A Finsler geometric surface model for membranes is studied by using the Monte Carlo simulation technique on connection-fixed triangle lattices with sphere topology. An inplane three-dimensional unit vector a is assumed to be the in-plane tilt variable of the surface. The interaction with a is described by the XY-model Hamiltonian. Since this variable a is considered as a vector field on the surface, a Finsler metric is defined by using a. We find that the model has three different phases. They change from the para-magnetic phase to the ferromagnetic and to the glass phases when the strength of the XY interaction increases. Both the para-magnetic and the glass phases are characterized by random configuration of a; the variable a randomly fluctuates in the para-magnetic phase while it is randomly frozen in the glass phase. We also find that the surface becomes spherical in both phases. On the contrary, in the ferro-magnetic phase the surface shape becomes tubular or discotic due to the anisotropic bending rigidity and surface tension coefficient, which are dynamically generated by ordered configurations of a.