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In summary, the Christoffel symbols of the first kind and those of the second kind are different ways of expressing the same concept, with the first kind being more useful for defining the Riemann curvature and the second kind for torsion and contorsion. This difference may be due to the fact that neither set of symbols are tensors. The first kind is used to describe the covariant derivative on tangent spaces, while the second kind describes the induced covariant derivative on dual tangent spaces. The two sets of symbols are related through a simple calculation. The definition of the induced covariant derivative in the cotangent bundle is also discussed, along with its properties and generalization to arbitrary tensors. Contrary to popular belief, the first and secondf

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If I am not wrong, and please correct this if it is not right, one set of Christoffel symbols describes the covariant derivative on the tangent spaces, the other describes the induced covariant derivative on the dual tangent spaces.

In this special case of a metric compatible affine connection, the difference in the Christoffel symbols seems somewhat trivial. But the underlying idea that there are two different connections on two different vector bundles is not.

From this point of view, one starts with the general definition of covariant derivative and then specializes to the case of a metric compatible connection.

For any affine connection, metric compatible or not, the covariant derivative extends to all tensors. The covariant derivative ##∇_{X}ω## of the 1 form ##ω## with respect to the tangent vector field ##X## is defined by the rule

##(∇_{X}ω)(Y)=X⋅ω(Y) -ω(∇_{X}Y)##

for any vector field ##Y##

In this formula ##X⋅ω(Y)## is the ordinary derivative of the function ##ω(Y)## with respect to the vector field ##X##.

If there is a metric ##<,>## then for any vector field ##Z## the inner product of vector fields with Z is a 1 form and the above formula becomes

##(∇_{X}<Z,>)(Y)=X⋅<Z,Y> -<Z,∇_{X}Y>##

and if the covariant derivative is metric compatible the formula simplifies to

##(∇_{X}<Z,>)(Y)=<∇_{X}Z,Y>##

In words this says that the covariant derivative of the inner product with a vector field Z is the inner product with the covariant derivative of Z. Without metric compatibility this would not be true.

Using this, it is a simple calculation to express the Christoffel symbols for the induced covariant derivative on the dual tangent spaces in term of the Christoffel symbols on the tangent spaces.

For a coordinate basis ##e_{i}##

##∇_{e_{i}}e_{j} = ΣΓ_{ij}^{k}e_{k}##

and ##(∇_{ei}<e_{j},>)=<Σ_{k}Γ_{ij}^{k}e_{k},>##

so the coefficients of this 1 form with respect to the dual basis vectors ##e_{l}^{*}## are

##<Σ_{k}kΓ_{ij}^{k}e_{k},e_{l}>## or using index notation this is

##Γ_{ij}^{k}g_{kl}##

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##(∇_{X}ω)(Y)=X⋅ω(Y) -ω(∇_{X}Y)##

Show that

1) ##∇_{X}ω## is an element of the cotangent space at each point of the manifold.

2) Given a tangent vector ##X_{p}## at a point ##p##, ##∇_{X_{p}}ω## is well defined

3) ##∇_{X}ω## defines a covariant derivative: There are several formal properties to verify

##∇_{X}ω## is linear in ##X##

##∇_{X}(ω+σ)= ∇_{X}ω+∇_{X}σ##

##∇_{X}(fω) = (X⋅f)ω + f∇_{X}ω##

4) ##∇_{X}ω## is actually a 1 form . Since in 1) it is shown to be an element of the cotangent space, it remains to show that it is smooth whenever ##X## is a smooth vector field.

5) Generalize this definition to arbitrary tensors.

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No. It is Christoffel symbols of the second kind that appear in the coordinate expressions for the covariant derivatives of both tangent and dual vectors.

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