  An increasingly common viewpoint is that protein dynamics data sets reside in
a non-linear subspace of low conformational energy. Ideal data analysis tools
for such data sets should therefore account for such non-linear geometry. The
Riemannian geometry setting can be suitable for a variety of reasons. First, it
comes with a rich structure to account for a wide range of geometries that can
be modelled after an energy landscape. Second, many standard data analysis
tools initially developed for data in Euclidean space can also be generalised
to data on a Riemannian manifold. In the context of protein dynamics, a
conceptual challenge comes from the lack of a suitable smooth manifold and the
lack of guidelines for constructing a smooth Riemannian structure based on an
energy landscape. In addition, computational feasibility in computing geodesics
and related mappings poses a major challenge. This work considers these
challenges. The first part of the paper develops a novel local approximation
technique for computing geodesics and related mappings on Riemannian manifolds
in a computationally feasible manner. The second part constructs a smooth
manifold of point clouds modulo rigid body group actions and a Riemannian
structure that is based on an energy landscape for protein conformations. The
resulting Riemannian geometry is tested on several data analysis tasks relevant
for protein dynamics data. It performs exceptionally well on coarse-grained
molecular dynamics simulated data. In particular, the geodesics with given
start- and end-points approximately recover corresponding molecular dynamics
trajectories for proteins that undergo relatively ordered transitions with
medium sized deformations. The Riemannian protein geometry also gives
physically realistic summary statistics and retrieves the underlying dimension
even for large-sized deformations within seconds on a laptop.
