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From the Fermi-Walker to the Cartan connection. (English) Zbl 1009.53019

Slovák, Jan (ed.) et al., The proceedings of the 19th Winter School “Geometry and physics”, Srní, Czech Republic, January 9-15, 1999. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 63, 149-156 (2000).
Let \(M\) be a \(C^\infty\)-manifold with a Riemannian conformal structure \(C\). Given a regular curve \(\gamma\) on \(M\), the authors define a linear operator on the space of (differentiable) vector fields along \(\gamma\), only depending on \(C\), called the Fermi-Walker connection along \(\gamma\). Then, the authors introduce the concept of Fermi-Walker parallel vector field along \(\gamma\), proving that such vector fields set up a linear space isomorphic to the tangent space at a point of \(\gamma\). This allows to consider the Fermi-Walker horizontal lift of \(\gamma\) to the bundle \(CO(M)\) of conformal frames on \(M\) and to define, for any conformal frame \(b\) at a point \(p\), a lift function \(k_b\) from the set of 2-jets of regular curves on \(M\) starting at \(p\) into the tangent space \(T_b(CO(M))\). Finally, using the lift functions \(k_b\), \(b\in CO(M) \), the authors construct a trivialization of the fiber bundle \(CO(M)_1\) over \(CO(M)\), \(CO(M)_1\), denoting the first prolongation of \(CO(M)\).
For the entire collection see [Zbl 0940.00040].

MSC:

53C05 Connections (general theory)
53C20 Global Riemannian geometry, including pinching
53C10 \(G\)-structures
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