Christoffel Symbols: Difference, Importance & Uses

What is the general difference or importance between using christoffel symbols of the first kind and those of the second kind in terms of geometry and their application. The christoffel symbols of the second are identical to those of the first except with the inverse metric tensor in front, correct?Why have the both the first and second kind? what are the first used for and the second used for in general? I know the importance of the second kind in terms of the covariant derivative but what use does the first kind have in physics and math.

This depends on what you mean by ”use”. Some things, such as arriving at the expression for the Christoffel symbols in terms of the metric, are easier to do using the first kind. Geometrically and computationally, the second kind is typically more intuitive and useful.

The reason for the distinction between the first and second kind may be that neither are tensors. If they were tensors then the difference would be the trivial lowering of an index, and we would regard them as manifestations of the same tensor. But they are not tensors, so we have to be careful and treat them separately. The first kind is useful for defining the Riemann curvature, second is useful for torsion and contorsion.

Michael Price said:

The reason for the distinction between the first and second kind may be that neither are tensors. If they were tensors then the difference would be the trivial lowering of an index, and we would regard them as manifestations of the same tensor. But they are not tensors, so we have to be careful and treat them separately. The first kind is useful for defining the Riemann curvature, second is useful for torsion and contorsion.

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}##

Some exercises for post #5 on the definition of the induced covariant derivative in the cotangent bundle,

##(∇_{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.

First kind and second kind are duals, right? One to compute covariant geometry, other one to compute contravariant geometry.

guru_4 said:

First kind and second kind are duals, right? One to compute covariant geometry, other one to compute contravariant geometry.

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.