Can Non-Riemannian Geometry Exist Without a Metric?

[zetafunction]

Non-Riemannian Geommetry ??

in Riemann Geommetry one needs a metric to define a distance so

$$ds^{2}= g_{i,j}dx^{i}dx^{j}$$ is a Bilinear form

the idea is can this be generalized to a non-metric Geommetry ? i mean, you define the distance via a function F so

$$ds^{2}= F(x_{i} , x_{j},dx_í} , dx_{j} )$$

so this time we do not have a Bilinear form or we do not have or depend on a metric $$g_{i,j}$$ is this the Non-Riemannian Geommetry ??

 

[quasar987]

The only example I know is Finsler geometry.

 

[Ben Niehoff]

The word "Riemannian" also usually implies that the quadratic form g is positive definite. Then Lorentz metrics constitute "non-Riemannian" geometry, but since changing the signature is not too big of a change, usually we just say "pseudo-Riemannian".

As Quasar mentions, Finsler geometry is another option. Finsler geometry is to Riemannian geometry as Banach space is to Hilbert space. That is, in Finsler geometry, you define a norm, but not an inner product. The norm satisfies the triangle inequality, but there is no notion of angles. There are some other properties that I can't remember.