Symmetric spaces are special Riemannian manifolds with involutive isometries at every point. They're crucial in geometry, Lie theory, and physics, showcasing constant curvature and rich symmetry properties.

This section dives into types, geometric properties, and examples of symmetric spaces. We'll explore their isometries, transvections, and structural features, connecting these concepts to the broader study of holonomy groups.

Symmetric Spaces and Types

Defining Symmetric Spaces

• Symmetric space represents a Riemannian manifold where each point serves as the center of an involutive isometry
• Riemannian symmetric space embodies a connected Riemannian manifold with an involutive isometry at every point preserving the metric
• Globally symmetric space encompasses a Riemannian manifold where the geodesic symmetry extends to a global isometry for all points
• Locally symmetric space characterizes a Riemannian manifold exhibiting constant sectional curvature in a neighborhood of each point
• Symmetric spaces play crucial roles in differential geometry, Lie group theory, and mathematical physics (string theory)

Geometric Properties of Symmetric Spaces

• Curvature tensor remains invariant under parallel transport in symmetric spaces
• Sectional curvature stays constant along parallel transported planes in symmetric spaces
• Geodesics in symmetric spaces exhibit periodic behavior, often forming closed loops
• Symmetric spaces possess rich isometry groups, allowing for extensive symmetry analysis
• Classification of symmetric spaces relates closely to the classification of simple Lie groups

Examples of Symmetric Spaces

• Euclidean spaces ($\mathbb{R}^n$) serve as the simplest examples of flat symmetric spaces
• Spheres ($S^n$) represent compact symmetric spaces with positive curvature
• Hyperbolic spaces ($H^n$) exemplify non-compact symmetric spaces with negative curvature
• Grassmannians (spaces of k-dimensional subspaces in $\mathbb{R}^n$) form important examples of symmetric spaces
• Lie groups equipped with bi-invariant metrics function as symmetric spaces

Symmetries and Isometries

Geodesic Symmetry and Its Properties

• Geodesic symmetry denotes an isometry that reverses geodesics passing through a given point
• Geodesic symmetry maps a point p to its antipodal point along any geodesic through p
• In symmetric spaces, geodesic symmetries exist globally and preserve the Riemannian metric
• Composition of two geodesic symmetries results in a transvection, a special type of isometry
• Geodesic symmetries generate the full isometry group of a symmetric space

Isometry Groups of Symmetric Spaces

• Isometry group of a symmetric space consists of all distance-preserving transformations
• Isometry groups of symmetric spaces are always Lie groups, allowing for algebraic analysis
• Connected component of the isometry group acts transitively on the symmetric space
• Isotropy subgroup at a point in a symmetric space relates to the fixed point set of the geodesic symmetry
• Isometry groups of symmetric spaces decompose as semidirect products of their identity component and discrete subgroups

Transvections and Their Role

• Transvection represents an isometry generated by composing two geodesic symmetries
• Transvections form a connected subgroup of the full isometry group
• Group of transvections acts transitively on the symmetric space
• Transvections preserve geodesics and parallel transport in symmetric spaces
• Study of transvections provides insights into the global geometry of symmetric spaces

Structure and Properties

Rank and Flat Submanifolds

• Rank of symmetric space defines the dimension of its maximal flat totally geodesic submanifold
• Flat totally geodesic submanifold represents a subspace with zero curvature and preserved by geodesics
• Rank relates to the algebraic structure of the isometry group of the symmetric space
• Higher rank symmetric spaces exhibit more complex geometric and algebraic properties
• Rank of a symmetric space determines many of its topological and geometric features (homology groups)

Curvature Properties of Symmetric Spaces

• Sectional curvature in symmetric spaces remains constant along parallel transported planes
• Ricci curvature of symmetric spaces exhibits simple expressions in terms of the Lie algebra of its isometry group
• Scalar curvature of symmetric spaces relates to the dimensions of certain subspaces in the associated Lie algebra
• Einstein equation is automatically satisfied for symmetric spaces, making them important in general relativity
• Curvature tensor of symmetric spaces possesses additional symmetries beyond those of general Riemannian manifolds

Algebraic Structure of Symmetric Spaces

• Symmetric spaces correspond to involutive automorphisms of semisimple Lie groups
• Cartan decomposition of the Lie algebra relates closely to the geometry of the symmetric space
• Root system of the associated Lie algebra determines the structure of flats and geodesics in the symmetric space
• Iwasawa decomposition provides a useful parameterization of points in non-compact symmetric spaces
• Killing form on the Lie algebra relates to the metric structure of the symmetric space