6.4 Homogeneous spaces

Homogeneous spaces are manifolds with high symmetry, where a Lie group acts transitively. They're key in Metric Differential Geometry, connecting algebra and geometry. Examples include spheres, projective spaces, and Grassmannians.

These spaces allow us to study geometric properties through algebraic structures. We'll explore their definition, Riemannian metrics, curvature, and classification. We'll also examine compact and noncompact cases, and their roles in various mathematical and physical applications.

Definition of homogeneous spaces

• Homogeneous spaces are an important class of manifolds that exhibit a high degree of symmetry and play a central role in the study of Metric Differential Geometry
• A homogeneous space is a manifold $M$ on which a Lie group $G$ acts transitively, meaning that for any two points $p,q \in M$, there exists an element $g \in G$ such that $g \cdot p = q$
• The study of homogeneous spaces provides a rich interplay between the algebraic structure of Lie groups and the geometric properties of manifolds

Transitive action of Lie groups

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• A Lie group $G$ acts transitively on a manifold $M$ if for any two points $p,q \in M$, there exists an element $g \in G$ such that $g \cdot p = q$
• Transitive actions allow us to study the geometry of the manifold $M$ through the algebraic properties of the Lie group $G$
• Examples of transitive actions include the action of the rotation group $SO(3)$ on the 2-sphere $S^2$ and the action of the Euclidean group $E(n)$ on $\mathbb{R}^n$

Isotropy subgroup and coset space

• For a point $p \in M$, the isotropy subgroup (or stabilizer) $H_p$ is the subgroup of $G$ that fixes $p$, i.e., $H_p = \{g \in G : g \cdot p = p\}$
• The coset space $G/H_p$ is the set of left cosets $\{gH_p : g \in G\}$, which can be identified with the orbit of $p$ under the action of $G$
• The coset space $G/H_p$ inherits a unique smooth structure from $G$ and $M$, making it a homogeneous space

Examples of homogeneous spaces

• The n-sphere $S^n$ is a homogeneous space under the action of the orthogonal group $O(n+1)$
• The projective space $\mathbb{RP}^n$ is a homogeneous space under the action of the projective linear group $PGL(n+1, \mathbb{R})$
• Grassmann manifolds $Gr(k,n)$, which parametrize k-dimensional subspaces of $\mathbb{R}^n$, are homogeneous spaces under the action of the general linear group $GL(n, \mathbb{R})$

Riemannian metrics on homogeneous spaces

• The study of Riemannian metrics on homogeneous spaces is a central theme in Metric Differential Geometry, as it allows us to investigate the interplay between the symmetries of the space and its geometric properties
• A Riemannian metric on a homogeneous space $M = G/H$ is called $G$-invariant if it is preserved under the action of $G$, i.e., the group action consists of isometries

Invariant metrics and reductive homogeneous spaces

• A homogeneous space $M = G/H$ is called reductive if the Lie algebra $\mathfrak{g}$ of $G$ admits a direct sum decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$, where $\mathfrak{h}$ is the Lie algebra of $H$ and $\mathfrak{m}$ is an $Ad(H)$-invariant complement
• Reductive homogeneous spaces allow for a natural construction of $G$-invariant Riemannian metrics by choosing an $Ad(H)$-invariant inner product on $\mathfrak{m}$
• Examples of reductive homogeneous spaces include symmetric spaces and naturally reductive spaces

Geodesics in homogeneous spaces

• Geodesics in a homogeneous space $M = G/H$ with a $G$-invariant Riemannian metric can be studied using the algebraic structure of the Lie group $G$
• Geodesics through the origin $eH \in M$ are given by the exponential map $\exp: \mathfrak{m} \rightarrow M$, where $\mathfrak{m}$ is the complement of $\mathfrak{h}$ in the reductive decomposition of $\mathfrak{g}$
• The homogeneity of the space allows for the extension of geodesics through any point using the group action

Curvature of homogeneous spaces

• The curvature tensor of a homogeneous space with a $G$-invariant Riemannian metric can be computed using the Lie bracket structure of the Lie algebra $\mathfrak{g}$
• Homogeneous spaces with constant sectional curvature, such as the sphere $S^n$ and the hyperbolic space $\mathbb{H}^n$, play a special role in geometry
• The study of curvature on homogeneous spaces leads to important results in Metric Differential Geometry, such as the classification of homogeneous Einstein manifolds

