Master Thesis

3.5 The Connection
3.5.1 Geometric interpretation
The connection can be introduced as a direction dependent linear transformation, which gives the change of a vector,
which is moved on a manifold:
ζ i =Γ i
k j ξj X k
(3.30)
Here theΓ i
k j are the connection coefficients or Christoffel symbols,ξj is the vector, which should be moved,ζ i is the
change of the vector and X k specifies the direction of the movement.
The ‘transformation law’ (3.30) has the geometrical meaning that it glues together adjacent tangent spaces. Vectors in
the tangent space Tp(M), which are transported from a point p∈M to the adjacent point q∈ M with tangent space Tq(M) are in the general curved space affected by an infinitesimal change. The change of the vector by shifting it from p to q
is characterized by the linear mappingΓ(X) i
j =Γi k jX k , where X= X k∂k∈T p(M) is the difference vector of p to q. One
sees, that the change, which is assumed to be linear in the vectorξand it’s movement X is direction dependent.
3.5.2 Definition andproperties
Definition 3.5.1 (Covariant Derivative). Let T∈� r
s ,Λ1 ∋Γ∈� 1
1
˙c(0)=X. The covariant derivative is the change of T along the X direction:
From this, one can see some important properties:
(∇X T) p := d
d t [c(t).Γ[T(c(t))]]
�
�
�
1. From the last comment of chapter 3.3, one concludes that∇ X commutes with contractions.
2.∇ X fulfills the Leibniz rule. This means: If A∈� a
b
a+b
c and B∈� then A⊗ B∈� d c+d and:
The most obvious consequences of (3.31) and (3.32) are:
3.
4.
and c(t)∈�1 0 be a smooth curve with c(0)= p and
� t=0
(3.31)
∇ X(A⊗ B)=(∇ X A)⊗ B+ A⊗(∇ X B) (3.32)
∇ X f = X .f , f∈� 0
0
∇ X Y= Y i ∇ X∂ i+ X .Y i ·∂ i, X∈� 1
0
(3.33)
(3.34)
As a result of the property that∇ X T is the change of T by walking along the curve c(t) one also gets from reparametrisation:
5.
Using the basis elements of T p(M) one gets:
∇ f X T=f∇ X T, f∈� (3.35)
∇ ∂k ∂ j=Γ i
k j ∂ i . (3.36)
Transforming this basis to another basis and using the rules (3.34) and (3.35) of the covariant derivative, one sees that
the Christoffel symbols transform not like a tensor does (see Eq. 3.16).
12 3.5 The Connection