  Given a semi-Riemannian manifold $(M,\langle \cdot,\cdot\rangle_g),$ we use
the transnormal functions defined on $M$ to reduce fully nonlinear first order
PDEs of the form \[ F(x,u,\langle \nabla_g u, \nabla_g u \rangle_g) = 0,\qquad
\text{on }M \] 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.
