A geometric reduction method for some fully nonlinear first-order PDEs on semi-Riemannian manifolds

Given a semi-Riemannian manifold $(M,⟨\cdot ,\cdot {⟩}_g),$ we use the transnormal functions defined on $M$ to reduce fully nonlinear first order PDEs of the form $$F(x,u,⟨{\mathrm{\nabla }}_{g}u,{\mathrm{\nabla }}_{g}u{⟩}_g)=0,\text{on }M$$ into ODEs and obtain local existence results of solutions which are constant along the level sets of the transnormal functions. In particular, we apply this reduction method to obtain new solutions to eikonal equations with a prescribed geometry.