  We are concerned in this article with a classical topic in spectral geometry
dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of
sectional curvature (resp. holomorphic sectional curvature) of a compact
Riemannian manifold (resp. K\"{a}hler manifold) can be completely determined by
the eigenvalues of its $p$-Laplacian for a \emph{single} integer $p$? We treat
this question under two conditions: gradient shrinking Ricci soliton for
Riemannian manifolds and cohomologically Einstein for K\"{a}hler manifolds. We
show that, with some sporadic unknown cases, this is true for each $p$.
Furthermore, we show that the condition of being isospectral can be relaxed to
a suitable almost-isospectral version.
