In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like primes in a number field.

That is, given a finite normal (i.e. ``Galois'') cover $M\to N$ of Riemannian manifolds, the preimage of prime geodesic $\mathfrak{p}$ in $N$ is a disjoint union $\mathfrak{P}_1,...,\mathfrak{P}_k$ of prime geodesics in $M$. The degree $f_i$ of the covering map $\mathfrak{P}_i\to \mathfrak{p}$ of circles should be thought of as the inertia degree of $\mathfrak{P}_i/\mathfrak{p}$, and the Frobenius $\sigma_{\mathfrak{P}_i}\in \text{Deck}(M/N)$ acts on $\mathfrak{P}_i$ by a $1/f_i$ rotation. Note $[M:N]=\sum f_i$. Other similar properties also hold, for instance there is also a Chebatorev density theorem.

My question is: is there an analogue of unique factorisation of ideals in this context?

I don't know what the right analogue of a non-prime ideal is here: perhaps $\mathfrak{p}^2$ should be $\mathfrak{p}$, but the geodesic loops round twice, but I can't think of the right definition for $I$ in general.



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