  We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new
model of random differential geometry, by considering a random, smooth
Riemannian metric on $\mathbb R^d$. We are motivated in our study by the random
geometry of first-passage percolation (FPP), a lattice model which was
developed to model fluid flow through porous media. By adapting techniques from
standard FPP, we prove a shape theorem for our model, which says that large
balls under this metric converge to a deterministic shape under rescaling. As a
consequence, we show that smooth random Riemannian metrics are geodesically
complete with probability one.
  In differential geometry, geodesics are curves which locally minimize length.
They need not do so globally: consider great circles on a sphere. For lattice
models of FPP, there are many open questions related to minimizing geodesics;
similarly, it is interesting from a geometric perspective when geodesics are
globally minimizing. In the present study, we show that for any fixed starting
direction $v$, the geodesic starting from the origin in the direction $v$ is
not minimizing with probability one. This is a new result which uses the
infinitesimal structure of the continuum, and for which there is no equivalent
in discrete lattice models of FPP.
