A characterization of Sasakian space forms by the spectrum (Q2407338)

Let $(M,g)$ and $(M',g')$ be two compact connected smooth Riemannian manifolds without boundaries. Denote by $\mathrm{Spec}^p(M,g)$ (resp. $\mathrm{Spec}^p(M',g')$) the spectrum of the Laplace-Beltrami operator on $p$-forms on $(M,g)$ (resp. $(M',g'))$. It is known that if $\mathrm{Spec}^p(M,g)=\mathrm{Spec}^p(M',g')$ for $p=0,1,2$ then $(M,g)$ is an Einstein manifold if, and only if, $(M',g')$ is an Einstein manifold; and $(M,g)$ is of constant sectional curvature $c$ if, and only if, $(M',g')$ is of constant sectional curvature $c$. \par The main result in the paper under review can be stated as follows. Suppose that $(M,g)$ and $(M',g')$ are Sasakian with $\dim(M)=2n+1\neq 15$ and $\dim(M')=2n'+1$, $n,n'\geq 1$. If $\mathrm{Spec}^2(M,g)=\mathrm{Spec}^2(M',g')$ then $n=n'$ and $M$ is a Sasakian space form with constant $\varphi$-sectional curvature $c$ and second Betti number zero if, and only if, $M'$ is so.