---
tag: HBDEPTRF11
kind: concept
name: Solvmanifold
msc: ["22E25", 22F30, 53C30]
specializes: [85TWR6E75P]
sources:
  - url: https://en.wikipedia.org/wiki/Solvmanifold
same_as:
  wikidata: Q7559043
  wikipedia: Solvmanifold
---

## Summary

A solvmanifold is a homogeneous space $G/H$ of a connected [[solvable Lie group|9CNSVFQ801]] $G$ by a closed subgroup $H$.
Tori, nilmanifolds and many mapping tori of torus maps are solvmanifolds.
The compact ones of the form $G/\Gamma$, with $\Gamma$ a lattice, are natural test spaces in geometry and dynamics.
For example, they give examples of symplectic manifolds without Kähler structure.

## Explanations

### L1 human

A solvable group is one built from commutative groups in finitely many steps.
A solvmanifold is a space on which such a group acts, so that any point can be moved to any other point.
The line is an example: the maps $x \mapsto ax + b$ already move any point of the line to any other, and their group is solvable.
The torus $\R^n / \Z^n$ is a compact example, with the group $\R^n$ acting by translations.