Master Thesis

3.7 Torsion andCurvature
The Torsion and Curvature are defined as tensor fields of higher rank:
T(X , Y) :=∇ X Y−∇ Y X−[X , Y] (3.38)
R(X , Y)Z :=∇ X(∇ Y Z)−∇ Y(∇ X Z)−∇ [X,Y]Z , (3.39)
where[X , Y]f := X .(Y.f)−Y.(X .f) is the commutator. As a result of the property T(X , Y)=−T(Y, X) and R(X , Y)=
−R(Y, X), we see that T, R∈Λ 2 for the torsion and the curvature.
Contracting by using the basis vectors one obtains the torsion tensor and the curvature tensor (also called Riemann
Tensor):
T(∂ j,∂ k).d x i := T i
jk
(3.40)
(R(∂k,∂ l)∂ j).d x i := R i
jkl , (3.41)
from which one can see, that T∈� 1
1
2 (M) and R∈� 3 (M).
The Riemann tensor fulfills Ri jkl = Rkli j and the first Bianchi identity (3.45). Together with the antisymmetry of
the curvature the number of independent components of the Riemann tensor reduces to F(n)= 1
12 n2 (n 2 − 1), which is
F(4)=20 in the four-dimensional space.
One obtains the Ricci tensor by contracting the first and third index of the Riemann tensor: Ri j := Rk ik j .4 A further
contraction of the Ricci tensor leads to the scalar curvature: R := Ri i .
3.8 Resulting relations
3.8.1 Identities
One sees by direct calculation, that the Jacobi identity holds:
�
[X ,[Y, Z]]= 0 . (3.42)
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Using this identity and (3.32) one can also verify the two following relations by a straightforward calculation of the most
left-hand term:
• First Bianchi identity:
• Second Bianchi identity:
� ��
(R(X , Y)Z)= T(T(X , Y), Z)+(∇X T)(Y, Z) �
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(3.43)
��
(∇X R)(Y, Z)+ R(T(X , Y), Z) � = 0 . (3.44)
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If the geometry is given by the Levi-Civita connection ( T(X , Y)=0 ) the Bianchi identities reduce to:
�
(R(X , Y)Z)= 0 (3.45)
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��
(∇X R)(Y, Z) � = 0 . (3.46)
4 As a result of the antisymmetry, there is only one way for contracting the indices which makes sense.
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14 3.7 Torsion and Curvature