Covariant Derivative: Definition & Meaning

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SUMMARY

The discussion focuses on the concept of covariant derivatives, specifically the notation involving semi-colons and upper indices. The expression A^{ab;c} = g^{cd} \nabla_d A^{ab} illustrates the relationship between the covariant derivative and the metric tensor. It is established that raising indices using the metric is a fundamental aspect of manipulating covariant derivatives. Additionally, the property \nabla_c g_{ab} = 0 indicates that the covariant derivative of the metric tensor vanishes, which is crucial in differential geometry.

PREREQUISITES
  • Understanding of Riemannian geometry
  • Familiarity with tensor notation
  • Knowledge of covariant derivatives
  • Basic principles of differential geometry
NEXT STEPS
  • Study the properties of Riemannian metrics in detail
  • Learn about the implications of \nabla_c g_{ab} = 0 in geometry
  • Explore the applications of covariant derivatives in physics
  • Investigate the role of Killing vectors in symmetries of spacetime
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Students and researchers in mathematics and physics, particularly those focusing on differential geometry, general relativity, and tensor calculus.

salparadise
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Hello everyone,

While studying properties of Riemann and tensor and Killing vectors, I found this notation/concept that I'm not sure of it's meaning.

What does it mean to have a covariant derivative, using semi-colon notation, showing in upper index position. Is it just a matter of raising the index via the metric of ordinary covariant derivative.

I tried to search in different textbooks for this, but couldn't find anything.

Thanks in advance for your help.
 
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A^{ab;c} = g^{cd} \nabla_d A^{ab} = \nabla_d (g^{cd} A^{ab})

Note that \nabla_c g_{ab} = 0.
 

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