Scalar Curvature on Compact Symmetric Spaces

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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,g₀) is a locally symmetric space of compact type with χ(M)≠ 0 and g is a Riemannian metric on M with scalg· g≥ scal₀· g₀, then g is a constant multiple of g₀. The previous results by Llarull and Goette, Semmelmann always needed the two inequalities g≥ g₀ and scalg≥ scal₀ in order to conclude g=g₀. Moreover, if (S²ᵐ,g₀) is the standard sphere, we improve this result further and show that any metric g on S²ᵐ of scalar curvature scalg≥ (2m-1)trg(g₀) is a constant multiple of g₀.