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Given a submanifold $N$ in a Riemannian manifold $(M, g)$, we have the notion of normal bundle $$\nu(N) := \{(p,v) \in TM \; | \; p \in N, \, g(v, w) = 0 \, \forall w \in T_p N\}.$$ It is well-known that a distance minimimizing geodesic from $N$ has to start in the normal bundle. That is, if $\gamma :[0,1] \to M$ is a unit speed geodesic satisfying $\gamma(0) \in N$ and $\text{dist}(N, \gamma(t)) = t$ for some time $t \in [0,t_0] \subset[0,1]$, then $\dot\gamma(0) \in \nu(N)|_{\gamma(0)}$. Here $\text{dist}(\_,\_)$ is the induced distance.

Now, suppose we have a Finsler manifold $(M,F)$ and a submanifold $N \subset M$. Is there any notion of 'normal bundle' of $N$ here, so that the initial velocity of any distance minimizing geodesic from $N$ must lie in this normal bundle?

Any references or comments regarding this will be highly appreciated. Cheers!

ChesterX
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  • There is a natural definition of the normal bundle in terms of the Finsler metric, and the initial velocity of any distance minimizing geodesic from the submanifold must lie in this normal bundle. –  Feb 20 '23 at 19:25
  • If you want to use minimizing geodesics, it is likely that you need smoothness of norms in tangent spaces. – Moishe Kohan Feb 20 '23 at 23:05
  • @MoisheKohan I am going by the usual convention of assuming $F : TM \to [0, \infty)$ to be continuous, and to be smooth outside the $0$-section. Isn't this enough to consider minimizing geodesic from a submanifold? – ChesterX Feb 21 '23 at 13:17
  • Just consider the $l_1$-metric on the plane and observe nonuniqueness of minimizing geodesics... – Moishe Kohan Feb 21 '23 at 13:18
  • @MoisheKohan I don't see why uniqueness is an issue here! I am not asking for unique minimizing geodesic. – ChesterX Feb 21 '23 at 13:27
  • So, it does not bother you that if $M\subset R^2$ is the $x$-axis, then the space of tangent directions to minimizing (to $M$) geodesics starting at $p=(0,0)$ is not a linear subspace of the tangent plane at $p$? – Moishe Kohan Feb 21 '23 at 13:40
  • @MoisheKohan I think this is what my question is about. In your example, let $\nu_p \subset T_p \mathbb{R}^2$ be the collection of all directions at $p \in M \subset \mathbb{R}^2$, along which we get minimizing (to $M$) geodesics. If $\nu_p$ is not a linear subspace, then what other properties can we expect? Can we characterize $\nu_p$ in any other way? – ChesterX Feb 21 '23 at 13:52

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