Classification of homogeneous spaces

• The classification of homogeneous spaces is a central problem in Metric Differential Geometry, as it provides a systematic way to understand the structure and properties of these spaces
• Several important classes of homogeneous spaces have been studied extensively, each with its own unique geometric and algebraic properties

Symmetric spaces

• A symmetric space is a connected Riemannian manifold $M$ such that for each point $p \in M$, there exists an isometry $\sigma_p: M \rightarrow M$ that fixes $p$ and reverses geodesics through $p$
• Symmetric spaces are homogeneous spaces with a rich algebraic structure and have been completely classified using the theory of semisimple Lie algebras
• Examples of symmetric spaces include Euclidean spaces, spheres, hyperbolic spaces, and Grassmann manifolds

Naturally reductive spaces

• A homogeneous space $M = G/H$ with a $G$-invariant Riemannian metric is called naturally reductive if there exists a reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$ such that $\langle [X,Y]_{\mathfrak{m}}, Z\rangle + \langle Y, [X,Z]_{\mathfrak{m}}\rangle = 0$ for all $X,Y,Z \in \mathfrak{m}$
• Naturally reductive spaces form a larger class than symmetric spaces and have geodesic symmetries at each point
• Examples of naturally reductive spaces include odd-dimensional spheres with the round metric and compact Lie groups with bi-invariant metrics

Normal homogeneous spaces

• A homogeneous space $M = G/H$ is called normal if the Lie algebra $\mathfrak{g}$ of $G$ admits a reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$ such that $[\mathfrak{h}, \mathfrak{m}] \subset \mathfrak{m}$
• Normal homogeneous spaces have a canonical connection called the Nomizu connection, which is closely related to the geometry of the space
• Examples of normal homogeneous spaces include isotropy irreducible spaces and generalized flag manifolds

Compact homogeneous spaces

• Compact homogeneous spaces are an important class of manifolds in Metric Differential Geometry, as they exhibit rich geometric and topological properties
• The study of compact homogeneous spaces often involves the interplay between Lie group theory, representation theory, and differential geometry

Relationship with Lie groups

• Every compact homogeneous space $M$ can be realized as a coset space $G/H$, where $G$ is a compact Lie group and $H$ is a closed subgroup of $G$
• The structure and properties of compact homogeneous spaces are closely related to the representation theory of compact Lie groups
• Examples of compact homogeneous spaces include spheres, projective spaces, and flag manifolds

Killing fields and isometries

• A Killing field on a Riemannian manifold $M$ is a vector field $X$ that generates a one-parameter family of isometries, i.e., the flow of $X$ preserves the Riemannian metric
• On a compact homogeneous space $M = G/H$, the Killing fields correspond to the elements of the Lie algebra $\mathfrak{g}$ of $G$
• The study of Killing fields and isometries on compact homogeneous spaces leads to important results in geometry and topology, such as the Bott periodicity theorem

Applications in geometry and physics

• Compact homogeneous spaces play a significant role in various areas of geometry and physics
• In Riemannian geometry, compact homogeneous spaces serve as model spaces and provide examples of manifolds with specific curvature properties (constant curvature, Einstein metrics, etc.)
• In mathematical physics, compact homogeneous spaces appear in the study of gauge theories, string theory, and quantum mechanics (e.g., as target spaces for nonlinear sigma models)

Noncompact homogeneous spaces

• Noncompact homogeneous spaces are another important class of manifolds in Metric Differential Geometry, with applications in various areas of mathematics and physics
• The study of noncompact homogeneous spaces often involves the interplay between Lie group theory, geometric analysis, and differential equations

Solvmanifolds and nilmanifolds

• A solvmanifold is a homogeneous space $M = G/H$, where $G$ is a solvable Lie group and $H$ is a closed subgroup
• A nilmanifold is a homogeneous space $M = G/H$, where $G$ is a nilpotent Lie group and $H$ is a closed subgroup
• Solvmanifolds and nilmanifolds provide important examples of noncompact homogeneous spaces with rich geometric and dynamical properties

