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Question: In which spaces can one define a geometrically meaningful notion of "direction"?

Here I should disambiguate that I mean something analogous to "relative direction", rather than "cardinal/absolute direction". Although answers addressing the latter would also be welcome.

Attempt: One obvious thing that occurs to me is that any notion of relative direction has to vary with the point under consideration. I.e., we would have to be looking for a notion for which, given a space $X$, there is a different set of directions $\mathcal{D}(x)$ for every point $x \in X$.

As discussed in this related question, vector spaces obviously have a well-defined notion of direction with respect to their additive identity. So a sufficient condition for a space to have a well-defined and geometrically meaningful notion of direction, seemingly, might be that there is a canonical vector space corresponding to each point of the space.

An obvious example of such a spaces are differentiable manifolds.

However, is this really a necessary condition for a geometrically meaningful notion of direction?

For example, on Riemannian manifolds, we can use the exponential map around a point $p$, to take directions in $T_pM$ (in the sense of vector spaces relative to their origins) and map them to (local) geodesics on the manifold $M$.

So would it not also be possible to define a notion of direction on the special class of metric spaces called length spaces, by defining some sort of geometrically meaningful equivalence relation on the set of all (local) geodesics leaving a given point $x$ in the space?

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  • $\begingroup$ Your link to relative direction is I think wrong do you mean en.wikipedia.org/wiki/Relative_direction or is this a third type ? Further still puzzling with your question , what of a geometry where vector addition is not communicative $a+b \not= b+a$ $\endgroup$ – Willemien Jul 16 '17 at 8:28
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    $\begingroup$ Sorry I think I made a mistake (for the direction part) for the vector addition part I was thinking about vector addition in the hyperbolic plane (hyperbolic geometry). But maybe you could argue vectors don't exist there $\endgroup$ – Willemien Jul 16 '17 at 12:18
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    $\begingroup$ Sorry you are over my head ,hope somebody else can help you $\endgroup$ – Willemien Jul 16 '17 at 13:16
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    $\begingroup$ As I read the wikipedia page, "relative direction" only makes sense with respect to an orthogonal frame, so (as you say) Riemannian manifolds are one natural setting. It's conceivable you can get by with less (e.g., a conformal structure). The wording of the question, however, suggests to me that you're trying to get at Platonic Truth...? If so, a working framework (e.g., Riemannian geometry) seems likelier to be of use in practice than trying to give "ultimate necessary and sufficient conditions". $\endgroup$ – Andrew D. Hwang Jul 16 '17 at 14:39
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    $\begingroup$ I will throw in a small comment. One can talk about (relative) directions if, say, when given any 2 points on a Riemannian manifold, there is a unique geodesic segment joining these 2 points. Then at either of the endpoints, one can talk about the direction of the other as the unique tangent vector "pointing" towards the other point. One class of such manifolds are Hadamard manifolds, if I remember correctly. $\endgroup$ – Malkoun Jul 16 '17 at 19:40
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This is kind of a crazy idea, but I want to write it down so I don't forget -- I might delete this community wiki answer later. (In particular, it being CW means that upvotes/downvotes to this question don't mean anything.)

Anyway, the primary idea, which may be incorrect, is the following:

the characterizing property of direction is that it is scale-invariant.

This is essentially how direction is defined in the case of vector spaces, so it should probably be the idea in other contexts as well.

Here is how this idea can be implemented in the case of an arbitrary metric space (this definition is similar to that for vector spaces, but does not always coincide -- especially when the vector space is not even metrizable).

Given a metric space $(X,d)$, for each point $x \in X$, let $Sim(X,x)$ denote the local similitude group centered at $x$ (the group of all surjective, equivalently bijective, similitudes which fix $x$).

By convention, with respect to itself, the point $x$ has no direction.

Then we say that, for each point $p$ in $X \setminus x$, the direction of $p$ relative to $x$ is the orbit of $p$ under the action of $Sim(X,x)$. (This is an equivalence class of points in $X$.)

(Being an orbit of $Sim(X,x)$ gives both scale-invariance and the result that the direction is relative to a certain point, since the elements of $Sim(X,x)$ are exactly those which "scale" the space $X$ by a positive constant while still fixing the point $x$, and per definition an orbit is invariant under the action of a group.)

I believe that this definition coincides with the usual one for $\mathbb{R}^n$ with the Euclidean metric -- I don't know if it coincides with the definition for Riemannian manifolds using geodesics instead of the tangent spaces.

On the other hand, I kind of like this definition because it seems very Kleinian.

Update: The above is (almost) COMPLETELY wrong.

In particular, we don't want directions to be invariant under rotations or reflections. However, rotations and reflections correspond to the non-trivial elements of the local isometry groups $Iso(X,x)$ which are obviously subgroups of the local similitude groups $Sim(X,x)$.

So, the "direction" as defined above would be invariant under the action of non-trivial elements of $Iso(X,x)$, which it inherently should not be.

I was confusing $Sim(X,x)$ with something like a "group of pure scalings centered at $x$" ("group of dilations centered at $x$"), which, if and when such a thing could be defined, would clearly be a subgroup, but in most cases a proper subgroup, of $Sim(X,x)$. So the definition above is too coarse of an equivalence relation to successfully define a notion of direction.

My guess is that, if and when a "pure scaling" or "dilation" group exists, denote it $Dil(x)$, then one would have that $$Sim(X,x)/Iso(X,x) \cong Dil(x) \,. $$

This is motivated by the example of Euclidean space, where $Sim(X,x)$ is the conformal group, $Iso(X,x)$ is the orthogonal group, and the group of dilations is isomorphic to $\mathbb{R}_{>0}$.

Now, obviously in the case of real vector spaces, a dilation group always exists, and it is always isomorphic to $\mathbb{R}_{>0}$. Therefore, the following definition generalizes the well-known definition/notion of direction used in real vector spaces.

By convention, the point $x$ has no direction with respect to itself. For any point $p \in X \setminus \{x\}$, the direction of $p$ relative to $x$ is the orbit of $p$ under the action of $Dil(x)$.

Again, since usually $Dil(x) \subsetneq Sim(X,x)$, this definition is strictly different than the originally proposed one.

Anyway, this is an even better definition (I think), because (1) it's still very Kleinian, and (2) it leads to a natural classification of geometric structures -- angle measures are the natural invariants of $Sim(X,x)$, rotations and reflections are the actions of $Iso(X,x)$, and directions are the natural invariants of $Dil(x) \cong Sim(X,x) / Iso(X,x)$, in addition to, of course, distance being the natural invariant of $Iso(X)$.

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The above is also an incorrect/not useful notion -- see this answer which shows that it is not useful on such a basic space as the sphere $\mathbb{S}^2$.

It might be possible to save the notion by changing the definition of local isometry group (and that of local similitude group analogously) -- see this follow-up question.

Also notions of direction in general metric spaces are discussed in Burago, Burago, Ivanov's book, Metric Geometry, on p. 100, p. 243, Section 9.1.8. pp. 318-324, p.352 and Section 10.9 pp. 390-398 (notion of direction in Alexandrov Spaces).

I am not sure if the notion in the aforementioned book is related to the one I mentioned, but the book mentions that for differentiable manifolds it is related to the standard notion of direction using tangent spaces. The notion of direction I was trying to define is also supposed to be related to the notion of angle measure I was trying to define. Burago et al's notion of direction is definitely related to the notion of angle (measure) they define. So whether my notion of direction is related to Burago et al's notion could be addressed also by figuring out/considering whether my notion of angle measure is related to Burago et al's notion of angle.

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