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Curvatures of the diagonal lift from an affine manifold to the linear frame bundle

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Kowalski O., Sekizawa M.

Open Mathematics

2012

tom 10

nr 3

837-843

We investigate the curvature of the so-called diagonal lift from an affine manifold to the linear frame bundle LM. This is an affine analogue (but not a direct generalization) of the Sasaki-Mok metric on LM investigated by L.A. Cordero and M. de León in 1986. The Sasaki-Mok metric is constructed over a Riemannian manifold as base manifold. We receive analogous and, surprisingly, even stronger results in our affine setting.

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Affine analogues of the Sasaki-Shchepetilov connection

100%

Robaszewska M.

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

2016

tom 15

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.

Ograniczanie wyników

1 Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

1 Open Mathematics

1 Kowalski O.

1 Robaszewska M.

1 Sekizawa M.

1 2016

1 2012

Wyniki wyszukiwania

Curvatures of the diagonal lift from an affine manifold to the linear frame bundle

Affine analogues of the Sasaki-Shchepetilov connection