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In this paper we put together some tools from differential topology andanalysis in order to study second order semi-linear partial differentialequations on a Riemannian manifold $M$. We look for solutions that areconstants along orbits of a given group action. Using some results obtained byHelgason in [J DIFFER GEOM,6(3), 411-419] we are able to write a (reduced)second order semi-linear problem on a submanifold $\Sigma$. This submanifoldis, in certain sense, transversal to the orbits of the group actions and itsexistence is assumed. We describe precise conditions on the Riemannian Manifold$M$ and the submanifold $\Sigma$ in order to be able to write the reducedequation on $\Sigma$. These conditions are satisfied by several particularcases including some examples treated separately in the literature such as thesphere, surfaces of revolution and others. Our framework also includes thesetup of polar actions or exponential coordinates. Using this procedure, we areleft with a second order semi-linear equation posed on a submanifold. Inparticular, if the submanifold $\Sigma$ is one-dimensional, we can use suitabletools from analysis to obtain existence and properties of solutions.

On polar actions invariant solutions of semilinear equations on manifolds