Covariant derivative and vector functions

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SUMMARY

The discussion centers on the identity for the Lie bracket of vector fields, specifically [V,W] = ∇V W - ∇W V, derived from O'Neil's "Linear Algebra" (5.1 #9). Participants express uncertainty about applying this identity to vector functions, particularly in the context of [xu, xv]. Clarification on the application of the covariant derivative to vector functions is sought, indicating a need for a more precise question to facilitate understanding.

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  • Understanding of covariant derivatives in differential geometry
  • Familiarity with vector fields and their properties
  • Knowledge of Lie brackets and their significance
  • Basic concepts from O'Neil's "Linear Algebra"
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  • Research the application of covariant derivatives to vector functions
  • Study the properties of Lie brackets in differential geometry
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Mathematicians, physicists, and students studying differential geometry, particularly those interested in the applications of covariant derivatives and vector fields.

tom.young84
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So given this identity:

[V,W] = \nablaVW-\nablaWV

^^I got the above identity from O'Neil 5.1 #9.

From this I'm not sure how to make the jump with vector functions, or if it is even possible to apply that definition to a vector function [xu,xv].
 
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You question is not clear. Can you be more precise?
 

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