**Question**
Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be the heat kernel associated to the metric $\mu_A$ resp. $\mu_B$, is then the formula
$$K_A(t;z,z) = K_B(t;z,z) \qquad \forall z \in U $$
true?

**Comments** If we denote by $h(t;z,z)$ the difference of the heat kernels on the open subset $U$, then, since $\Delta_{A|U} = \Delta_{B|U}$, it is a solution of the heat equation $(\partial_t + \Delta)h(t;z) = 0$ with initial condition $h(0;z)=0$ for any $z\in U$; but I don't derive any boundary condition to make the solution of the problem above unique.

I ask this question because I remember having read something similar somewhere, and I would like it to be true; but the comment above seems to indicate that the case is hopeless, is it really the case?

Thank you in advance for your interest!

**Note** This question has originally been posted on Math.SE.