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We study overdetermined problems for fully nonlinear elliptic equations insubdomains $\O$ of the Euclidean sphere $\mathbb{S}^{N}$ and the hyperbolicspace $\mathbb{H}^{N}$. We prove, the existence of a classical solution to theunderlined equation forces $\O$ to be a geodesic ball in the ambient space. Ourresult extends to fully nonlinear equations, a similar result in the case ofsemilinear equations with the Laplace operator due to Kumaresan and Prajapat.

Overdetermined problems for fully nonlinear equations in space forms