The group of bijective linear transformations of the plane to itself (real matrices with non-zero determinant) naturally induces bijections of the space of lines in the plane to itself, which form a group of self-homeomorphisms of the space of lines. Hence the same group forms a group of self-homeomorphisms of the Möbius band described in the previous paragraph. But there is no metric on the space of lines in the plane that is invariant under the action of this group of homeomorphisms. In this sense, the space of lines in the plane has no natural metric on it.