In Riemannian geometry the metric tensor is a function of positions only whereas in Finsler geometry it is a function of postions as well as directional arguments both.

Can you explain the reason behind taking different fundamental functions and advantages of doing that ?

Finsler geometry is a pure mathematical field and it is regarded as the generalization of Riemannian one. There are some geometrical models which could be explained only with the help of Finsler metrics. For instance, model of slope of a mountain, models of anisotropic media. You may find several works on these topics. Here, I refer and enclose a famous book (Eds: Bao, et al.) which consists of answer to your question up to some extent (pl. see pp. 200-203).

Here is an intuitive explanation, that I am sure has variants in other countries.

I live in a small city in the east of Holland, a rather small country: the geographic distance from east to west, is about 200km. Geographic distance is what Riemannian distance is modeling and, of course the distance from west to east of the country is the same distance, i.e. about 200 km.

Many of my friends, live in the west, as indeed most of the people in Holland do. I and most of my friends in the east regularly drive to our friends in the west. Because we easterners are used to regular drives to the west, the perception is that it is not so far. My friends in the west, however, are used to most people living near in the west and seldom drive to the east. In their perception, it is actually quite far away. This shows that the _perception_ of distance is more like a Finsler geometry:

d_{percept}(east, west) < d_{percept}(west, east).

There is a second aspect of Finsler geometry. Some Finsler distance functions are symmetric but instead of being defined locally (i.e on the tangent space) by an Euclidean norm it is defined by a Minkowski p-norm.

Lets take p = \infty. Then imagine walking through New York and being at a crossing. Somebody says that your destination is less than three blocks away. Given the New York Grid like street plan that is roughly a "ball" in the max norm i.e. a square. Of course the meaning of three blocks also depends on where you are: the three blocks are relative to your position and if you have an other distance lying around (here geographical distance) a "ball" with radius three blocks in the financial district down town Manhattan has a different size than in Soho.

The metric structure

No precise question - no precise answer.

The standard slogan: ' Objects in Finsler geometry typically depend not only on the position, but also on the direction.' More precisely, some of the possible frameworks for Finsler geometry are:

(1) The double tangent bundle TTM of a manifold M (Grifone's approach).

(2) The vertical subbundle of TTM (Bejancu, Abate-Patrizio,...).

(3) Matsumoto's Finsler bundle (this is a principal bundle).

(4) The pull-back bundle of the tangent bundle by its own projection (perhaps the most natural and surely the most popular approach; see Akbar-Zadeh, P.Dazord, P.Dombrowski (MR 37,#3469), M.Crampin, Bao-Chern-Shen,...).

The different frameworks suggest fundamental differences between Finsler geometry and Riemannian geometry.

Now I recall that a Finsler manifold is a smooth manifold M together with a Finsler function F on its tangent manifold. Requirements:

(F1) F is continuous on TM, smooth on the slit tangent manifold.

(F2) F is positive-homogeneous of degree 1.

(F3) F is positive-definite in the sense that F(v)>0 whenever v is a nonzero tangent vector to M.

(F4) Let E:= 1/2( F square) the energy function associated to F. Then the Lagrange 2-form attached to E is (pointwise) non-degenerate.

We obtain a more illuminating answer to the question if we characterize Riemannian manifolds among Finsler manifolds. One of the simplest result of this kind is the following:

Proposition: A Finsler function F on TM arises from a Riemannian metric on M if, and only if, the energy function associated to F is two times differentiable (and hence it is smooth) on the whole tangent manifold TM.

For a proof see, e.g., [CSF], Prop.9.2.26.

Reference

[CSF] J.Szilasi, R.L.Lovas and D.Cs.Kertesz, Connections, Sprays and Finsler Structures, World Scientific 2014.

Dear Prof. József Szilasi

Thanks for the nice explanation. I have your book in our University Library. I will see the proof of the proposition.