3. Parallel derivatives and Christoffel tensors

We begin by remembering that each basis has its own parallel derivative

In:= Out= defined so that

In:= Out= This derivative is always flat, and has a vanishing Riemann

In:= Out= But it has torsion, unless the basis is coordinated,

In:= Out= In:= Out= In xTensor` we can define the Christoffel tensor connecting two covariant derivatives (cf. Section 6 of xTensorDoc.nb). In particular, this works with the PD of our bases

In:=  Out= Notice how the Christoffel connecting any covariant derivative to PD is defined with the derivative, in our case by DefBasis.

In:= Out= Any derivative of any Basis object can be translated into a component of  Christoffel tensors relating that derivative to the PDs of the bases involved

In:= Out= In:= Out= In:= Out= With another parallel derivative,

In:= Out= In:= Out= Complex bases have FRiemann and AChristoffel tensors, as explained in xTensorDoc, section 6.8.

In:= Global`AChristoffelPDcomp

 Dagger[AChristoffelPDcomp]^=AChristoffelPDcomp† DependenciesOfTensor[AChristoffelPDcomp]^={M3} Info[AChristoffelPDcomp]^={nonsymmetric AChristoffel tensor ,} MasterOf[AChristoffelPDcomp]^=PDcomp ServantsOf[AChristoffelPDcomp]^={AChristoffelPDcomp†} SlotsOfTensor[AChristoffelPDcomp]^={InnerC,-TangentM3,-InnerC} SymmetryGroupOfTensor[AChristoffelPDcomp]^=StrongGenSet[{},GenSet[]] TensorID[AChristoffelPDcomp]^={AChristoffel,PDcomp,PD} xTensorQ[AChristoffelPDcomp]^=True

The FRiemann is also zero

In:= Out= Derivatives of complex bases are also replaced by the corresponding Christoffels

In:= Out= In:= Out= In:= Out= In:=   Out= Both basis indices must belong to the same vbundle:

In:=  Created by Mathematica  (May 16, 2008) 