# 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 $\displaystyle{ F }$, a 2 dimensional Riemannian manifold $\displaystyle{ M }$, and a local orthogonal Cartesian coordinate system $\displaystyle{ C }$ on $\displaystyle{ M }$, we introduce an assignment $\displaystyle{ a: M \to F }$ satisfying

 $\displaystyle{ \frac{da}{ds} = \mu \cos \theta + a \lambda \sin \theta }$ (1)

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

## Arithmetical perspective

each point is a canonical form of arithmetical expression.