  To date, most methods for simulating conditioned diffusions are limited to
the Euclidean setting. The conditioned process can be constructed using a
change of measure known as Doob's $h$-transform. The specific type of
conditioning depends on a function $h$ which is typically unknown in closed
form. To resolve this, we extend the notion of guided processes to a manifold
$M$, where one replaces $h$ by a function based on the heat kernel on $M$. We
consider the case of a Brownian motion with drift, constructed using the frame
bundle of $M$, conditioned to hit a point $x_T$ at time $T$. We prove
equivalence of the laws of the conditioned process and the guided process with
a tractable Radon-Nikodym derivative. Subsequently, we show how one can obtain
guided processes on any manifold $N$ that is diffeomorphic to $M$ without
assuming knowledge of the heat kernel on $N$.
  We illustrate our results with numerical simulations and an example of
parameter estimation where a diffusion process on the torus is observed
discretely in time.
