Differential-Geometry of Manifolds Appendix to Connections and Covariant Differentiation, guaranteed and verified 100% Pass

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Appendix to Connections and Covariant Differentiation




Γ𝑖𝑗𝑘 is not a tensor, but how do the components change under a change of
coordinates?




Proposition:
̅ be overlapping coordinate patches on a manifold, 𝑀, with
Let 𝑈 and 𝑈
local coordinates (𝑥 1 , … , 𝑥 𝑛 ) and (𝑥̅ 1 , … , 𝑥̅ 𝑛 ) respectively, then

𝜕𝑥 𝑟 𝜕𝑥 𝑙 𝜕𝑥̅ 𝑘 𝑚 𝜕 2 𝑥 𝑚 𝜕𝑥̅ 𝑘
Γ̅𝑖𝑗𝑘 = 𝑗 Γ +
𝜕𝑥̅ 𝜕𝑥̅ 𝑖 𝜕𝑥 𝑚 𝑙𝑟 𝜕𝑥̅ 𝑖 𝜕𝑥̅ 𝑗 𝜕𝑥 𝑚


𝜕 𝑘
Proof: ∇ 𝜕 (𝜕𝑥̅ 𝑗 ) = Γ̅𝑖𝑗 𝜕̅ 𝑘 .
̅𝑖
𝜕𝑥



By the Chain Rule:


𝑛 𝑛
𝜕 𝜕𝑥 𝑙 𝜕 𝜕 𝜕𝑥 𝑚 𝜕
=∑ ; = ∑
𝜕𝑥̅ 𝑖 𝜕𝑥̅ 𝑖 𝜕𝑥 𝑙 𝜕𝑥̅ 𝑗 𝜕𝑥̅ 𝑗 𝜕𝑥 𝑚
𝑙=1 𝑚=1



𝜕 𝜕𝑥 𝑚 𝜕
so ∇ 𝜕 (𝜕𝑥̅ 𝑗 ) = ∇ 𝜕𝑥𝑙 𝜕 ( 𝜕𝑥̅ 𝑗 ).
𝜕𝑥 𝑚
̅𝑖
𝜕𝑥 ̅ 𝜕𝑥𝑙
𝜕𝑥 𝑖

, 2


By Property #3 in the definition of a connection we get:


𝜕 𝜕𝑥 𝑙 𝜕 𝜕𝑥 𝑚 𝜕 𝜕𝑥 𝑚 𝜕
Γ̅𝑖𝑗𝑘 𝜕𝑥̅ 𝑘 = ( ) + ∇ 𝜕𝑥𝑙 𝜕 (𝜕𝑥 𝑚 ) .
𝜕𝑥̅ 𝑖 𝜕𝑥 𝑙 𝜕𝑥̅ 𝑗 𝜕𝑥 𝑚 𝜕𝑥̅ 𝑗
̅ 𝑖 𝜕𝑥𝑙
𝜕𝑥




By Property #1 we get:


𝜕𝑥 𝑙 𝜕 2 𝑥 𝑚 𝜕 𝜕𝑥 𝑚 𝜕𝑥 𝑙 𝜕
= 𝜕𝑥̅ 𝑖 + ∇ 𝜕 (𝜕𝑥 𝑚 )
𝜕𝑥 𝑙 𝜕𝑥̅ 𝑗 𝜕𝑥 𝑚 𝜕𝑥̅ 𝑗 𝜕𝑥̅ 𝑖
𝜕𝑥𝑙




𝜕𝑥 𝑙 𝜕 2 𝑥 𝑚 𝜕 𝜕𝑥 𝑚 𝜕𝑥 𝑙 𝑡 𝜕
= 𝑚 + Γ𝑙𝑚 .
𝜕𝑥̅ 𝑖 𝜕𝑥 𝑙 𝜕𝑥̅ 𝑗 𝜕𝑥 𝜕𝑥̅ 𝑗 𝜕𝑥̅ 𝑖 𝜕𝑥 𝑡



Reindex the second term by replacing 𝑚 with 𝑟 and 𝑡 with 𝑚:



𝜕𝑥 𝑙 𝜕 2 𝑥 𝑚 𝜕 𝜕𝑥 𝑟 𝜕𝑥 𝑙 𝜕
= 𝜕𝑥̅ 𝑖 + 𝜕𝑥̅ 𝑗 Γ𝑙𝑟𝑚 𝜕𝑥 𝑚
𝜕𝑥 𝑙 𝜕𝑥̅ 𝑗 𝜕𝑥 𝑚 𝜕𝑥̅ 𝑖



𝜕 𝜕𝑥 𝜕 𝑥 𝑙 2 𝑚
𝜕𝑥 𝑟 𝜕𝑥 𝑙 𝜕
Thus Γ̅𝑖𝑗𝑘 𝜕𝑥̅ 𝑘 = (𝜕𝑥̅ 𝑖 𝜕𝑥 𝑙 𝜕𝑥̅ 𝑗 + 𝜕𝑥̅ 𝑗 Γ𝑙𝑟𝑚 ) 𝜕𝑥 𝑚 .
𝜕𝑥̅ 𝑖



Now apply both sides to 𝑥̅ 𝑘 :


𝜕𝑥̅ 𝑘 𝜕𝑥 𝜕 𝑥 𝑙 2 𝑚
𝜕𝑥̅ 𝜕𝑥 𝑘 𝑟 𝜕𝑥 𝑙 𝜕𝑥̅ 𝑘
Γ̅𝑖𝑗𝑘 𝜕𝑥̅ 𝑘 = 𝜕𝑥̅ 𝑖 𝜕𝑥 𝑙 𝜕𝑥̅ 𝑗 𝜕𝑥 𝑚 + 𝜕𝑥̅ 𝑗 Γ𝑙𝑟𝑚 .
𝜕𝑥̅ 𝑖 𝜕𝑥 𝑚