Spin(9) geometry of the octonionic Hopf fibration
- First Online:
- 159 Downloads
We deal with Riemannian properties of the octonionic Hopf fibration S15 → S8, 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 S1 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.
Unable to display preview. Download preview PDF.
- [BrGr72]R. B. Brown, A. Gray, Riemannian manifolds with holonomy group Spin(9), in: Differential Geometry, in Honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 41–59.Google Scholar
- [BG99]C. Boyer, K. Galicki, 3-Sasakian manifolds, in: Surveys in Differential Geometry: Essays on Einstein Manifolds, Int. Press, Boston, MA, 1999, pp. 123–184.Google Scholar
- [CP99].D. M. J. Calderbank, H. Pedersen, Einstein−Weyl geometry, in: Surveys in Differential Geometry: Essays on Einstein Manifolds, Int. Press, Boston, MA, 1999, pp. 387–423.Google Scholar
- [Par01a]M. Parton, Hermitian and special structures on products of spheres, PhD thesis, Dipartimento di Matematica, Università di Pisa, 2001, www.sci.unich.it/~parton/rice/main.pdf.