Twisting procedure on torogonal structures (Q1327115)

Let \((E,Q)\) be a pseudo-Euclidean vector space and denote by \((E',Q')\) its complexification. The Clifford group \(\Gamma(Q')\) is the subgroup of the Clifford algebra \(C(Q')\) consisting of the elements \(g\) which satisfy \(\alpha(g)xg^{-1}\in E'\) for all \(x\in E'\): \[ 1\to C^*\to \Gamma(Q')@> p>> O(Q')\to 1. \] Denote by \(N: \Gamma(Q')\to C\) the spinor norm morphism. The groups \(\text{Pin}(Q)\) and \(\Delta(Q)\) are defined by \(\text{Pin}(Q)= \text{Ker}(N)\cap p^{-1}(O(Q'))\), \(\Delta(Q)= N^{- 1}(S^ 1)\cap p^{-1}(O(Q))\). The torogonal group \(\Delta(Q)\) is a central extension of \(\text{Pin}(Q)\) by \(S^ 1\). In the present paper, the author studies torogonal structures on pseudo-Riemannian manifolds (existence, classification, connections and examples).