Trace of Fischer–Marsden Equation on a Riemannian Space

Among the important differential equations on a Riemannian space $\left(M,g\right)$ of dimension n are the static perfect fluid equation (SPFE), namely $fRic-Hess\left(f\right)={\displaystyle \frac{1}{n}}\left(f\tau -\Delta f\right)g$, and the Fischer–Marsden equation (FME), namely $\left(\Delta f\right)g+fRic=Hess\left(f\right)$, where $Ric$ is the Ricci tensor, $\tau$ is the scalar curvature of $\left(M,g\right)$ and $Hess\left(f\right)$ and $\Delta f$ are the Hessian and the Laplacian of the smooth function f. The trace of the FME is $\Delta f=-{\displaystyle \frac{\tau }{n-1}}f$, which we call the TFME, and if we combine the TFME with the SPFE, we observe that it reduces to the FME. Thus, in the presence of the TFME on the Riemannian space $\left(M,g\right)$ the fundamental differential equations SPFE and FME are equivalent. In this article, we consider the presence of the TFME on a Riemannian space $\left(M,g\right)$ and study its impact on the Riemannian space $\left(M,g\right)$. The importance of this study follows from the fact that results obtained for Riemannian spaces admitting solutions to the TFME automatically are generalizations of corresponding results on spaces admitting solutions to the FME. First, we show that for a connected and compact Riemannian space $(M,g)$, $dimM=n>1$, with scalar curvature $\tau$ that admits a nontrivial solution f to the TFME, with the Ricci operator Q satisfying $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$, and with the integral of $Ric\left(\nabla f,\nabla f\right)$ having a suitable lower bound, it is necessary and sufficient that $(M,g)$ is an n-sphere ${\mathbf{S}}^n\left(c\right)$. In addition, we show that a compact and connected space $(M,g)$, $dimM=n>1$, admits a nontrivial solution f to the TFME such that the scalar curvature $\tau$ satisfies $(n-1)c<\tau \le n(n-1)c$ for some constant $c>0$, and the Ricci curvature $Ric(\nabla f,\nabla f)$ is bounded below by $(n-1)c$, if and only if $(M,g)$ is an n-sphere ${\mathbf{S}}^n\left(c\right)$. Finally, it is shown that a connected and compact Riemannian space $(M,g)$, $dimM=$$n>1$, with constant scalar curvature $\tau$ admits a nontrivial solution f to the TFME, with the Ricci operator Q satisfying $Q\left(\nabla f\right)={\displaystyle \frac{\tau }{n}}\nabla f$, if and only if $(M,g)$ is the sphere ${\mathbf{S}}^n\left(c\right)$.