of study in differential geometry.

Symmetric spaces are closely related to Lie groups and can be classified into compact, noncompact, and Euclidean types. Their curvature properties play a crucial role in determining their geometry and topology, distinguishing them from general Riemannian manifolds.

Definition of symmetric spaces

• Symmetric spaces are a special class of Riemannian manifolds that possess a high degree of symmetry, making them important objects of study in differential geometry
• The concept of symmetric spaces generalizes the notion of Euclidean spaces and spheres, which exhibit a large group of isometries that preserve the geometry of the space

Riemannian manifolds with symmetry

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• A Riemannian manifold is a smooth manifold equipped with a Riemannian metric, which is a positive definite inner product on each tangent space
• Symmetric spaces are Riemannian manifolds with an additional symmetry property: for each point $p$ in the manifold, there exists an isometry (a distance-preserving map) that fixes $p$ and reverses geodesics through $p$
• This symmetry condition implies that the geometry of the manifold is the same in all directions around each point

Geodesic symmetries

• In a symmetric space, for each point $p$ and each geodesic $\gamma$ passing through $p$, there exists an isometry $s_p$ called the geodesic symmetry at $p$ that fixes $p$ and reverses the direction of $\gamma$
• Geodesic symmetries are a crucial tool in the study of symmetric spaces, as they relate the local geometry around a point to the global geometry of the space
• The existence of geodesic symmetries at every point is a strong condition that distinguishes symmetric spaces from general Riemannian manifolds

Involutive isometries

• An involutive isometry is an isometry $\sigma$ such that $\sigma^2 = \text{id}$, meaning that applying the isometry twice yields the identity map
• In a symmetric space, the geodesic symmetries at each point are involutive isometries
• The involutive property of geodesic symmetries is essential in the classification and structure theory of symmetric spaces, as it relates to the algebraic properties of the isometry group

Lie groups and symmetric spaces

• Lie groups are smooth manifolds that are also groups, where the group operations (multiplication and inversion) are smooth maps
• Symmetric spaces are closely related to Lie groups, as the isometry group of a symmetric space is a Lie group, and many symmetric spaces can be realized as homogeneous spaces of Lie groups

Lie groups as symmetric spaces

• Every Lie group $G$ can be viewed as a symmetric space by considering the map $\sigma_g: G \to G$ given by $\sigma_g(h) = gh^{-1}g$ for $g, h \in G$
• The map $\sigma_g$ is an involutive isometry of $G$ that fixes the point $g$, making $G$ a symmetric space with respect to any left-invariant Riemannian metric
• This construction provides a rich source of examples of symmetric spaces and highlights the interplay between the group structure and the geometric structure

Lie algebra of isometries

• The Lie algebra $\mathfrak{g}$ of a Lie group $G$ is the tangent space at the identity element, equipped with a Lie bracket operation
• In the context of symmetric spaces, the Lie algebra of the isometry group plays a crucial role in understanding the infinitesimal symmetries of the space
• The Lie algebra of isometries decomposes into a direct sum of the Lie algebra of the isotropy subgroup (the subgroup fixing a point) and its orthogonal complement, which can be identified with the tangent space at the fixed point

Exponential map and symmetries

• The exponential map $\exp: \mathfrak{g} \to G$ is a smooth map that relates the Lie algebra to the Lie group
• In a symmetric space, the exponential map can be used to construct geodesics and to relate the infinitesimal symmetries (Lie algebra elements) to global symmetries (isometries)
• The geodesic symmetries in a symmetric space can be expressed in terms of the exponential map and the Lie algebra of the isometry group, providing a powerful tool for studying the geometry of the space

Classification of symmetric spaces

• Symmetric spaces can be classified into three main types based on their curvature and completeness properties: compact type, noncompact type, and Euclidean type
• The classification of symmetric spaces is a major result in differential geometry and has important implications for the study of these spaces and their applications

Compact type

• A symmetric space is of compact type if it is complete and has non-negative sectional curvature
• Examples of compact type symmetric spaces include spheres, projective spaces, and Grassmannians
• Compact type symmetric spaces have a rich geometry and topology, and they are related to important objects in algebraic topology and representation theory, such as homology and cohomology groups and representations of compact Lie groups

