Geodesic
Affine analogues of the Sasaki-Shchepetilov connection
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, no. 15 (2016).

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For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.
Keywords: connection on a vector bundle, associated vector bundle, connection form, locally symmetric connection 
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     author = {Robaszewska, Maria},
     title = {Affine analogues of the {Sasaki-Shchepetilov} connection},
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Robaszewska, Maria. Affine analogues of the Sasaki-Shchepetilov connection. Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica, no. 15 (2016). http://geodesic.mathdoc.fr/item/AUPCM_2016_15_a1/