Riemannian geometry was first put forward in generality by Bernhard Riemann in the nineteenth century and was used by Albert Einstein in his general relativity theory. It deals with a broad range of geometries whose metric properties vary from point to point, as well as two standard types of Non-Euclidean geometry, spherical geometry and hyperbolic geometry, as well as Euclidean geometry itself.

Finsler geometry is an old area in differential geometry, and developed very slowly in the last 70 years, due to the complexity of Finsler structures. Roughly speaking, Finsler geometry is to study metric spaces without quadratic restriction on its metric function. One of the fundamental problems in Finsler geometry is to study the characteristic of projectively flat metrics on an open domain in a Euclidean space. This is the Hilbert’s 4th problem in regular case. I’ve discussed the sufficient and necessary condition for a kind of polynomial Finsler metrics, which is defined by a Riemannian metric and a 1-form, to be locally projectively flat. And also, I’ve discussed the sufficient and necessary condition for a special kind of polynomial Finsler metrics to be locally projectively flat, obtained the non-trivial special solution and completely determined the locally structure of those with constant flag curvature.