# Arithmetic expression space

**Arithmetic expression space** is a Riemannian manifold equipped with an assignment satisfying the flow equation. In such a space, a thread-able expression can be viewed as a walk geometrically, and then all the walks formed a group, and we may discuss some special canonical form of arithmetic expression in such a Riemannian manifold.

## Formal definition

Given a number field [math]\displaystyle{ F }[/math], a 2 dimensional Riemannian manifold [math]\displaystyle{ M }[/math], and a local orthogonal Cartesian coordinate system [math]\displaystyle{ C }[/math] on [math]\displaystyle{ M }[/math], we introduce an assignment [math]\displaystyle{ a: M \to F }[/math] satisfying

[math]\displaystyle{ \frac{da}{ds} = \mu \cos \theta + a \lambda \sin \theta }[/math] |
(1) |

where [math]\displaystyle{ ds }[/math] is the trace length of a moving point, and [math]\displaystyle{ \theta }[/math] is the angle against the local additive axis of the moving point in the orthogonal frame [math]\displaystyle{ C }[/math], [math]\displaystyle{ \mu }[/math] is the additive generator and [math]\displaystyle{ e^\lambda }[/math] is the multiplicative generator. This differential equation is the flow equation.

## Properties

## Examples

## Arithmetical perspective

each point is a canonical form of arithmetical expression.