  The off-diagonal heat-kernel expansion of a Laplace operator including a
general gauge-connection is computed on a compact manifold without boundary up
to third order in the curvatures. These results are used to study the
early-time expansion of the traced heat-kernel on the space of transverse
vector fields satisfying the differential constraint $D^\mu v_\mu = 0$. It is
shown that the resulting Seeley-deWitt coefficients generically develop
singularities, which vanish if the metric is flat or satisfies the Einstein
condition. The implications of our findings for the evaluation of the
gravitational functional renormalization group equation are briefly discussed.
