  We consider the geometric inverse problem of determining a closed Riemannian
manifold from measurements of the heat kernel in an open subset of the
manifold. In this paper we analyze the stability of this problem in the class
of $n$-dimensional Riemannian manifolds with bounded diameter and sectional
curvature. It is well-known that a sequence in this class of manifolds can
collapse to a lower dimensional stratified space when the injectivity radius of
the sequence of manifolds goes to zero. We prove the uniqueness of the inverse
problem on the limiting spaces of the collapsing manifolds. As a result, we
obtain stability results for the inverse problem in the class of manifolds with
bounded diameter and sectional curvature.
