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Clifford approach to metric manifolds. (English) Zbl 0752.53014
Geometry and physics, Proc. Winter Sch., Srni/Czech. 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 26, 123-133 (1991).
[For the entire collection see Zbl 0742.00067.]
For the purpose of providing a comprehensive model for the physical world, the authors set up the notion of a Clifford manifold which, as mentioned below, admits the usual tensor structure and at the same time a spin structure. One considers the spin space generated by a Clifford algebra, namely, the vector space spanned by an orthonormal basis \(\{e_ j: j=1,\dots,n\}\) satisfying the condition \(\{e_ i,e_ j\}\equiv e_ ie_ j=e_ je_ i=2I\eta_{ij}\), where \(I\) denotes the unit scalar of the algebra and (\(\eta_{ij}\)) the nonsingular Minkowski metric of signature (\(p,q\)), (\(p+q=n\)). Then, for a raw manifold structure with local chart (\(x^ i\)), one assigns the vector basis \(\{e_ \mu(x): \mu=1,\dots,n\}\), by the rule \(e_ \mu(x)=h_ \mu^ i(x)e_ i\), \((\text{det}(h_ \mu^ i)\neq 0)\), so that \(g_{\lambda\mu}(x)=h^ i_{\lambda}(x)h^ j_ \mu(x)e_{ij}\) becomes a metric. A differentiable manifold constructed in this way may be called a Clifford manifold. Introducing a linear connection, such that \(D_ \mu e_ \nu\equiv\partial_ \mu e_ \nu-\Gamma^ \sigma_{\mu\nu}e_ \sigma + [G_ \mu,e_ \nu]=0\) where \(G_ \mu={1\over 2}\sum e_ ie_ j \eta^{ik}h^ \sigma_ k(D_ \mu h^ j_ \sigma)+S_ \mu I\), \(D_ \mu h^ i_ \sigma=\partial_ \mu h_ \sigma^ i-\Gamma^ \rho_{\mu\sigma} h_ \rho^ i\) and \(S_ \mu\) is an arbitrary covector field, one proves, that \(D_ \rho g_{\lambda\mu}=0\), which implies, that the connection endows the Clifford manifold with a Riemannian structure.
Reviewer: T.Okubo (Victoria)

53B30 Local differential geometry of Lorentz metrics, indefinite metrics