I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space.

One can get a formula for the Fubini-Study metric on the complex projective plane via a pullback of the scalar product $\langle A , B \rangle = \mathrm{Tr}\, AB$ on the space of hermitian matrices along the mapping $x \mapsto \frac{x x^\dagger} {x^\dagger x}$.

I have extended this calculation to the case of the octonionic projective plane plane and in the process I've noticed that it actually works also for the hyperbolic plane and indefinite signatures in general. One just has to use $G$-hermitian matrices, for some symmetric bilinear form $G$. Explicitely, one can obtain the hyperbolic metric via pullback along $$ x \mapsto \frac{x(Gx)^\dagger}{(Gx)^\dagger x}. $$ The target space is the space of matrices that satisfy $A^\dagger G = GA$ and the scalar product $\langle A , B \rangle = \mathrm{Tr}\, AB$ is no longer necessarily positive definite.

I suspect this is well known in the classical case but it's very hard to google references because there is so much material. Moreover, I suspect this first appeared in the 19th century.

Q1: What should I cite as a reference for the classical case of (real or complex) hyperbolic plane?

Q2: Has this construction been actually used in the octonionic (or quaternionic) setting before?

  • $\begingroup$ It seems to me that the question title (addressed by @IgorRivin in an answer) is different from the actual text-question. That is, the Fubini-Study metric on the projective space (= the Borel-HarishChandra compactification of the hyperbolic plane, yes, ...) does not restrict to be the hyperbolic geometry on the hyperbolic plane. What is actually wanted? $\endgroup$ – paul garrett Sep 8 '15 at 18:40
  • $\begingroup$ @paulgarrett: I've edited the question. Is it more clear now? I don't want to go into details too much. I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space. $\endgroup$ – Vít Tuček Sep 8 '15 at 19:35
  • $\begingroup$ If you are still interested in this: For the construction of the octonionic hyperbolic plane along these lines, see Mostow's book "Strong rigidity of locally symmetric spaces". I think it is more complicated than your description and uses Jordan algebras to do the "linear algebra" in this situation. $\endgroup$ – Misha Jul 10 '16 at 16:32
  • $\begingroup$ @Misha Thank you, I am aware of Mostow's book. (And it was not easy to get that text. :() But IIRC he writes down the metric distance function and appeals to some general principles. I haven't found the simple elementary calculation that I'm proposing here. $\endgroup$ – Vít Tuček Jul 10 '16 at 16:56

I believe that this idea goes back to Minkowski, where he defines the symmetric space of SL(n) as the positive semi-definite cone (with action by similarity) - in the case of n=2, the set det = 1 is precisely the hyperbolic plane.

  • $\begingroup$ Thank you! Would you happen to now a specific paper or a monograph which I can cite? $\endgroup$ – Vít Tuček Sep 8 '15 at 19:37
  • $\begingroup$ Well, this is certainly not the best, but it's something: math.umbc.edu/~gowda/tech-reports/trGOW11-03.pdf $\endgroup$ – Igor Rivin Sep 8 '15 at 19:41
  • $\begingroup$ @VítTuček A (not THE) canonical paper is Selberg's classic harmonic analysis and discontinuous groups, collected works, vol 1, p. 432 $\endgroup$ – Igor Rivin Sep 8 '15 at 21:57
  • $\begingroup$ I'm sorry, but Selberg doesn't seem to contain what I'm looking for. $\endgroup$ – Vít Tuček Sep 10 '15 at 13:35

Certainly the general idea goes back at least to Minkowski, and/or Poincare, maybe Beltrami. (Poincare's model of the hyperbolic plane had geodesics arcs of circles orthogonal to the boundary, while Beltrami's geodesics were Euclidean straight lines (intersected with the disc).)

Various of C.L. Siegel's IAS notes talk about the classical non-compact homogeneous spaces of hermitian type at great length, c. 1960, as though this were all well-known, and, indeed, his 1939 paper on what we would call "Siegel modular forms" seems to take the general idea for granted.

Jacques Tits has several papers (look at MathSciNet) that use the octonions to model the exceptional domains and groups. I remember first seeing such examples in the book of Walter Bailey, Jr. "Introductory Lectures on Automorphic Forms".

See also J.P. May's little book on "Exceptional Lie groups/algebras"...

The Wiki entry on "classical groups" gives a good outline of many things... One point is that, although E. ("Poppa") Cartan and Killing had models of the "classical" Lie groups and related things c. 1890, the exceptional groups and domains did not have known models until perhaps Chevalley in the late 1940s.

In courses of G. Shimura at Princeton in the mid-1970s, the classical groups and domains were mentioned as common-places... without references, mostly.

  • 1
    $\begingroup$ BTW, Poincare model is due to Beltrami. $\endgroup$ – Anton Petrunin Sep 9 '15 at 0:18
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    $\begingroup$ @AntonPetrunin, is that so? I had the impression that Beltrami had the straight-line idea slightly earlier, and then Poincare' the circle model a few years later... but this is only a vague recollection. $\endgroup$ – paul garrett Sep 9 '15 at 0:35
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    $\begingroup$ Many have such impressions ;) --- read his "Teoria fondamentale degli spazii di curvatura costante", Annali. di Mat., ser II, 2 (1868), 232--255. $\endgroup$ – Anton Petrunin Sep 9 '15 at 13:19
  • $\begingroup$ @AntonPetrunin, thanks for the reference! :) $\endgroup$ – paul garrett Sep 9 '15 at 13:38
  • $\begingroup$ @paulgarrett Thank you! Unfortunately neither Baily nor May are in our library. I'll check Tits next week as I don't have acces to mathscinet right now. $\endgroup$ – Vít Tuček Sep 10 '15 at 13:53

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