# Scalar Curvature on Compact Symmetric Spaces - Mathematics > Differential Geometry

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Abstract: A classic result by Gromov and Lawson states that a Riemannian metric ofnon-negative scalar curvature on the Torus must be flat. The analogousrigidity result for the standard sphere was shown by Llarull. Later Goette andSemmelmann generalized it to locally symmetric spaces of compact type andnontrivial Euler characteristic. In this paper we improve the results byLlarull and Goette, Semmelmann. In fact we show that if $M,g 0$ is a locallysymmetric space of compact type with $\chi M eq 0$ and $g$ is a Riemannianmetric 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 andGoette, 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 andshow that any metric $g$ on $S^{2m}$ of scalar curvature $\mathrm{scal} g\geq2m-1\mathrm{tr} gg 0$ is a constant multiple of $g 0$.

Author: ** Mario Listing**

Source: https://arxiv.org/