Noncompact type

• A symmetric space is of noncompact type if it is complete, simply connected, and has non-positive sectional curvature
• Examples of noncompact type symmetric spaces include hyperbolic spaces and symmetric spaces associated with semisimple Lie groups
• Noncompact type symmetric spaces have important applications in number theory, representation theory, and the study of discrete subgroups of Lie groups, such as lattices and arithmetic groups

Euclidean type

• A symmetric space is of Euclidean type if it is complete and has zero sectional curvature
• Euclidean spaces and flat tori are examples of Euclidean type symmetric spaces
• Euclidean type symmetric spaces are the simplest type of symmetric spaces and serve as a building block for the study of more general symmetric spaces through the process of taking products and quotients

Curvature of symmetric spaces

• The curvature of a symmetric space plays a crucial role in determining its geometry and topology
• Symmetric spaces have special curvature properties that distinguish them from general Riemannian manifolds and make them more tractable objects of study

Constant curvature spaces

• A space has constant curvature if its sectional curvature is the same at every point and for every plane section
• Examples of constant curvature spaces include Euclidean spaces (zero curvature), spheres (positive curvature), and hyperbolic spaces (negative curvature)
• Symmetric spaces of compact type and noncompact type generalize the notion of constant curvature spaces, as they have non-negative and non-positive sectional curvature, respectively

Sectional curvature

• Sectional curvature is a measure of the curvature of a Riemannian manifold that assigns a real number to each plane section (a two-dimensional subspace of the tangent space) at each point
• In a symmetric space, the sectional curvature determines the local geometry and the behavior of geodesics
• The sign of the sectional curvature (positive, negative, or zero) is an important invariant of a symmetric space and is used in the classification of these spaces

Ricci curvature

• Ricci curvature is another measure of the curvature of a Riemannian manifold, obtained by taking the trace of the sectional curvature tensor
• In a symmetric space, the Ricci curvature is closely related to the mean curvature of geodesic spheres and the volume growth of balls
• The Ricci curvature of a symmetric space is used in the study of harmonic functions, heat kernels, and the spectrum of the Laplace-Beltrami operator

Examples of symmetric spaces

• There are many important examples of symmetric spaces that arise in various branches of mathematics and physics
• These examples illustrate the richness and diversity of symmetric spaces and their applications

Spheres and hyperbolic spaces

• Spheres $S^n$ and hyperbolic spaces $H^n$ are the most basic examples of symmetric spaces of compact and noncompact type, respectively
• These spaces have constant positive (spheres) or negative (hyperbolic) sectional curvature and are the model spaces for the study of geometry in positive and negative curvature
• Spheres and hyperbolic spaces play a central role in topology, geometry, and the study of discrete groups, such as Kleinian groups and Fuchsian groups

Grassmannians and flag manifolds

• The Grassmannian $Gr(k, n)$ is the space of $k$-dimensional linear subspaces of $\mathbb{R}^n$ (or $\mathbb{C}^n$)
• Flag manifolds are generalizations of Grassmannians, where instead of a single subspace, one considers a sequence of nested subspaces (a flag)
• Grassmannians and flag manifolds are important examples of symmetric spaces of compact type and have applications in algebraic geometry, representation theory, and the study of vector bundles and characteristic classes

Symmetric spaces of exceptional type

• In addition to the classical examples of symmetric spaces (such as spheres, Grassmannians, and symmetric spaces associated with classical Lie groups), there are also exceptional symmetric spaces that do not fit into these families
• Examples of exceptional symmetric spaces include the Cayley projective plane (associated with the exceptional Lie group $\text{E}_6$) and the octonionic projective plane (associated with the exceptional Lie group $\text{E}_8$)
• These exceptional symmetric spaces have unique geometric properties and are related to the study of exceptional Lie groups, which play a significant role in the classification of simple Lie algebras and the study of symmetries in high dimensions

Geometry of symmetric spaces

• The geometry of symmetric spaces is a rich and well-developed area of differential geometry, with many deep connections to other areas of mathematics
• The study of geodesics, Jacobi fields, and the cut locus provides insight into the global structure of symmetric spaces and their singularities

