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Homogeneous spaces with geodesic orbit Riemannian metrics and with integrable invariant distributions

V. N. Berestovskii, Yu. G. Nikonorov

TL;DR

The paper investigates homogeneous spaces $G/H$ with compact stabilizers in two classes: spaces with integrable invariant distributions and GO-spaces where every geodesic is an orbit of a 1-parameter subgroup, and it analyzes when GO holds for all invariant metrics. A main result is a negative answer to whether every invariant metric on a space with integrable invariant distributions is GO, supported by explicit counterexamples including Einstein solvmanifolds with negative Ricci curvature. The authors develop a framework based on Levi decompositions, strong subalgebras, and metric Lie algebras of solvable type to construct new examples, notably solvmanifolds with commutative nilradicals that admit integrable distributions yet fail the GO property. These findings delineate the boundary between integrable distributions and GO, highlight the existence of rigid GO-spaces among integrable-distribution spaces, and enrich the catalogue of non-GO Einstein solvmanifolds.

Abstract

We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary invariant Riemannian metric on the space, every geodesic is an orbit of a 1-parameter subgroup of the isometry group. We found several homogeneous spaces of the first type that are not spaces of the second type. Among them there are several homogeneous spaces that admit invariant Einstein metrics.

Homogeneous spaces with geodesic orbit Riemannian metrics and with integrable invariant distributions

TL;DR

The paper investigates homogeneous spaces with compact stabilizers in two classes: spaces with integrable invariant distributions and GO-spaces where every geodesic is an orbit of a 1-parameter subgroup, and it analyzes when GO holds for all invariant metrics. A main result is a negative answer to whether every invariant metric on a space with integrable invariant distributions is GO, supported by explicit counterexamples including Einstein solvmanifolds with negative Ricci curvature. The authors develop a framework based on Levi decompositions, strong subalgebras, and metric Lie algebras of solvable type to construct new examples, notably solvmanifolds with commutative nilradicals that admit integrable distributions yet fail the GO property. These findings delineate the boundary between integrable distributions and GO, highlight the existence of rigid GO-spaces among integrable-distribution spaces, and enrich the catalogue of non-GO Einstein solvmanifolds.

Abstract

We consider homogeneous spaces of Lie groups with compact stabilizer subgroups of two types: those with integrable invariant distributions and those with geodesic orbit invariant Riemannian metrics. The latter means that for an arbitrary invariant Riemannian metric on the space, every geodesic is an orbit of a 1-parameter subgroup of the isometry group. We found several homogeneous spaces of the first type that are not spaces of the second type. Among them there are several homogeneous spaces that admit invariant Einstein metrics.
Paper Structure (4 sections, 24 theorems, 18 equations)

This paper contains 4 sections, 24 theorems, 18 equations.

Table of Contents

  1. Homogeneous spaces with integrable invariant distributions: some known results
  2. Geodesic orbit manifolds and the main problem
  3. Some examples and counterexamples
  4. One class of solvmanifolds

Key Result

Proposition 1

Let $(\mathfrak{g}, [\cdot, \cdot])$ be the Lie algebra of the Lie group $G$, $\mathfrak{h}\subset \mathfrak{g}$ be the Lie algebra of the Lie subgroup $H$. Then $G/H$ is a homogeneous manifold with integrable invariant distributions if and only if A) every vector subspace of $\mathfrak{q}\subset \m

Theorems & Definitions (49)

  • Remark 1
  • Definition 1
  • Proposition 1: Ber1989
  • Corollary 1: Ber1989
  • Proposition 2: Ber1989
  • Proposition 3
  • Theorem 1: Ber1992Ber1995
  • Definition 2: Gor2008BerGor2014
  • Definition 3: Gor2008BerGor2014
  • Theorem 2: BerGor2014
  • ...and 39 more