Scalar Curvature on Compact Symmetric Spaces

Abstract

A classic result by Gromov and Lawson states that a Riemannian metric of non--negative scalar curvature on the Torus must be flat. The analogous rigidity result for the standard sphere was shown by Llarull. Later Goette and Semmelmann generalized it to locally symmetric spaces of compact type and nontrivial Euler characteristic. In this paper we improve the results by Llarull and Goette, Semmelmann. In fact we show that if $(M,g_0)$ is a locally symmetric space of compact type with $χ(M)\neq 0$ and $g$ is a Riemannian metric on $M$ with $\mathrm{scal}_g\cdot g\geq \mathrm{scal}_0\cdot g_0$, then $g$ is a constant multiple of $g_0$. The previous results by Llarull and Goette, Semmelmann always needed the two inequalities $g\geq g_0$ and $\mathrm{scal}_g\geq \mathrm{scal}_0$ in order to conclude $g=g_0$. Moreover, if $(S^{2m},g_0)$ is the standard sphere, we improve this result further and show that any metric $g$ on $S^{2m}$ of scalar curvature $\mathrm{scal}_g\geq (2m-1)\mathrm{tr}_g(g_0)$ is a constant multiple of $g_0$.