Antonio J. Di Scala

Dipartimento di Matematica. Politecnico di Torino. Corso Duca degli Abruzzi 24, 10129 Torino - Italy

e-mail: antonio.discala@polito.it

Abstract. The first part of this article presents the definition of Lie Bracket related to commuting flows of vector fields. In the second part, basic definitions and of connections and curvature are given in order to emphasize the link between Lie Brackets and curvature. Finally, by using locally-defined connections, we give a short and original proof of a classical theorem of Beltrami. The article is addressed to a non specialist in local differential geometry.

Keywords and phrases. Lie Bracket, curvature tensor, a±ne connection.

2000 Mathematics Subject Classification. Primary: 53B20. Secondary: 53B21.

Resumen. La primera parte del artículo presenta al corchete de Lie asociado al problema de la comutatividad de dos flujos. En la segunda parte se introducen las definiciones básicas de conexión y curvatura en fibrados vectoriales, subrayando la relación corchete-curvatura. Finalmente, usando conexiones afines localmente definidas, se da una demostración original y sencilla de un teorema de Eugenio Beltrami. Este artículo apunta a un lector no especialista (e.g. un estudiante de doctorado en matemática o física, etc) en geometría diferencial local.

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(Recibido en febrero de 2005. Aceptado en noviembre de 2005)