0034-7426 0034-7426 S0034-74262005000200004 Italy 00 12 2005 00 12 2005 39 2 113 131

Corchete y curvatura

 

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)

]]> 2003 1865 1 7 7 185-204 2003 1928 1975 2001 47 1-2 1-2 57-63 1949 1963 1 1999 149 1 1 149-181 2003 1997 45 3 3 575-592 1958 32 215-223 2005 161 1 1 579-588 1979 2 1979 3 1974 29 699-713 2004 140 1675-1678