Comments on La forme élégante du plan projectif complexe

Ruxor (2016-11-24T21:50:41Z)

@jonas (contd.): Concerning your question on the "other direction" (i.e., the complex structure) on the complex projective plane, yes, it can be defined purely locally from the Riemannian structure (except for orientation): namely, given one direction at a point, you choose the other direction such that the sectional curvature between the two directions is at great as possible, where "sectional curvature" of two directions means (at least roughly) the Gaussian curvature of the surface element defined around the point being considered by starting geodesics from that point in the plane defined by the two directions. (More generally, the sectional curvature is directly connected to the holomorphic angle between the two directions.) This can probably be reformulated purely in metric terms (avoid differential topology), but it would be messy and I'd probably get it wrong.

Concerning your other question, the real lines, complex lines and real planes of the complex projective plane are exactly the subsets of the complex projective plane which are isometric (as metric subspaces for the induced metric) to the abstractly defined real line (i.e., circle with circumference π), complex line and real plane: these are automatically submanifolds, and they are totally geodesic in the sense that the distance given by the induced Riemannian metric coincides with the induced distance on the subspace (so there is no ambiguity as to what the distance means). As for what the embedding looks like on a coordinate system, another thing we can say is that there is always an isometry (indeed, a motion) of the global space which places them in the position of the "standard" real line (meaning the set of (u:v:w) such that w=0 and v/u is real), complex line (w=0) or real plane (v/u, w/v and w/u all real), so we can also define real lines, complex lines and real planes as images of these "standard" ones under the unitary group.

Ruxor (2016-11-24T15:45:22Z)

@jonas: You're right with your counterexample, I overlooked something in my previous answer.

Here's a statement of Wang's result that I hope is correct this time: if X is a compact connected two-point homogeneous metric space with metric d, then (1) there exists a (different) metric D on X, which is equivalent to d in the sense that there exists a continuous increasing function f for which D(x,y) = f(d(x,y)) (in particular, the identity gives a homeomorphism from (X,d) to (X,D)), and with the additional property that (X,D) is "convex" as a metric space, meaning that for any two distinct points x,y, there exists z, different from x and y, such that D(x,y)=D(x,z)+D(z,y), (2) this D is unique up to a constant factor (given (X,d) a compact connected two-point homogeneous space, there exists a unique up to a scale factor convex metric D on X which is equivalent to d) and (3) (X,D) (which obviously is still compact connected two-point homogeneous) is isometric (up to a scale factor) to a sphere or to a real/complex/quaternionic/octonionic projective space.

So the fix for metric spaces is to add the "convex" hypothesis, which is harmless since it can always be made so, but which is essential in fixing the metric. (Or agree to consider metric spaces up to equivalence of metrics. I must admit that I don't have a very clear view of what this equivalence relations gives in general, and to what extend we can find a distinguished representative in a more general context. But here, "convex" will do.) This "convex" metric is the one given by the Riemannian structure. We can define a "geodesic arc" between x and y to be a curve γ:[0;ℓ]→X such that is an isometric embedding (D(γ(u),γ(v))=|u−v| for all u,v): then not only is X (the compact connected two-point homogeneous space with a convex metric) convex but also there exists a geodesic arc between any two points.

It should be possible to define every concept at the metric space level: for example, define a "totally geodesic sphere" to be a subspace Y of X (where X has its convex metric) which for the induced metric is isometric to a Euclidean sphere with its Riemannian distance-on-the-surface (so it's also convex), i.e., an isometric embedding of a sphere in X; note in particular that a geodesic on the sphere Y connecting two points of Y is still a geodesic in X. Then unless I'm mistaken, the totally geodesic 1-spheres, resp. 2-spheres, resp. 4-spheres and resp. 8-spheres in the real/complex/quaternionic/octonionic projective spaces are exactly the real lines, resp. complex lines, resp. quaternionic lines, resp. octonionic lines. But I won't try to state all the concepts like that, because I would make many mistakes; but I'm pretty sure "everything" can be defined purely in terms of the metric (once we make it convex, and using its convexity to our advantage). I find it much clearer to do so at the Riemannian level, or even better, at the coordinate level.

The way I see it, once we have the classification result that all connected Riemannian manifolds that are homogeneous and isotropic, or all compact connected metric spaces with a convex metric that are two-point homogeneous, are isometric to <…list…>, and since we understand the (self-)isometry group of all the spaces of the list (and how they behave in terms of a coordinate description), we are then quite justified in studying these spaces from their coordinates (or even, forget about the classification result which justifies their interest and look at them for their own sake).