Homogeneous Einstein metrics

• An Einstein metric on a Riemannian manifold $M$ is a metric whose Ricci curvature is proportional to the metric tensor, i.e., $Ric = \lambda g$ for some constant $\lambda$
• The study of homogeneous Einstein metrics on noncompact homogeneous spaces is an active area of research in Metric Differential Geometry
• Examples of noncompact homogeneous spaces admitting Einstein metrics include solvmanifolds, nilmanifolds, and symmetric spaces of noncompact type

Role in geometric analysis and relativity

• Noncompact homogeneous spaces play a significant role in geometric analysis and relativity
• In geometric analysis, noncompact homogeneous spaces appear in the study of harmonic analysis, heat kernels, and geometric flows (e.g., Ricci flow)
• In relativity, noncompact homogeneous spaces serve as models for cosmological spacetimes and provide examples of solutions to the Einstein field equations (e.g., Bianchi spacetimes)

Connections and curvature on homogeneous spaces

• The study of connections and curvature on homogeneous spaces is a central theme in Metric Differential Geometry, as it provides a way to understand the intrinsic geometry of these spaces
• Homogeneous spaces often admit a natural class of connections that are compatible with the symmetries of the space, leading to simplifications in the computation of curvature

Invariant connections and Nomizu construction

• An invariant connection on a homogeneous space $M = G/H$ is a connection that is preserved under the action of the Lie group $G$
• The Nomizu construction provides a way to obtain invariant connections on reductive homogeneous spaces using the Lie algebra structure of $\mathfrak{g}$ and the reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$
• The canonical connection on a reductive homogeneous space, obtained via the Nomizu construction, plays a crucial role in the study of the geometry of the space

Ricci curvature and scalar curvature

• The Ricci curvature and scalar curvature are important geometric invariants that measure the amount of curvature in a Riemannian manifold
• On homogeneous spaces, the Ricci and scalar curvatures can be computed using the structure constants of the Lie algebra $\mathfrak{g}$ and the invariant metric
• The study of Ricci and scalar curvatures on homogeneous spaces leads to important results in Metric Differential Geometry, such as the classification of homogeneous Einstein manifolds and the Alekseevsky conjecture

Homogeneous structures and holonomy

• A homogeneous structure on a manifold $M$ is a reduction of the frame bundle of $M$ to a subgroup $G$ of the general linear group $GL(n, \mathbb{R})$
• The holonomy group of a connection on a manifold $M$ is the group of parallel transports along loops based at a point
• Homogeneous spaces with invariant connections provide examples of manifolds with reduced holonomy groups, which are of great interest in Metric Differential Geometry and mathematical physics (e.g., Calabi-Yau manifolds, $G_2$ manifolds)

Homogeneous spaces in Lie theory

• The study of homogeneous spaces is closely related to Lie theory, as homogeneous spaces are constructed using Lie groups and their subgroups
• Lie theory provides powerful tools for understanding the structure and properties of homogeneous spaces, and conversely, homogeneous spaces serve as important examples and applications of Lie theoretic concepts

Relationship with representation theory

• The representation theory of Lie groups plays a crucial role in the study of homogeneous spaces
• The isotropy representation of a homogeneous space $M = G/H$ is the representation of the isotropy subgroup $H$ on the tangent space $T_{eH}M$, which determines the local geometry of the space
• The study of spherical representations and multiplicity-free spaces provides a deep connection between representation theory and the geometry of homogeneous spaces

Homogeneous spaces as manifolds with symmetry

• Homogeneous spaces are manifolds equipped with a transitive action of a Lie group, which provides a natural notion of symmetry
• The presence of symmetry on homogeneous spaces allows for the use of powerful techniques from Lie theory and representation theory to study their geometry and topology
• Examples of manifolds with symmetry that can be studied using the theory of homogeneous spaces include symmetric spaces, flag manifolds, and isotropy irreducible spaces

Applications in harmonic analysis and PDEs

• Homogeneous spaces play a significant role in harmonic analysis and the study of partial differential equations (PDEs)
• The symmetries of homogeneous spaces allow for the use of Fourier analysis and representation theory to study function spaces and differential operators on these manifolds
• Important examples of PDEs that can be studied using the theory of homogeneous spaces include the Laplace-Beltrami equation, the heat equation, and the wave equation on symmetric spaces and other homogeneous manifolds