Recently I saw a definition of a Bryant sphere, which is a sphere equipped with a non-Riemannian Finsler metric on $S^n$ that satisfies the following properties:

- $ K=1 $,
- All geodesics are great circles,
- All prime geodesics have the same length $ 2π $.

The antipodal map $\Phi$ for a Bryant sphere is $\Phi(x) = −x$, $\forall x ∈S^n ⊂\mathbb{R}^n+1$.

I felt that perhaps a Bryant sphere in the case $n=2$ is the same as the standard sphere $S^2$ which imbeds in $\mathbb{R}^3$, specifically because they have the same antipodal map. But these two spheres are completely different in the case $n=2$. I don't know much about Finsler metrics, but do you think the metric makes a big difference or, perhaps it is better to ask what causes this difference?