I'll try to answer your other questions later.

jonas (2016-11-24T12:28:11Z)

It's not that I don't like differential geometry. Indeed, it sounds like differential geometry is probably the best way to study this question. The problem is rather that I don't know much about differential geometry, and about Riemannian manifolds in particular. This surprised me, because your mathematical blog posts are usually written with so much introduction that much of it is accessible to people with fewer higher maths knowledge. In this article in particular, you start with the easier topic of introducing the line incidence structure of the real projective space, so I didn't guess that knowledge of Riemannian geometry would be useful to make sense of the statements. Thus, any of the specific problems I still have might just be because I don't have the differential geometry background.

Suppose I try to use only metric spaces. In the reply you state that any compact connected metric space that is two-point homogeneous is automatically a Riemannian manifold. But the way I interpret this naively, this seems false. Consider a radius 1/2 sphere in the euclidean 3-space, with the usual metric of the 3-space restricted on the sphere. This is a (compact connected) metric space, and is two-point homogenous in the metric sense. This space looks very much like the complex projective line with the metric you give: they're homeomorphic as topological spaces, and the metrics are locally similar in that for any two points with distance at most $ \\epsilon $ in either of them, the distances in the two spaces differ by at most $ O(\\epsilon^2) $. But the two metrics aren't the same globally, because the diameter of this sphere is 1, not \\pi/2. It doesn't seem likely that you could get this metric on the sphere as the arc-length metric of any Riemannian manifold on this topology. Perhaps there is a sense how this space is still equivalent to the real projective line, but that equivalence appears to require differential geometry, so any properties that use the metric globaly will not automatically transfer.

Now, in the reply, you say that

> real lines are geodesics so they have to be preserved by isometries, complex lines are totally geodesic spheres of maximal curvature (namely 4), and real planes are totally geodesics sphere of minimal curvature (namely 1)

That sounds like you are defining those from local properties of the metric, or better, directly from the Riemannian structure. But I don't know what a geodesic sphere means, and I'm probably far from seeing why these definitions work and what properties are easy to prove from them. So, I have further questions.

In the complex projective space, I have a point and a direction in it. If you take the real line in this direction, and then the unique complex line through this real line, and then this complex line defines another direction in the point that is orthogonal to the first direction. You then use this second direction to define the hermitian angle. Is this second direction a local second order property of the space as a Riemannian manifold, in that you can define it locally from the curveture, without using the global structure of the space?

Secondly, I'd still like to know in what sense the real planes and complex lines of the complex projective space look like the real projective plane and the complex projective line respectively. If we're working with Riemannian manifolds, this should eventually claim an isomorphism between two Riemannian manifolds. But my problem here is that even for (compact boundaryless connected real) smooth manifolds, I don't know how to define that a 2-dimensional manifold is a subspace of a 4-dimensional manifold. Can you point me to a definition of how to restrict the charts of a 4-dimensional manifold for a suitable subspace of a 4-dimensional manifold to get a 2-dimensional submanifold, to be able to define such a subspace? Or does such a definition not exist when you define a smooth manifold with smooth charts, but only with the more involved coordinateless definitions? Even in that case, can you point to what such a subspace means? Eventually I'd need such a subspace relation even for the (compact connected) Riemannian manifolds you're talking about here. Without such a definition, it is hard to make sense of why you're calling those subsets real space and complex line in particular.

The easy way out might be if you study these geometries in a global sense from the start, such as equipped a point-line incidence structure, or with an isometry group, in both cases still equipped with a topology. Perhaps if you're talking only about homogenous and isotropic spaces, it is still possible to give a definition of what these spaces are that you're interested in. But in that case, if you don't know the classification theorem for the Riemannian case in first place, there seems to be a danger that you lose something because this global definition could allow fewer spaces.

Ruxor (2016-11-24T10:45:26Z)

@jonas: You're right, I swept a lot of dust under the rug. It turns out that for many imprecise statements, there are a lot of ways to make them precise, but I suppose I should have been clearer.

