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Geodesic orbit pseudo-Riemannian H-type nilmanifolds: case of minimal admissible Clifford modules

Kenro Furutani, Irina Markina, Yurii Nikonorov

Abstract

We investigate the geodesic orbit property of pseudo-Riemannian nilmanifolds, specifically those known in the literature as pseudo $H$-type Lie groups -- i.e., 2-step nilpotent Lie groups of Heisenberg type equipped with a left invariant pseudo-Riemannian metric. The study of homogeneous geodesics on Riemannian $H$-type Lie groups was completed by C.~Riehm in 1984. In this work, we extend these results to the pseudo-Riemannian $H$-type Lie groups and provide a complete characterization of the geodesic orbit property for the case where the underlying Lie algebras are constructed from the admissible Clifford modules of minimal dimension.

Geodesic orbit pseudo-Riemannian H-type nilmanifolds: case of minimal admissible Clifford modules

Abstract

We investigate the geodesic orbit property of pseudo-Riemannian nilmanifolds, specifically those known in the literature as pseudo -type Lie groups -- i.e., 2-step nilpotent Lie groups of Heisenberg type equipped with a left invariant pseudo-Riemannian metric. The study of homogeneous geodesics on Riemannian -type Lie groups was completed by C.~Riehm in 1984. In this work, we extend these results to the pseudo-Riemannian -type Lie groups and provide a complete characterization of the geodesic orbit property for the case where the underlying Lie algebras are constructed from the admissible Clifford modules of minimal dimension.

Paper Structure

This paper contains 23 sections, 36 theorems, 172 equations, 3 tables.

Table of Contents

  1. Introduction and main results
  2. Auxiliary results
  3. Pseudo-Riemannian $H$-type nilmanifolds
  4. Integral basis, periodicity, and the automorphism groups of pseudo $H$-type Lie algebras
  5. Pseudo-Riemannian $H$-type nilmanifolds $N_{r,s}$ with $1\leq r+s\leq 3$
  6. Pseudo-Riemannian $H$-type nilmanifold $N_{0,1}$
  7. Pseudo-Riemannian $H$-type nilmanifold $N_{1,1}$
  8. Pseudo-Riemannian $H$-type nilmanifold $N_{0,2}$
  9. Pseudo-Riemannian $H$-type nilmanifold $N_{1,2}$
  10. Pseudo-Riemannian $H$-type nilmanifold $N_{2,1}$
  11. Pseudo-Riemannian $H$-type nilmanifold $N_{0,3}$
  12. On naturally reductive pseudo-Riemannian $H$-type nilmanifolds
  13. On totally geodesic submanifolds of geodesic orbit pseudo-Riemannian manifolds
  14. Pseudo-Riemannian $H$-type nilmanifolds $N_{r,s}$ with $r+s> 3$, $(r,s)\neq (3,4)$
  15. Pseudo-Riemannian $H$-type nilmanifolds $N_{r,s}$ with $r+s=0\mod 4$
  16. ...and 8 more sections

Key Result

Proposition 1

Let $(M= G/H, g)$ be a homogeneous reductive pseudo-Riemannian manifold, with the corresponding reductive decomposition $\mathfrak{g} = \mathfrak{h}\oplus \mathfrak{m}$. Then $M$ is a $G$-geodesic orbit space if and only if, for any $T\in \mathfrak{m}$, there exist $P = P(T) \in \mathfrak{h}$ and $ where $\langle \cdot\,, \cdot \rangle$ denotes the inner product on $\mathfrak{m}$ defined by $g$,

Theorems & Definitions (48)

  • Definition 1
  • Proposition 1: Geodesic Lemma DK2007
  • Definition 2
  • Theorem 1: Rie84
  • Theorem 2
  • Proposition 2
  • Proposition 3: Bar, Theorem 3.3
  • Corollary 1
  • Remark 1
  • Theorem 3
  • ...and 38 more