Spin(9) and almost complex structures on 16-dimensional manifolds

Abstract

Abstract. For a Spin(9)-structure on a Riemannian manifold M 16 we write explicitly the matrix ψ of its Kähler 2-forms and the canonical 8-form Φ Spin(9) . We then prove that Φ Spin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Φ Spin(9) and Φ 2 Spin(9) in the special case of holonomy Spin(9).

On a characteristic of the first eigenvalue of the Dirac operator on compact spin symmetric spaces with a Kähler or Quaternion-Kähler structure

“…This canonical 8-form was first introduced in [BG72] by means an integral formula. Explicit algebraic expressions of this form are far from being simple [BPT85], [AM96], [LGM10], [PP12]. Roughly speaking, the form is constructed by means of the Kähler 2-forms associated to the almost complex structures J αβ := I α • I β , [LGM10], [PP12].…”

“…Explicit algebraic expressions of this form are far from being simple [BPT85], [AM96], [LGM10], [PP12]. Roughly speaking, the form is constructed by means of the Kähler 2-forms associated to the almost complex structures J αβ := I α • I β , [LGM10], [PP12]. To avoid an explicit expression, we will use here the fact that it may be expressed in terms of "higher Casimir operators", see §125 and §126 in [Žel73] or [Hom04].…”

Spheres with more than 7 vector fields: All the fault of Spin(9)

“…287-289] and [9,17]), so that the 36 compositions I α I β are complex structures on R 16 . We will see how the eight complex structures J 1 , .…”

On the Cohomology of Some Exceptional Symmetric Spaces

“…Namely, if ν l denotes the volume form on the line l = {(x, mx)} or l = {(0, y)} in O 2 , a computation shows that , cf. [19], pp. 338-339.…”

“…Aim of the present paper is to describe how, basing on the recent work [19], [21], [22] about Spin(9), Spin(10) · U(1) and further even Clifford structures, one can construct canonical differential 8-forms on the symmetric spaces E III, E VI, E VIII, F II. Their classes are one of the cohomology generators, namely the one corresponding to their octonionic Kähler structure.…”