By "homogeneous and isotropic space", I mean a connected Riemannian manifold (i.e., [paracompact] differentiable manifold with a — smoothly varying — positive definite quadratic form on the tangent space; this can then be made into a metric space in a standard way) such that the group of [self-]isometries (A)acts transitively on the points (i.e., given two points, there is an isometry taking the first to the seecond), and (B)acts transitively on the unit vectors at a point (i.e., given two unit vectors at the same point, there is an isometry fixing this point and whose differential takes the first vector to the second). It turns out (it's not hard to show but not completely trivial either) that this is equivalent to asking that the Riemannian manifold is "two-point homogeneous" in the sense that (*)for any two pairs of points with the same distance, there is an isometry taking the first pair to the second pair (as ordered pairs). Here, "isometry" means any map preserving distance, and no other structure; an isometry of metric spaces is obviously continuous, and an isometry of Riemannian manifolds is necessarily infinitely differentiable, which helps explain why (B) above makes sense.

Now if you don't like differential geometry, the property (*) also makes sense for arbitrary metric spaces. I don't think we can say much about (*) alone on metric spaces, but if we add the assumptions that the metric space is compact and connected, then it turns out (this is a result by Wang in 1952) that it is automatically a Riemannian manifold with the properties above. (But I don't know if we can formulate the non-compact / hyperbolic case without ever speaking of manifolds.)

If we consider a connected Riemannian manifold which is homogeneous and isotropic (or equivalently, two-point homogeneous) as explained above, then the result by Wang (for the compact / positively curved case) and Tits (for the general case) is that it is necessarily [isometric to] a Euclidean space, or a sphere, or a real projective space, or a complex projective space, or a quaternionic projective space, or the octonionic projective plane, or a real hyperbolic space, or a complex hyperbolic space, or a quaternionic hyperbolic space, or the octonionic hyperbolic plane. And as I just said, the same result holds for compact connected metric spaces which are two-point homogeneous (but then we only get the "sphere" and "projective space" cases).

But additionally, we have a complete description of the isometry groups in each of these cases (they're all very friendly Lie groups), these isometry groups preserve not only the distance but also just about any structure you can imagine on these spaces: incidence structures, real projective subspaces, etc. (Essentially the only "catch" is that the complex projective/hyperbolic spaces have anti-holomorphic isometries, which take the coordinates to their complex conjugates: in the quaternionic/octonionic case you can't do that, and in the real case there is nothing to be done; but even these anti-holomorphic isometries preserve the subspaces one is interested in.) Also, these isometries preserve all kinds of angles, at least if we are willing to forget about orientation (anti-holomorphic isometries will take holomorphic angles to their opposite, for example).

I don't know if there is a very simple proof (of the fact that isometries must preserve various kinds of subspaces and various kinds of angles) that avoids the classification, but you can get a feeling of why isometries of the complex projective plane must preserve real lines, complex lines and real planes: real lines are geodesics so they have to be preserved by isometries, complex lines are totally geodesic spheres of maximal curvature (namely 4), and real planes are totally geodesics sphere of minimal curvature (namely 1); as all of this is defined purely in terms of curvature (hence, distance), it has to be invariant by isometries.

There may be more dust swept under the rug in various places, but I hope this more or less clears up your questions.

And yes, "gnomonique" is pronounced with /gn/ in French (_not_ /ɲ/). At least, the word "gnomon" (the rod on a sundial) is pronounced like that, I'm not sure I've ever actually heard someone say "gnomonique" out loud. For what it's worth, "gnome" and "gnou" are also pronounced with /gn/.

jonas (2016-11-24T01:26:54Z)

Thinking further, there's a few more statements you'd need to make sure that some of what you say isn't abusing the notation.

If you're talking about metric spaces, you'd have to prove that the hermetian angle you define on the complex projective line and the complex projective space are indeed metric. This is probably not very hard to prove directly, but it has to be stated separately, because I'm not very familiar with a hermetian angle in first place. In any case, this must follow from the embeddings to larger dimensional real spheres you give, but that might give an unnecessarily complicated proof.

Next, you'd need to prove that any complex line in the complex projective space is isometric to the complex projective line with the metric you defined. This is also not clear a priori, again because I don't know anything about the hermetian angle. Similarly, you'd need to prove that any real plane in the complex projective space is isometric to the real projective space with the elliptic metric you defined.

If that were true, it would probably give a way to answer some of my earlier questions. You say in the article that in the complex projective plane, a complex line is the set of points polar to a fixed point, where polar means the distance is exactly $ \\pi/2 $. It is possible then that you can define a real line of the complex projective plane as the set of points polar to two fixed points that are polar from each other. Is that true? If so, that would show that a real line of the complex projective plane is also a property that depends only on the metric. You'd still have to show why a real plane can also defined from the metric, as well as that you can define the geometric angle and the hermitian angle (as defined later) from just the metric.

jonas (2016-11-24T00:32:15Z)

In chapter “Le plan projectif complexe”, “Si $ w \\ne 1 $” is a typo for “Si $ w \\ne 0 $“.

