us a Minkowski norm on Tx M.
Therefore, a Finsler structure of M may be viewed as a smoothly varying family of Minkowski norms. Basically, this family has rather limited differentiability along the Zero section of the tangant bundle TM. Here, concern ourselves with certain geometrical aspects of Minkowski norms is discussed.
In this paper, two important features of Minkowski norm will be discussed. actually. Every n-dimensional vector space is linearly isomorphic to Rn, whose elements y have the form (y1, … yn). then, there is no loss of generality in confing our discussion to Minkowski norms on Rn.
The function F (x1, …, xn; y1, …, yn) is positive unless all the yi are zero. It is also homogeneous of degree one in y. Take optics for instance. Keep in mind that in an anisotropic medium, the speed of light depends on its direction of travel. At each location x, visualize y as an arrow that emanates from x. Now measure the amount of time it takes light to travel from x to the tip of y, and call the result F(x,y). The hypothesized homogeneity allows us to rewrite the displayed integral as ∫_a^b▒F(x,dx) . This them represents the total time it takes light to traverse a given possibly curved path in this medium. Actually, mathematical ecology provides more esoteric examples. Therefore, we have explicit mathematical examples.