Riemannian Geometry Unit 6 Review: Isometries and Homogeneous Spaces

Key Concepts and Definitions

• Riemannian manifold $(M, g)$ consists of a smooth manifold $M$ equipped with a Riemannian metric $g$
• Riemannian metric $g$ assigns an inner product to each tangent space $T_pM$ that varies smoothly with respect to the point $p \in M$
• Isometry is a distance-preserving map between Riemannian manifolds
  • Formally, an isometry is a diffeomorphism $f: (M, g) \to (N, h)$ such that $f^*h = g$, where $f^*$ denotes the pullback of the metric
• Lie group is a smooth manifold that is also a group, with group operations (multiplication and inversion) being smooth maps
• Homogeneous space is a manifold on which a Lie group acts transitively by isometries
  • Transitive action means for any two points $p, q \in M$, there exists an element $g$ in the Lie group $G$ such that $g \cdot p = q$
• Isotropy subgroup (or stabilizer) at a point $p \in M$ is the subgroup $H_p = \{g \in G : g \cdot p = p\}$ that fixes the point $p$
• Coset space $G/H$ is the set of left cosets $\{gH : g \in G\}$ of a subgroup $H$ in a group $G$

Isometries in Riemannian Geometry

• Isometries preserve the Riemannian metric and, consequently, geometric properties such as lengths of curves, angles between vectors, and curvature
• The set of all isometries of a Riemannian manifold $(M, g)$ forms a Lie group, denoted by $\text{Isom}(M, g)$
• Isometries can be characterized by their action on the tangent spaces
  • An isometry $f: (M, g) \to (M, g)$ induces a linear isometry $df_p: T_pM \to T_{f(p)}M$ between tangent spaces for each point $p \in M$
• The Lie algebra of $\text{Isom}(M, g)$ consists of Killing vector fields, which are vector fields $X$ satisfying $\mathcal{L}_Xg = 0$, where $\mathcal{L}_X$ denotes the Lie derivative
• Isometries preserve geodesics, meaning that if $\gamma(t)$ is a geodesic in $(M, g)$, then $f \circ \gamma(t)$ is also a geodesic for any isometry $f$
• The Myers-Steenrod theorem states that the group of isometries of a Riemannian manifold is a Lie group with respect to the compact-open topology
• Isometries play a crucial role in understanding the symmetries and geometric structure of Riemannian manifolds

Types of Isometries

• Isometries can be classified into different types based on their fixed points and the induced action on tangent spaces
• Trivial isometry (identity map) fixes every point on the manifold and induces the identity map on each tangent space
• Involutive isometry (reflection) satisfies $f \circ f = \text{id}$ and has a fixed-point set of codimension one (hypersurface)
• Rotations are isometries that fix a point $p$ and induce a rotation on the tangent space $T_pM$
• Translations are isometries that have no fixed points and preserve a given direction (parallel vector field) on the manifold
• Glide reflections are a combination of a reflection and a translation in the direction perpendicular to the reflection hypersurface
• Isometries can also be classified as orientation-preserving (direct isometries) or orientation-reversing (indirect isometries)
  • Orientation-preserving isometries have determinant +1 when expressed in local coordinates
  • Orientation-reversing isometries have determinant -1 when expressed in local coordinates
• The classification of isometries depends on the geometry and topology of the underlying Riemannian manifold

Homogeneous Spaces: An Introduction

• A homogeneous space is a manifold $M$ on which a Lie group $G$ acts transitively by isometries
  • Transitive action means that any two points on the manifold can be mapped to each other by an element of the group
• Homogeneous spaces are a generalization of symmetric spaces, which are Riemannian manifolds with a symmetry at each point (involutive isometry)
• The isotropy subgroup (or stabilizer) $H_p$ at a point $p \in M$ is the subgroup of $G$ that fixes the point $p$
  • The isotropy subgroups at different points are conjugate to each other, i.e., $H_q = gH_pg^{-1}$ for some $g \in G$ such that $g \cdot p = q$
• A homogeneous space can be identified with the coset space $G/H$, where $H$ is the isotropy subgroup at a fixed point $p_0 \in M$
  • The coset space $G/H$ is the set of left cosets $\{gH : g \in G\}$ of $H$ in $G$
• The natural projection $\pi: G \to G/H$ is a smooth submersion and induces a unique smooth structure on $G/H$ such that $\pi$ is a smooth map
• The action of $G$ on $M \cong G/H$ is given by left multiplication on cosets: $g \cdot (xH) = (gx)H$ for $g \in G$ and $xH \in G/H$
• Examples of homogeneous spaces include Euclidean spaces, spheres, projective spaces, and Grassmannians