Geodesics and parallel transport

• Geodesics are the generalization of straight lines to curved spaces and are the shortest paths between points in a symmetric space
• In a symmetric space, geodesics are intimately related to the symmetries of the space, as the geodesic symmetries at each point map geodesics to geodesics
• Parallel transport is a way of moving vectors along curves (such as geodesics) while preserving their angle and length, and it plays a crucial role in the study of the curvature and holonomy of symmetric spaces

Jacobi fields and conjugate points

• Jacobi fields are vector fields along a geodesic that describe the behavior of nearby geodesics
• In a symmetric space, Jacobi fields can be used to study the stability of geodesics and the existence of conjugate points (points where nearby geodesics intersect)
• The study of Jacobi fields and conjugate points is important in understanding the global geometry of symmetric spaces and their singularities, such as the cut locus

Cut locus and injectivity radius

• The cut locus of a point $p$ in a symmetric space is the set of points where geodesics starting from $p$ cease to be minimizing (i.e., they are no longer the shortest paths)
• The injectivity radius of a symmetric space is the largest radius for which the exponential map at each point is a diffeomorphism (a smooth bijection with a smooth inverse)
• The cut locus and injectivity radius provide important information about the global geometry of a symmetric space and are related to the study of geodesic convexity, the behavior of balls and spheres, and the singularities of the space

Harmonic analysis on symmetric spaces

• Harmonic analysis is the study of functions and their Fourier transforms on symmetric spaces
• The tools of harmonic analysis, such as invariant differential operators and spherical functions, provide a powerful framework for understanding the geometry and representation theory of symmetric spaces

Invariant differential operators

• Invariant differential operators are differential operators on a symmetric space that commute with the action of the isometry group
• Examples of invariant differential operators include the Laplace-Beltrami operator and the Dirac operator
• The study of invariant differential operators is central to the harmonic analysis on symmetric spaces, as they provide a way to construct and analyze eigenfunctions and to study the spectrum of the space

Spherical functions and Fourier transform

• Spherical functions are special functions on a symmetric space that are eigenfunctions of invariant differential operators and are invariant under the action of the isotropy subgroup at a point
• The spherical Fourier transform is a generalization of the classical Fourier transform to symmetric spaces, using spherical functions as the basis for the transform
• The spherical Fourier transform provides a way to analyze functions on symmetric spaces in terms of their frequency components and is a powerful tool in harmonic analysis and representation theory

Representations of isometry groups

• The isometry group of a symmetric space is a Lie group that acts transitively on the space
• Representations of the isometry group (i.e., homomorphisms from the group to the group of linear transformations on a vector space) play a crucial role in the harmonic analysis on symmetric spaces
• The study of representations of isometry groups is related to the construction of spherical functions, the decomposition of $L^2$ spaces on symmetric spaces, and the study of the spectrum of invariant differential operators

Applications of symmetric spaces

• Symmetric spaces have numerous applications in various areas of mathematics and physics, showcasing their importance and versatility
• These applications range from the study of automorphic forms and representation theory to integrable systems and mathematical physics

Representation theory and automorphic forms

• Symmetric spaces are closely related to the representation theory of Lie groups and the study of automorphic forms
• Automorphic forms are functions on symmetric spaces that satisfy certain transformation properties with respect to discrete subgroups of the isometry group (such as arithmetic groups)
• The study of automorphic forms is a central topic in number theory and has important applications in the theory of $L$-functions, the Langlands program, and the study of modular forms and Shimura varieties

Integrable systems and soliton equations

• Symmetric spaces have important applications in the theory of integrable systems and soliton equations
• Many integrable systems, such as the Korteweg-de Vries (KdV) equation and the nonlinear Schrödinger equation, can be viewed as infinite-dimensional symmetric spaces
• The geometry of symmetric spaces provides a framework for understanding the integrability and soliton solutions of these equations, as well as their symmetries and conservation laws

Symmetric spaces in physics

• Symmetric spaces arise naturally in various areas of mathematical physics, such as general relativity, string theory, and quantum field theory
• In general relativity, symmetric spaces are used to model homogeneous and isotropic spacetimes, such as the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which describes the large-scale structure of the universe
• In string theory and supergravity, symmetric spaces appear as the target spaces for nonlinear sigma models and as the moduli spaces of supersymmetric solutions
• The geometry and topology of symmetric spaces play a crucial role in understanding the symmetries and dualities of these physical theories, as well as their quantum corrections and nonperturbative effects