Ivancevic_Applied-Diff-Geom

468 Applied Differential Geometry: A Modern IntroductionFrom (3.289) it is possible to deduce the transformation property of ˆΓ µνρˆδξ ˆΓµν ρ = ( ˆX ξ ˆΓµν ρ )−( ˆX (∂µξ α )ˆΓ αν ρ )−( ˆX (∂νξ α )ˆΓ µα ρ )+( ˆX (∂αξ ρ )ˆΓ µν α )−( ˆ∂ µ ˆ∂νˆξρ ).The metric Ĝµν is defined as a symmetric tensor of rank two. It can beobtained for example by a set of vector–fields ʵ a , a = 0, . . . , 3, where a isto be understood as a mere label. These vector–fields are called vierbeins.Then the symmetrized product of those vector–fields is indeed a symmetrictensor of rank twoĜ µν := 1 2 (ʵ a Ê ν b + Êν b Ê µ a )η ab .Here η ab stands for the usual flat Minkowski space metric with signature(− + ++). Let us assume that we can choose the vierbeins ʵ a such thatthey reduce in the commutative limit to the usual vierbeins e µ a . Then alsothe metric Ĝµν reduces to the usual, undeformed metric g µν .The inverse metric tensor we denote by upper indicesĜ µν Ĝ νρ = δ ρ µ.We use Ĝµν respectively Ĝµν to raise and lower indices.The curvature and torsion tensors are obtained by taking the commutatorof two covariant derivatives 15which leads to the expressions[ ˆD µ , ˆD ν ] ˆV ρ = ˆR µνρα ˆVα + ˆT µνα ˆDα ˆVρˆR µνρ σ = ˆ∂ ν ˆΓµρ σ − ˆ∂ µˆΓνρ σ + ˆΓ νρβ ˆΓµβ σ − ˆΓ µρβ ˆΓνβσˆT µν α = ˆΓ νµ α − ˆΓ νµ α .If we assume the torsion–free case, i.e.,ˆΓ µν σ = ˆΓ νµ σ ,we find a unique expression for the metric connection (Christoffel symbols)defined (by means of ˆDα Ĝ βγ = 0 ) in terms of the metric and its inverseˆΓ αβ σ = 1 2 ( ˆ∂ α Ĝ βγ + ˆ∂ β Ĝ αγ − ˆ∂ γ Ĝ αβ )Ĝγσ .15 The generalization of covariant derivatives acting on tensors is straight forward [Aschieriet. al. (2005)].