# Can a definition make a good question?

There is a question I would like to ask at MO, but it seems somewhat unorthodox compared most others, so I want to get some support (or discouragement) here first. What I want to do is to give a new definition, justify why it is a natural extension of a well known concept, and ask if someone has seen it or has something to say about it. I am trying to extend a theory that I understand to a setting outside my main expertise, and I therefore risk making a trivial question. I have put effort in trying to find research or notes about this new object but to no avail. I do have genuine research interest in this thing, so it is not only a matter of curiosity.

Would something like this make a good question? (I don't see why not, but I have not encountered such questions here yet, so I'm slightly worried.) Is there something that I should be particularly careful about when formulating a question of this kind? Are there examples of good and bad questions like this to learn from?

I chose to make this meta question quite general in order to focus discussion on the idea of definitions as questions rather than the particulars of the question I had in mind. But if you want a short description of my question, I can give one. The full question is lengthy, so I will not produce it here.

A short version of the question I had in mind:

Periodic geodesics on compact Lie groups can be described algebraically, without any reference to minimizing arc length: a periodic geodesic is a mapping $S^1\ni t\mapsto x\phi(t)\in G$ where $\phi:S^1\to G$ is a nontrivial homomorphism and $x\in G$. By analogue, we can define a geodesic on a finite group by replacing $S^1$ with a finite cyclic group. The geodesic flow on finite groups can be seen as a discrete time dynamical system. Have geodesics on finite groups been studied before, perhaps under another name? Does this structure look familiar to anyone? A problem in the field of inverse problems asks whether a function on a closed manifold is uniquely determined by its integrals over all closed geodesics. This problem has been studied quite a lot, also on Lie groups. A natural generalization of the question now asks whether a function on a finite group is uniquely determined by its sums over all geodesics. Has this problem been studied before? Is it known to have applications, abstract or concrete? I have obtained some results on this finite generalization. I am fairly confident that this problem is new to the inverse problems community, but I am not sure if it is well known in another field. It would be great if I could motivate the question or relate it to existing literature. Any ideas, references or analogues could be helpful in understanding and solving the problem.

• It's a bit old but here is one: mathoverflow.net/questions/22050/… – François G. Dorais Oct 15 '14 at 13:24
• "... and ask if someone has seen it or has something to say about it": do you have in mind a reference request, or are you rather interested in people's opinions, or in hints on further directions of research people may give you, or where else is the focus of your question? – Stefan Kohl Oct 15 '14 at 13:27
• @StefanKohl, it is partially a reference request. I would like to know if my object and a related problem appear somewhere else. That would give more context for studying the problem and might also help find useful tools. But useful insight need not come in the form of references. At the moment the problem is merely an abstract generalization of something that has been studied in my field. – Joonas Ilmavirta Oct 15 '14 at 13:32
• On the face of it, it sounds fine to me. But it would help to know what the question is before being sure. – Todd Trimble Oct 15 '14 at 14:09
• @ToddTrimble, I added a short version of my planned question. – Joonas Ilmavirta Oct 15 '14 at 14:48
• I think this is a sophisticated and well-motivated question (or set of questions). – Joseph O'Rourke Oct 15 '14 at 23:14
• In my viewer, the question content appears as one paragraph. I think the content deserves three. Also, I think more success will be had if you add something like "as an analog to thm A in reference R, I have specific result or conjecture RC; Can anyone refer me to something resembling RC or a variant?" Here RC could be an affirmation of the points determined by geodesics for groups of prime power order, and is specific enough to get a response. I feel your post needs something that concrete and specific. – The Masked Avenger Oct 16 '14 at 5:12
• @TheMaskedAvenger, the version above is not the final one. The full question is longer and provides more details, and will of course be split to several paragraphs. I will also give examples of specific groups that I am interested in. Thanks for reminding me that the question should be sufficiently concrete and specific -- it is easy to get carried away presenting everything in full generality. – Joonas Ilmavirta Oct 16 '14 at 7:24
• @JoonasIlmavirta according to your statement " A problem in the field of inverse problems asks whether a function on a closed manifold is uniquely determined by its integrals over all closed geodesics", what type of closed manifolds satisfy this property, some elementary examples?(Any way, I can learn a lot from this your post and I think your Idea is very interesting) – Ali Taghavi Oct 20 '14 at 9:58
• @AliTaghavi, the property is satisfied eg. by Anosov manifolds and thus also by manifolds with negative sectional curvature. It is an open problem to study which other manifolds satisfy this. See the references in the paper I linked in my question. – Joonas Ilmavirta Oct 20 '14 at 11:12