2013
DOI: 10.1007/s00031-013-9233-x
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Spin(9) geometry of the octonionic Hopf fibration

Abstract: We deal with Riemannian properties of the octonionic Hopf fibration S 15 → S 8 , in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S 1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.2… Show more

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Cited by 19 publications
(19 citation statements)
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“…In Section 7, we revisit W 4 in the G = Spin(9) case, following Ref. [Fri01] as well as our previous work [Orn+13].…”
Section: Proposition 23 ([Cor92]mentioning
confidence: 99%
“…In Section 7, we revisit W 4 in the G = Spin(9) case, following Ref. [Fri01] as well as our previous work [Orn+13].…”
Section: Proposition 23 ([Cor92]mentioning
confidence: 99%
“…However, a specific (and for us convenient) excellent account to the group Spin(10) has been given by R. Bryant in the file [Bry99]. Since the representation of Spin(9) ⊂ SO(16) in Bryant's notes is slightly different from the one used in R. Harvey's book [Har90], and since we used this latter both in our previous papers [PP12, PP13,OPPV13] and in the previous Sections, we need first to rephrase in our context some arguments.…”
Section: The Lie Algebra Spin(10) ⊂ Su(16)mentioning
confidence: 99%
“…Also, the M. Atiyah and J. Berndt paper in Surveys in Differential Geometry [12] shows interesting connections with classical algebraic geometry. Coming to very recent contributions, it is worth mentioning the work by N. Hitchin [13] based on a talk for R. Penrose's 80th birthday, which deals with Spin(9) in relation to further groups of interest in octonionic geometry.The aim of the present article is to give a survey of our recent work on Spin(9) and octonionic geometry, in part also with L. Ornea and V. Vuletescu, and mostly contained in the references [14][15][16][17][18][19][20].Our initial motivation was to give a construction, as simple as possible, of the canonical octonionic 8-form Φ Spin(9) that had been defined independently through different integrals by M. Berger [21] and by R. Brown and A. Gray [22].…”
mentioning
confidence: 99%
“…Here, the Friedrich approach to Spin(9) allows to recognize both the non-existence of nowhere zero vertical vector fields and some simple properties of locally conformally parallel Spin(9)-structures (here, see Theorem 4, Section 7, and Ref. [14]). We then discuss the broader contexts of Clifford structures and Clifford systems, that allow us to deal with the complex Cayley projective plane, whose geometry and topology can be studied by referring to its projective algebraic model, known as the fourth Severi variety.…”
mentioning
confidence: 99%
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