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Open AccessArticle

The Role of Spin(9) in Octonionic Geometry

1
Dipartimento di Economia, Università di Chieti-Pescara, Viale della Pineta 4, I-65129 Pescara, Italy;
2
Dipartimento di Matematica, Sapienza—Università di Roma, Piazzale Aldo Moro 2, I-00185 Roma, Italy
*
Author to whom correspondence should be addressed.
This paper is dedicated to the memory of Thomas Friedrich.
Axioms 2018, 7(4), 72; https://doi.org/10.3390/axioms7040072
Received: 20 September 2018 / Revised: 4 October 2018 / Accepted: 6 October 2018 / Published: 12 October 2018
(This article belongs to the Special Issue Applications of Differential Geometry)
Starting from the 2001 Thomas Friedrich’s work on Spin ( 9 ) , we review some interactions between Spin ( 9 ) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin ( 9 ) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin ( 9 ) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin ( 9 ) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley–Rosenfeld planes and to three series of Grassmannians. View Full-Text
Keywords: Spin(9); octonions; vector fields on spheres; Hopf fibration; locally conformally parallel; Clifford structure; Clifford system; symmetric spaces Spin(9); octonions; vector fields on spheres; Hopf fibration; locally conformally parallel; Clifford structure; Clifford system; symmetric spaces
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Parton, M.; Piccinni, P. The Role of Spin(9) in Octonionic Geometry. Axioms 2018, 7, 72.

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