G-Invariant Riemannian Metrics

• G-Invariant Riemannian Metrics are defined on manifolds with Lie group actions, where a metric is uniquely determined by an Ad(H)-invariant inner product at a base point.
• These metrics are parametrized by scaling factors corresponding to irreducible isotropy components, facilitating explicit classifications on flag manifolds and Ledger-Obata spaces.
• They enable the reduction of Einstein equations and the analysis of geodesic flows, spectral properties, and natural reductivity in homogeneous geometric settings.

A G-invariant Riemannian metric is a Riemannian metric on a manifold equipped with a smooth action of a Lie group G by isometries. The theory of such metrics is foundational in homogeneous, symmetric, and nearly homogeneous geometries, and has deep implications in Lie group theory, global analysis, and geometric flows. On a homogeneous space M = G/H, where H is a closed subgroup, a G-invariant Riemannian metric is completely determined by its value at a single point, typically the coset of the identity, via an Ad(H)-invariant inner product on the complement of Lie algebras 𝔤 = 𝔥 ⊕ 𝔪. The classification, parametrization, and geometric analysis of these G-invariant metrics form a rich subject with connections to natural reductivity, geodesic orbit manifolds, Einstein metrics, and the spectral geometry of the Laplacian.

1. Structural Foundations of G-Invariant Metrics

Let G be a Lie group acting transitively by isometries on a manifold M, so that M ≅ G/H for some closed subgroup H. The tangent space at the base point (identity coset) can be identified with a complementary subspace 𝔪 to 𝔥 = Lie(H) inside 𝔤 = Lie(G), satisfying the reductive condition [𝔥, 𝔪] ⊂ 𝔪. Any G-invariant Riemannian metric is then determined by an Ad(H)-invariant inner product on 𝔪, invariant under the adjoint action of the isotropy. By Schur's Lemma, when 𝔪 splits into a direct sum of pairwise inequivalent irreducible Ad(H)-modules, the metric is diagonalizable along these summands, each scaled by a positive real parameter.

For example, on classical flag manifolds associated with compact Lie groups, 𝔪 always admits such a canonical decomposition, and the space of G-invariant metrics is parametrized by positive numbers, one per irreducible isotropy summand (Pulemotov, 2015, Alves et al., 2014, Grajales et al., 2024).

2. Parametrization and Families of G-Invariant Metrics

The full family of G-invariant metrics on M = G/H is described by the structure of the isotropy representation:

• Isotropy Decomposition: 𝔪 = ⊕_{i=1}^s 𝔪_i, with each 𝔪_i irreducible and Ad(H)-invariant.
• Metric Parameters: The associated metric is

$g = \sum_{i=1}^s x_i B|_{\mathfrak{m}_i}$

for some Ad(G)-invariant inner product B on 𝔤 and positive scalars x_i (Pulemotov, 2015, Grajales et al., 2024).

The number of independent parameters equals the number of non-isomorphic irreducible submodules of the isotropy representation. In flag manifolds of classical groups, explicit counts and labelings are given in terms of the number of "t-roots," with precise formulas for each Lie type (Alves et al., 2014).

Special cases, such as Ledger-Obata spaces $F^m/diag(F)$, also permit a block-diagonal description of invariant metrics, either via so-called "coordinate-zero" complements or orthogonal complements defined relative to the Killing form. Here, possible metrics are characterized by positive-definite matrices subject to symmetry and positivity constraints (Nikolayevsky et al., 2017).

3. Natural Reductivity and Geodesic Orbit Criteria

A central concept is that of naturally reductive metrics: a G-invariant metric is naturally reductive with respect to G if there exists a reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$ such that

$$\langle [X, Y]_{\mathfrak{m}}, Z \rangle + \langle Y, [X, Z]_{\mathfrak{m}} \rangle = 0, \quad \forall X \in \mathfrak{h},\ Y,Z \in \mathfrak{m}.$$

The normal homogeneous case corresponds to the inner product being the restriction of an Ad(G)-invariant inner product on 𝔤.

On Ledger-Obata spaces, the exhaustive classification is achieved: an invariant metric is naturally reductive if and only if it is generated by either a diagonal positive-definite matrix on the (m-1)-fold sum of 𝔣, or as the restriction of a certain ad(𝔤)-invariant quadratic form, subject to an explicit sign constraint on coefficients (Nikolayevsky et al., 2017).

It is also proved that on Ledger-Obata spaces, a metric is a geodesic orbit (g.o.) metric if and only if it is naturally reductive. In the fundamental case $m=3$, any invariant metric is automatically naturally reductive, exhibiting a rigidity phenomenon for small m (Nikolayevsky et al., 2017).

