Journal of Generalized Lie Theory and Applications

A connection whose curvature is the Lie bracket

Kent E. Morrison

Full-text: Open access

Abstract

Let $G$ be a Lie group with Lie algebra $\g$. On the trivial principal $G$-bundle over $\g$ there is a natural connection whose curvature is the Lie bracket of $\g$. The exponential map of $G$ is given by parallel transport of this connection. If $G$ is the diffeomorphism group of a manifold $M$, the curvature of the natural connection is the Lie bracket of vectorfields on $M$. In the case that $G=\SO(3)$ the motion of a sphere rolling on a plane is given by parallel transport of a pullback of the natural connection by a map from the plane to $\so(3)$. The motion of a sphere rolling on an oriented surface in $\R^3$ can be described by a similar connection.

Article information

Source
J. Gen. Lie Theory Appl., Volume 3, Number 4 (2009), 311-319.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.jglta/1281106598

Digital Object Identifier
doi:10.4303/jglta/S090404

Mathematical Reviews number (MathSciNet)
MR2602993

Zentralblatt MATH identifier
1181.53040

Subjects
Primary: 53C05: Connections, general theory 53C29: Issues of holonomy

Keywords
Differential geometry Connections Issues of holonomy

Citation

Morrison, Kent E. A connection whose curvature is the Lie bracket. J. Gen. Lie Theory Appl. 3 (2009), no. 4, 311--319. doi:10.4303/jglta/S090404. https://projecteuclid.org/euclid.jglta/1281106598


Export citation

References

  • W. Ambrose and I. M. Singer. A theorem on holonomy. Trans. Amer. Math. Soc., 75 (1953), 428–443.
  • D. Bleecker. Gauge Theory and Variational Principles. Global Analysis Pure and Applied Series A 1, Addison-Wesley Publishing Co., Reading, Mass., 1981.
  • B. D. Johnson. The nonholonomy of the rolling sphere. Amer. Math. Monthly, 114 (2007), 500–508.
  • S. Kobayashi and K. Nomizu. Foundations of Differential Geometry. Vol. I. John Wiley & Sons, New York, 1996.
  • S. Kobayashi and K. Nomizu. Foundations of Differential Geometry. Vol. II. John Wiley & Sons, New York, 1996.
  • H. Samelson. Lie bracket and curvature. Enseign. Math. (2), 35 (1989), 93–97.
  • R. W. Sharpe. Differential Geometry. Cartan's Generalization of Klein's Erlangen Program. Graduate Texts in Math. 166, Springer-Verlag, New York, 1997. }