Are There Special Properties of Geodesics in a Lie Group?

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SUMMARY

This discussion focuses on the relationship between geodesics in Lie groups and their properties as Riemannian manifolds. It establishes that the nature of the metric, particularly whether it is bi-invariant, significantly influences the behavior of geodesics. When the metric is bi-invariant, geodesics correspond to the exponential map. In cases where the metric is not bi-invariant, as detailed in Section 17.6 of the referenced material, geodesics align with integral curves of left-invariant vector fields.

PREREQUISITES
  • Understanding of Lie groups and their structure
  • Familiarity with Riemannian geometry concepts
  • Knowledge of Christoffel symbols and their applications
  • Basic comprehension of the exponential map in differential geometry
NEXT STEPS
  • Study Chapter 17 of the provided resource on metrics on Lie groups
  • Explore the properties of bi-invariant metrics in Lie groups
  • Learn about Cartan connections and their implications for geodesics
  • Investigate left-invariant vector fields and their role in differential geometry
USEFUL FOR

Mathematicians, physicists, and students of differential geometry interested in the interplay between Lie groups and Riemannian metrics, particularly those studying geodesics and their properties.

Andre' Quanta
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Suppose to have a Lie group that is at the same time also a Riemannian manifold: is there a relation between Christoffel symbols and structure constants? What can i say about the geodesics in a Lie group? Do they have special properties?
 
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It depends on what sort of metric the Lie group admits. http://www.seas.upenn.edu/~jean/diffgeom.pdf seems to have a lot of the explanation that you need. Chapter 17 discusses metrics on Lie groups. If the metric is bi-invariant (see the text for the definition), then the geodesics correspond to the exponential map. Section 17.6 discusses Cartan connections, which can be defined when the metric is not bi-invariant, for which the geodesics coincide with integral curves of left-invariant vector fields.
 
Wow, "class notes" consisting of 807 pages. Some prof!
 

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