Publication: From the Fermi-Walker to the Cartan connection
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2000
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Springer
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Abstract
Let M be a differentiable manifold and C ={e2org / a : M -> R } a Riemannian conformal structure on M. Given any regular curve in M, 7 : I -> M, there is a natural way of defining an operator, D/dt: £(7) -> £(7), the Fermi-Walker
connection along 7, which only depends on the conformal structure C, and such that it coincides with the Fermi-Walker connection along 7 -in the classical sense- of any
g € C such that g("y'(t),y'(t)) = 1 Vt G I. This Fermi- Walker connection enables us to construct a lift-function Kb : T*M -> TbCO(M) for every b G CO(M), and p = n(b), n : CO(M) —> M being the usual projection. In some sense, Kb combines all the different lift-functions TPM -> T6CO(M) given by the Levi-Civita connections of the compatibles metrics g € C. But over all, Kb determines the conformal structure C over M, so that it may be used to know about the normal Cartan connection and the Weyl conformal curvature tensor.
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Jan Slovák and Martin Čadek (eds.): Proceedings of the 19th Winter School "Geometry and Physics". Circolo Matematico di Palermo, Palermo, 2000. Rendiconti del Circolo Matematico di
Palermo, Serie II, Supplemento No. 63. pp. 149--156.
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