Lie group, Riemannian metric, and connection

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SUMMARY

The discussion centers on computing the Riemannian connection compatible with a left-invariant Riemannian metric on a Lie group. The unique symmetric connection can be derived using the structure constants C(ij, k) and the metric g(ij). Key references include John Milnor's "Morse Theory" (pages 48-49) and "Riemannian Geometry" by Manfredo P. do Carmo (page 55), which provide essential formulas for this computation.

PREREQUISITES
  • Understanding of Lie groups and their properties
  • Familiarity with Riemannian metrics and connections
  • Knowledge of structure constants in differential geometry
  • Basic proficiency in tensor calculus
NEXT STEPS
  • Study the unique symmetric connection in Riemannian geometry
  • Explore the derivation of the Riemannian connection using structure constants
  • Review John Milnor's "Morse Theory" for advanced concepts
  • Examine Manfredo P. do Carmo's "Riemannian Geometry" for practical applications
USEFUL FOR

Mathematicians, theoretical physicists, and graduate students specializing in differential geometry and Lie groups will benefit from this discussion.

marton
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hello, i have met with a problem. please help me.
A Lie group,with a left-invariant Riemannian metric, i want to compute the connection compatible with the Riemannian metric. C(ij, k) are the structure constants, g(ij) are the metric, then how to compute the Riemannian connection in terms of g(ij) and C(ij, k)?

thanks a lot.
 
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there is a unique symmetric connexion compatible with a riemannian metric. see pages 48-49 of john milnors morse theory, for the formulas.
 
Also, check out p. 55 in Riemannian Geometry by Do Carmo. The equation in the middle of the page is precisely the way to find the answer to your question.
 

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