Properties of Homogeneous Spaces

• Homogeneous spaces inherit a unique Riemannian metric (up to scalar multiplication) from the Lie group $G$ that is invariant under the group action
  • This metric is called the Killing metric and is determined by the Lie algebra of $G$
• The Riemannian curvature tensor of a homogeneous space is constant, i.e., it does not depend on the point on the manifold
  • This property follows from the transitivity of the group action and the invariance of the metric
• Geodesics in a homogeneous space are orbits of one-parameter subgroups of the isometry group $G$
  • Given a tangent vector $v \in T_pM$, there exists a unique geodesic $\gamma(t)$ starting at $p$ with initial velocity $v$, and it is given by $\gamma(t) = \exp(tX) \cdot p$, where $X \in \mathfrak{g}$ (Lie algebra of $G$) satisfies $d\pi_e(X) = v$
• Homogeneous spaces are complete Riemannian manifolds, meaning that geodesics can be extended indefinitely
• The isometry group of a homogeneous space $M \cong G/H$ contains the Lie group $G$ as a subgroup, but it may be larger
  • The full isometry group is determined by the normalizer of $H$ in $G$, i.e., $N_G(H) = \{g \in G : gHg^{-1} = H\}$
• Homogeneous spaces have a rich algebraic structure that interacts with their geometric properties, making them an important object of study in differential geometry and representation theory

Examples and Applications

• Spheres $S^n$ are homogeneous spaces under the action of the orthogonal group $O(n+1)$
  • The isotropy subgroup at a point is isomorphic to $O(n)$, so $S^n \cong O(n+1)/O(n)$
• Projective spaces $\mathbb{RP}^n$ and $\mathbb{CP}^n$ are homogeneous spaces under the action of the projective linear groups $PGL(n+1, \mathbb{R})$ and $PGL(n+1, \mathbb{C})$, respectively
  • For $\mathbb{RP}^n$, the isotropy subgroup is isomorphic to $GL(n, \mathbb{R})$, so $\mathbb{RP}^n \cong PGL(n+1, \mathbb{R})/GL(n, \mathbb{R})$
• Grassmannians $Gr(k, n)$, the space of $k$-dimensional subspaces of $\mathbb{R}^n$, are homogeneous spaces under the action of the orthogonal group $O(n)$
  • The isotropy subgroup is isomorphic to $O(k) \times O(n-k)$, so $Gr(k, n) \cong O(n)/(O(k) \times O(n-k))$
• Lie groups themselves are homogeneous spaces under the action of left (or right) multiplication
  • For a Lie group $G$, the isotropy subgroup at the identity element is the trivial subgroup $\{e\}$, so $G \cong G/\{e\}$
• Homogeneous spaces have applications in physics, such as in the study of symmetries in general relativity and gauge theories
  • For example, the space-time of special relativity is a homogeneous space under the action of the Poincaré group
• Homogeneous spaces also appear in the study of dynamical systems with symmetries, where the phase space is often a homogeneous space under the action of the symmetry group

Connections to Other Areas of Mathematics

• Homogeneous spaces are closely related to the theory of Lie groups and Lie algebras
  • The classification of homogeneous spaces often relies on the classification of Lie algebras and their subalgebras
• The study of invariant differential operators on homogeneous spaces is a central topic in representation theory and harmonic analysis
  • Invariant differential operators are closely related to the representation theory of the isometry group and its Lie algebra
• Homogeneous spaces appear in algebraic geometry as quotients of algebraic groups by algebraic subgroups
  • The study of algebraic homogeneous spaces often involves techniques from algebraic geometry and commutative algebra
• The topology of homogeneous spaces is related to the theory of principal bundles and fiber bundles
  • A homogeneous space $G/H$ can be viewed as the base space of a principal $H$-bundle over $G$
• Homogeneous spaces have connections to number theory and arithmetic geometry, particularly in the study of locally symmetric spaces and Shimura varieties
  • These spaces are important in the Langlands program and the study of automorphic forms
• The geometry and analysis on homogeneous spaces have applications in mathematical physics, particularly in the study of integrable systems and quantum mechanics
  • Many integrable systems, such as the rigid body and the Toda lattice, can be described using the geometry of homogeneous spaces

Problem-Solving Techniques

• To determine if a given Riemannian manifold is a homogeneous space, look for a transitive action by isometries
  • Start by identifying the isometry group of the manifold and checking if it acts transitively
• When working with a homogeneous space $M \cong G/H$, use the quotient map $\pi: G \to G/H$ to relate properties of $M$ to properties of $G$ and $H$
  • For example, use the differential of $\pi$ to relate tangent vectors and geodesics on $M$ to elements of the Lie algebra of $G$
• Utilize the classification of Lie algebras and their subalgebras to classify homogeneous spaces
  • The Lie algebra of the isometry group and the Lie algebra of the isotropy subgroup can provide insights into the structure of the homogeneous space
• When computing curvature or geodesics on a homogeneous space, use the invariance properties to simplify calculations
  • The Riemannian curvature tensor is constant, and geodesics are orbits of one-parameter subgroups, which can simplify computations
• Use representation theory to study invariant differential operators and harmonic analysis on homogeneous spaces
  • Decompose the space of functions on the homogeneous space into irreducible representations of the isometry group to analyze invariant differential operators
• When faced with a geometric problem on a homogeneous space, consider if the problem can be reduced to a problem on the isometry group or the isotropy subgroup
  • Many geometric properties of homogeneous spaces can be understood through the properties of the Lie groups involved
• Utilize connections to other areas of mathematics, such as algebraic geometry, topology, and mathematical physics, to gain insights and apply techniques from these fields to the study of homogeneous spaces