In the chapter “Droites complexes, plans réels et droites réelles” I believe “ssi si ces points sont sur un même plan réel” is a typo for “ssi ces points sont sur un même plan réel”.

And a stupid question: in French, is “gnomonique” pronounced with a /gn/ at start?

Now for my more substantial problem about the middle part of this article.

You say that the complex projective plane is homogeneous and isotropic, and define the meaning of those properties from some isometries. What is the definition of an isometry here? I don't understand what structure an isometry has to preserve. Is it only the metric on these set of points? But if it were so, then you'd have to prove that the terms you talked about above, namely the real lines and the real planes of the complex projective plane, can also be defined from the metric. That would be equivalent to proving that an isometry of the complex projective space takes a real line to another real line, and a real plane to another real plane.

Even if you could prove that though, it would not clear all my worries. You say that the complex projective plane is an important example, because it's one of the simplest spaces that are homogeneous and isotropic but aren't maximally symmetric. I don't understand what kind of spaces you're considering in first place. Can you give a formal definition for a homogeneous and isotropic space, at least in some narrow sense that fits the spaces mentioned in this article? That is, this definition would be such that at least the complex projective plane, the complex projective line, the real projective plane, and hopefully also the real projective space of any finite dimension satisfies it. Ideally, you'd also have to define an isometric mapping such a space to another space, so you can talk about the isometry groups of a space, embedding a space to a larger one, and such that all the properties you care about are invariant to isometry. Now there's a small chance that I wouldn't understand some of the definitions involved here, if they require such differential geometry that goes over my head, but I hope this is not the case.

Ruxor (2016-11-23T08:54:08Z)

@jonas: There's no particular logic to the capitalization of Arccos (but capitalization of math operators is generally inconsistent: some people write "Ker" and "Im" where others write "ker" and "im", for instance; some people distinguish "Im" for the imaginary part of a complex and "im" for the image of a function, but others don't, and sometimes it's the other way around…). I just thought it stood out better. (The other think you point out is, indeed, a typo, and is now fixed.)

jonas (2016-11-22T23:16:41Z)

(I've only read the introduction part so far.)

Why are you writing $ \\mathrm{Arccos} $ for the inverse cosine function, when the books I've seen call it $ \\mathrm{arccos} $ instead? See eg. chapter 4.4 in the classical Abramowitz-Stegun handbook <URL: http://www.convertit.com/Go/ConvertIt/Reference/AMS55.asp?Res=150&Page=79&Submit=Go >, or chapter 4.23 of the handbook <URL: http://dlmf.nist.gov/4.23 >.

Under “La droite projective (ou elliptique) complexe, ou sphère de Riemann”, ”$ \\surd((|u|^2+|v|^2)\\cdot(|y'|^2+|v'|^2)) $” is probably a typo because there's no $ y' $ in scope.

Bob (2016-11-22T17:27:53Z)

Merci David, comme d'habitude tu traites d'un sujet fondamental d'une façon originale et avec une multitude de points de vue. C'est vraiment agréable et instructif.

Surt (2016-11-21T12:50:47Z)

J'adore !

Coquille dans "les transformations du plan elliptique sont toutes les … tandis que celles du plan elliptique …"

Fred le marin (2016-11-20T23:03:03Z)

J'ai vu de la lumière, et je suis entré…

Le fait qu'il existe des matrices de translation est une certaine surprise, car c'est a priori impossible pour celui qui ignore tout des coordonnées homogènes.
(il connait par ex. les matrices de rotation, mais dans un espace vectoriel - non affine).
Incontournable cependant pour la 3D (en informatique etc.).
Ce qui donne déjà (au moins) une application utile de ces bestioles abstraites.
Est-ce que des théories de physique (supercordes, supergravité) à bcp de dimensions (ex: 11) utilisent indirectement des espaces projectifs ?
La M.Q. utilise bien, dans son formalisme de calcul, des opérateurs hermitiens : alors rêvons.
Au bas mot, les notions d'homogénéité/isotropie semblent cruciales en physique pour énoncer (par exemple) la conservation de l'énergie totale.
(les lois de l'Univers sont invariantes par translation en temps <=> il y a conservation de l'énergie : un fameux corolaire d'un théorème de Noether)
En espérant que les habitants de la droite à l'infini ne verront pas trop de bourdes dans mon laïus nocturne (certes peu généreux face à la copieuse entrée du jour).


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