Videolibrary
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 Video Library Archive Most viewed videos Search RSS New in collection

Geometric structures on complex manifolds
October 4, 2011 10:20, Moscow

Some curiosities on $\mathrm{Spin}(9)$ and the sphere $S^{15}$

Paolo Piccinni

Sapienza Universita' di Roma
 Video records: Flash Video 1,569.5 Mb Flash Video 257.9 Mb MP4 257.9 Mb

Abstract: Although holonomy $\mathrm{Spin}(9)$ is only possible for the two $16$-dimensional symmetric spaces $\mathbb OP^2$ and $\mathbb OH^2$, weakened holonomy $\mathrm{Spin}(9)$ conditions have been proposed and studied, in particular by Th. Friedrich. A basic problem is to have a simple algebraic formula for the canonical $8$-form $\Phi_{\mathrm{Spin}(9)}$, similar to the usual definition of the quaternionic $4$-form $\Phi_{\mathrm{Sp}(n)\cdot\mathrm{Sp}(1)}=\omega_I^2+\omega_J^2+\omega_K^2$, witten in terms of local compatible almost hypercomplex structures $(I,J,K)$.
In the talk, a simple formula for $\Phi_{\mathrm{Spin}(9)}$ is presented, discussing a family of local almost hypercomplex structures associated with a $\mathrm{Spin}(9)$-manifold $M^{16}$. Some of these complex structures, now on model spaces $\mathbb R^{16^q}$, are then used to give an approach through $\mathrm{Spin}(9)$ to the very classical problem of writing down a maximal system of tangent vector fields on spheres $S^{N-1}\subset\mathbb R^N$. If time permits, some properties of manifolds equipped with a locally conformal parallel $\mathrm{Spin}(9)$ metric will be also discussed.