The Role of Spin(9) in Octonionic Geometry

Abstract

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.Axioms 2018, 7, 72 2 of 32 Coming into the new millennium, since its very beginning, new interest in dealing with different aspects of octonionic geometry appeared, and new features of structures and weakened holonomies related to Spin(9) were pointed out. Among the references, there is, notably, the J. Baez extensive Bulletin AMS paper on octonions [8] as well as the not less extensive discussions on his webpage [9]. Next, and from a more specific point of view, there is the Thomas Friedrich paper on "weak Spin(9)-structures" [10], which proposes a way of dealing with a Spin(9) structure, and this was later recognized by A. Moroianu and U. Semmelmann [11] to fit in the broader context of Clifford structures. 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]. Our construction of Φ Spin(9) uses the already mentioned definition of a Spin(9)-structure proposed by Thomas Friedrich and has a strong analogy with the construction of a Sp(2) · Sp(1)-structure in dimension 8 (see Section 3 as well as [15]). By developing our construction of Φ Spin(9) , we realized that some features of the S 15 sphere can be conveniently described through the same approach that we used. The fact that S 15 is the lowest dimensional sphere that admits more than seven global linearly independent tangent vector fields is certainly related to the Friedrich point of view. Namely, by developing a convenient linear algebra, we were able to prove that the full system of maximal linearly independent vector fields on any S n sphere can be written in terms of the unit imaginary elements in C, H, O and the complex structures that...