By this paper https://www.mat.univie.ac.at/~michor/rie-met.pdf for any given manifold $M$, there is a Hilbert manifold (I believe its a Hilbert manifold, if not it is still an infinite dimensional manifold for which a Riemannian metric can be endowed on) $\mathcal{M}(M)$ of all Riemannian metrics on $M$. I am wondering if there is perhaps a finite dimensional submanifold $N$ of $\mathcal{M}(M)$ for which either or both (preferably both) of the following properties apply:

• The geodesic equation in $N$ (which is Riemannian via the pullback metric) has an explicit solution
• For all $g \in N$ the geodesic equation for $(M, g)$ has an explicit solution

I would mostly be interested in the case for which $M = \mathbb{R}^n$ but other manifolds would be interesting as well. I am doing this for a computational algorithm (that needs to run in an efficient amount of time) so the main properties I am looking for is that the geodesic equation for $N$ and the geodesic equation for $(M, g)$ for all $g \in N$ is efficiently computable (possibly using autograd libraries) and $N$ is a big enough family that it can represent a very diverse set of Riemannian metrics. Also, I am not attached to the metric on $\mathcal{M}(N)$ used in the paper, I would be interested in a different metric if it applies to this problem well.