Covariant derivative

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Concept of differential geometry

Covariant derivative is a mathematical term published in 1993.

Chinese name: Covariant derivative
Foreign name: covariant derivative

Discipline: differential geometry
Time of publication: 1993
Properties: Mathematical noun

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Definition 1

yes Smooth manifold M upper smooth Vector bundle E， Let Γ (E) be smooth section 。 be Covariant derivative by Linear mapping R: Γ (E) → Γ (T * M ⨂ E), meeting

∇(fσ)=df⨂σ+f∇σ，

Where f ∈ C ∞ (M)，σ∈Γ(E)。^[2]

Definition 2

Let 𝓗 be ξ=π: E → M liaison , with Contact mapping κ。 Given that f: N → M, ξ is the cross section X along f, u ∈ TN, then X is relative to the Covariant derivative Is u X:=κX * u∈E。

When N=M, f=1 M ,? Called 𝓗 Covariant derivative operator 。^[4]

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? can be regarded as: Γ (E) ⨂ Γ (TM) → Γ (E), that is

∇ V σ:=∇σ(V)，V∈T x M。

R is tensor for V and linear for σ.^[3]

If u ∈ T p N， Then u X∈E f(p) 。 So for U ∈𝖃 N, R U X is the cross section of ξ along f, R U X(p):=∇ U(p) 10. X is parallel along f, if and only if for ∀ U ∈𝖃 N, R U X=0。

∇ u (X+Y)=∇ u X+∇ u Y。

∇ au+v X=a∇ u X+∇ v X，a∈ℝ。

∇ u hX=u(h)X(p)+h(p)∇ u X，h∈𝓕N。

If g: L → N, w ∈ TL, then w (X∘g)=∇ g*ω X。^[4]