  We approximate the heat kernel $h(x,y,t)$ on a compact connected Riemannian
manifold $M$ without boundary uniformly in $(x,y,t)\in M\times M\times [a,b]$,
$a>0$, by $n$-fold integrals over $M^n$ of the densities of Brownian bridges.
Moreover, we provide an estimate for the uniform convergence rate. As an
immediate corollary, we get a uniform approximation of solutions of the Cauchy
problem for the heat equation on $M$.
