Title: On the spectral geometry of manifolds with conic singularities,
Abstract: In the previous article we derived a detailed asymptotic expansion of the
heat trace for the Laplace-Beltrami operator on functions on manifolds with
conic singularities. In this article we investigate how the terms in the
expansion reflect the geometry of the manifold. Since the general expansion
contains a logarithmic term, its vanishing is a necessary condition for
smoothness of the manifold. In the two-dimensional case this implies that the
constant term of the expansion contains a non-local term that determines the
length of the (circular) cross section and vanishes precisely if this length
equals $2\pi$, that is, in the smooth case. We proceed to the study of higher
dimensions. In the four-dimensional case, the logarithmic term in the expansion
vanishes precisely when the cross section is a spherical space form, and we
expect that the vanishing of a further singular term will imply again
smoothness, but this is not yet clear beyond the case of cyclic space forms. In
higher dimensions the situation is naturally more difficult. We illustrate this
in the case of cross sections with constant curvature. Then the logarithmic
term becomes a polynomial in the curvature with roots that are different from
1, which necessitates more vanishing of other terms, not isolated so far.