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?

  • $\begingroup$ What is a "non-Riemannian metric"? a distance? a Finsler metric? $\endgroup$ – YCor Aug 5 '19 at 7:20
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    $\begingroup$ Yes, exactly. I forgot to write it was a non-Riemannian Finsler metric $\endgroup$ – Maryam Ajorlou Aug 5 '19 at 7:31
  • $\begingroup$ My intuition for Finsler geometry is that you try to draw the unit sphere in each tangent space, which in this case you can compare to the usual unit sphere. But that doesn't seem to give much intuition, in particular as to how the Gauss-Finsler curvature manages to be constant. In my mental picture of a surface, it is always embedded in Euclidean 3-space, which makes me a bad topologist. If it has a metric, I imagine it isometrically embedded, which makes me a bad differential geometer. $\endgroup$ – Ben McKay Aug 5 '19 at 10:45
  • $\begingroup$ @Ben McKay, Thank you for sharing your intuition. Do you mean unit balls? I only know little Finsler, but apparently, on Finsler manifolds, the additional structure, F(x,-), Minkowski function helps to define a kind of the norm in each tangent spaces which the norm induces a metric. ( While we have the inner product and so induced metric ) I guess in the end they will have a normed metric vector space in each tangent space. So they have unit balls in each tangent space and some possibilities like having the continuous linear operators. Please correct my word and guess. $\endgroup$ – Maryam Ajorlou Aug 5 '19 at 17:12
  • $\begingroup$ Can the Finsler metric / distance function be approximated by Riemannian one? $\endgroup$ – Vít Tuček Aug 5 '19 at 18:52

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