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The off-diagonal heat-kernel expansion of a Laplace operator including ageneral gauge-connection is computed on a compact manifold without boundary upto third order in the curvatures. These results are used to study theearly-time expansion of the traced heat-kernel on the space of transversevector fields satisfying the differential constraint $D^\mu v_\mu = 0$. It isshown that the resulting Seeley-deWitt coefficients generically developsingularities, which vanish if the metric is flat or satisfies the Einsteincondition. The implications of our findings for the evaluation of thegravitational functional renormalization group equation are briefly discussed.

Off-diagonal heat-kernel expansion and its application to fields with differential constraints