  Path integrals constitute powerful representations for both quantum and
stochastic dynamics. Yet despite many decades of intensive studies, there is no
consensus on how to formulate them for dynamics in curved space, or how to make
them covariant with respect to nonlinear transform of variables. In this work,
we construct rigorous and covariant formulations of time-slicing path integrals
for quantum and classical stochastic dynamics in curved space. We first
establish a rigorous criterion for correct time-slice actions of path integrals
(Lemma 1). This implies the existence of infinitely many equivalent
representations for time-slicing path integral. We then show that, for any
dynamics with second order generator, all time-slice actions are asymptotically
equivalent to a Gaussian (Lemma 2). Using these results, we further construct a
continuous family of equivalent actions parameterized by an interpolation
parameter $\alpha \in [0,1]$ (Lemma 3). The action generically contains a
spurious drift term linear in $\Delta \boldsymbol x$, whose concrete form
depends on $\alpha$. Finally we also establish the covariance of our
path-integral formalism, by demonstrating how the action transforms under
nonlinear transform of variables. The $\alpha = 0$ representation of time-slice
action is particularly convenient because it is Gaussian and invariant, as long
as $\Delta \boldsymbol x$ transforms according to Ito's formula.
