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If $(M,g)$ is a compact riemannian manifold such that:

$$\Delta_g (g)=\lambda g$$

where

$$\Delta_g=\sum_{i=1}^n \nabla_{e_i}\nabla_{e_i}-\nabla_{\nabla_{e_i}e_i}$$

with $\nabla$ the Levi-Civita connection, $(e_i)$ is an orthonormal basis and $\lambda$ is a number.

Then is $M$ locally a sphere?

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