<abstract xmlns="http://www.w3.org/1999/xhtml">

For two-dimensional manifold <italic>M</italic> with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle <italic>TM</italic> ⊕ <italic>E</italic>, where <italic>E</italic> is a trivial bundle. The metrizable case, when <italic>M</italic> is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: <bold>36</bold> (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on <italic>TM</italic> ⊕ (ℝ × <italic>M</italic>) and two on <italic>TM</italic> ⊕ (ℝ2 × <italic>M</italic>) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝ<italic>N</italic>, after suitable local affine embedding of (<italic>M,</italic>∇) into ℝ<italic>N</italic>.
</abstract>

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.

Affine analogues of the Sasaki-Shchepetilov connection

Institute of Mathematics, Pedagogical University of Cracow, Podchorążych 2, 30-084 Kraków,

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

{"article-title":"Affine analogues of the Sasaki-Shchepetilov connection"}

For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector...

Affine analogues of the Sasaki-Shchepetilov connection

Published Online: Dec 23, 2016

Page range: 37 - 49

Received: Jul 25, 2015

Accepted: Jul 08, 2016

DOI: https://doi.org/10.1515/aupcsm-2016-0004

Keywords
connection on a vector bundle, associated vector bundle, connection form, locally symmetric connection

© 2016 Maria Robaszewska, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Affine analogues of the Sasaki-Shchepetilov connection

Published Online: Dec 23, 2016

Page range: 37 - 49

Received: Jul 25, 2015

Accepted: Jul 08, 2016

DOI: https://doi.org/10.1515/aupcsm-2016-0004

Keywordsconnection on a vector bundle, associated vector bundle, connection form, locally symmetric connection

© 2016 Maria Robaszewska, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Keywords
connection on a vector bundle, associated vector bundle, connection form, locally symmetric connection