In real flag manifolds of $G_2$-type, explicit algebraic criteria are given for a metric to be geodesic orbit, in terms of the metric parameters satisfying certain relations arising from Souris' criterion (Grajales et al., 2024).

4. Einstein and Ricci Prescribed Metrics

G-invariant metrics facilitate the reduction of Einstein's equations to an algebraic system. On a flag manifold or homogeneous space, the Ricci tensor with respect to a G-invariant metric remains diagonal in the isotropy splitting, so the Einstein equations become a set of polynomial equations for the scaling parameters {x_i}. An explicit formula for Ricci components on each irreducible summand 𝔪_i is

$$\mathrm{Ric}_i = \frac{1}{2 x_i} \sum_{j,k}\biggl[ -c_{ij}^k \frac{x_i^2}{x_j x_k} + d_{ij}^k \frac{x_j}{x_k} + e_{ij}^k \frac{x_k}{x_j} \biggr]$$

(Pulemotov, 2015), with the structure constants determined by the commutator algebra of 𝔤.

A fundamental existence theorem asserts: given a nonzero, positive semi-definite G-invariant symmetric (0,2)-tensor T on M, there exist positive parameters {x_i} and scalar c > 0 such that Ric(g) = c T, provided H is a maximal connected subgroup of G. The prescribed Ricci problem yields homogeneous Einstein metrics when T = g, and its solvability is determined by a variational approach based on the scalar curvature functional S(g) restricted to appropriate parameter loci (Pulemotov, 2015).

In classical flag manifolds, the number of independent Einstein equations equals the number of isotropy summands, and the resulting algebraic system admits at least one solution when G/H is a standard homogeneous space for compact G and maximal H (Alves et al., 2014).

5. Geometric Properties and Isometry Groups

The decomposition properties of G-invariant metrics admit effective recognition of reducibility and symmetry. On Ledger-Obata spaces $F^m/diag(F)$, an invariant metric is reducible if and only if the associated metric endomorphism on 𝔤 allows an orthogonal splitting compatible with the group structure—concretely, if the underlying positive semi-definite matrix splits into block-diagonal components under a certain combinatorial "tree" algorithm (Nikolayevsky et al., 2017).

Any invariant metric on $F^m/diag(F)$ decomposes as a Riemannian product of irreducible Ledger-Obata factors, and the full connected isometry group of each irreducible factor is $F^{m_i}$ acting by left multiplication, with no hidden symmetries (Nikolayevsky et al., 2017). Invariant metrics on spheres and projective spaces are similarly classified by the homogeneity properties of G, with Clifford–Wolf homogeneity indicating maximal symmetry at the round metric (Berestovskii et al., 2012).

6. Spectral and Variational Aspects

The study of G-invariant metrics is intertwined with spectral geometry. For example, on compact Lie groups and certain spherical manifolds, the set of G-invariant metrics of unit volume exhibits unbounded first Laplacian eigenvalue under explicit deformations, via Killing field perturbations or Urakawa’s scaling. This demonstrates the richness and flexibility of G-invariant geometry even within constant-volume classes (Cernea, 2010).

The scalar curvature function S(g) on the space of G-invariant metrics is a key tool in the classification and existence theory for Einstein metrics, as critical points of S(g) often correspond to solutions of the Einstein or Ricci-prescribed equations (Pulemotov, 2015).

7. Examples and Explicit Classifications

Explicit families of G-invariant metrics on spheres, flag manifolds, Ledger-Obata and group homogeneous spaces have been classified in detail. For odd-dimensional spheres S^{2k+1}, the entire set of connected, almost effective compact G acting transitively yields a parametric family of G-invariant metrics, with normal and generalized normal (G-8-homogeneous) cases precisely described in terms of allowed parameter ranges (Berestovskii et al., 2012). Sectional curvature formulas and orbit structures of unit Killing fields are also explicit in these examples.

For classical flag manifolds, algebraic systems for Einstein metrics are given in closed form, with the number of variables and equations governed by the explicit enumeration of isotropy components (Alves et al., 2014). The geometry of $G_2$-type flag manifolds admits similarly explicit parametrizations, and qualitative descriptions of Ricci flow trajectories (Grajales et al., 2024). Invariant metric geometry on GL(n) with additional right-O(n)-invariance has also been characterized, with geodesic formulas of matrix-logarithmic type and applications to non-linear elasticity (Martin et al., 2014).