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Notes: A recent meta discussion showed green light to asking a question of this kind. There is an earlier question about geodesics on graphs, but it does not answer my question.

Notes: A recent meta discussion showed green light to asking a question of this kind. There is an earlier question about geodesics on graphs, but it does not answer my question.

Notes: A recent meta discussion showed green light to asking a question of this kind. There is an earlier question about geodesics on graphs, but it does not answer my question.

Notes: A recent meta discussion showed green light to asking a question of this kind. There is an earlier question about geodesics on graphs, but it does not answer my question.

• Every subgroup is totally geodesic in both cases (Lie and finite).
• On every nontrivial group there exists at least one geodesic in both cases.
• Geodesics are invariant under left and right translations in both cases.
• A finite abelian group is a product of cyclic groups, and a compact, connected, abelian Lie group is a product of copies of $S^1$ (a torus).

I wanted to consider the corresponding problem on finite groups, where of course the integral is replaced with a sum. That is, is a function on a finite group determined by its sums over all geodesics? It seems that I can give a decent answer, but not a complete classification of groups where the answer is affirmative. My current classification covers, for example, all Abelian, symmetric, alternating, dihedral and dicyclic groups.

If you can answer any of the questions with a modified definition of geodesics, please do so. I am looking for a reasonable definition — or several if there are several — and I certainly do not want to forbid other definitions than mine. A possible variant is described below.

Geodesics as dynamical systems: (For those prefer this point of view.) Geodesics on Lie groups (and all other Riemannian manifolds) can be realized as continuous time dynamical systems. There is a Hamiltonian flow on the cotangent bundle $T^*M$ (or the cosphere bundle $S^*M$) so that the projections of flow lines to $M$ are the geodesics.

A variant: Peter Michor suggested a different formulation in the comments below. I defined geodesics to be translations of cyclic subgroups, but it also makes sense to define geodesics as translations of maximal cyclic subgroups. In this setting a "Lie subgroup" $H$ of the finite group $G$ is such a subgroup that any maximal cyclic subgroup on it is also maximal in $G$. Then there are also non-Lie subgroups but all Lie subgroups are totally geodesic, analogously with the positive dimensional Lie groups. Answers to my questions with this kind of geodesics are also very welcome. (I have not looked at the integral geometry problem in this setting yet.)

• Every subgroup is totally geodesic in both cases (Lie and finite).
• In every nontrivial group there exists at least one geodesic in both cases.
• Geodesics are invariant under left and right translations in both cases.
• A finite abelian group is a product of cyclic groups, and a compact, connected, abelian Lie group is a product of copies of $S^1$ (a torus).

I wanted to consider the corresponding problem on finite groups, where of course the integral is replaced with a sum. That is, is a function on a finite group determined by its sums over all geodesics? It seems that I can give a decent answer, but not a complete classification of groups where the answer is affirmative. My current classification covers, for example, all abelian, symmetric, alternating, dihedral and dicyclic groups.

If you can answer any of the questions with a modified definition of geodesics, please do so. I am looking for a reasonable definition — or several if there are several — and I certainly do not want to forbid other definitions than mine. One possible variant is described below.

Geodesics as dynamical systems: (For those who prefer this point of view.) Geodesics on Lie groups (and all other Riemannian manifolds) can be realized as continuous time dynamical systems. There is a Hamiltonian flow on the cotangent bundle $T^*M$ (or the cosphere bundle $S^*M$) so that the projections of flow lines to $M$ are the geodesics.

A variant: Peter Michor suggested a different formulation in the comments below. I defined geodesics to be translations of cyclic subgroups, but it also makes sense to define geodesics as translations of maximal cyclic subgroups. In this setting a "Lie subgroup" $H$ of the finite group $G$ is such a subgroup that any maximal cyclic subgroup on it is also maximal in $G$. Then there are also non-Lie subgroups but all Lie subgroups are totally geodesic, analogously with the positive dimensional Lie groups. Answers to my questions with this kind of geodesics are also very welcome.