Honam Mathematical J. 42 (2020), No. 1, pp. 75–91 https://doi.org/10.5831/HMJ.2020.42.1.75 ON THE CONSTRUCTION OF PSEUDO-FINSLER EIKONAL EQUATIONS Muradiye C ¸ imdiker ∗ and Cumali Ekici Abstract. In this study, we have generalized pseudo-Finsler map by introducing the concept of semi-Riemannian map and have found pseudo-Finsler eikonal equations using pseudo-Finsler map. After that, we have obtained some sufficient theorems on pseudo-Finsler manifolds for the existence of solutions to the eikonal equation. At the same time, we have introduced a natural definition for the affine maps between pseudo-Finsler manifolds and have reached the affine solutions of them. 1. Introduction Finsler geometry is a kind of differential geometry and originated by P. Finsler (1894-1970) in 1918. It is considered a generalization of Riemannian geometry. In this geometry, tangent spaces carry Minkowski norms instead of the inner products and the geometric objects on tangent vectors depend not only on the base but also on the fibre component. In the last decades, Finsler geometry produced remarkable development. Many papers and books published on this topic [6, 15, 17, 18]. In pseudo-Finsler manifolds, the Finsler metric is only nondegener- ate (rather than on the particular case of Finsler manifolds where the metric is positive definite). This is absolutely necessary for physical and biological applications of the subject. A more restrictive definition of Lorentz-Finsler manifold was given by Beem and essentially adapted by mathematicians working in pseudo-Finsler geometry [4, 5]. Eikonal equation is a relatively simple first-order non-linear partial differential equation which describes many phenomena in the physical Received April 2, 2019. Revised May 6, 2019. Accepted May 16, 2019. 2010 Mathematics Subject Classification. 53C50, 58B20, 78A05. Key words and phrases. Pseudo-Finsler manifold, eikonal, affine map. *Corresponding author