3. Parallel derivatives and Christoffel tensors

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

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defined so that

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This derivative is always flat, and has a vanishing Riemann

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But it has torsion, unless the basis is coordinated,

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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

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

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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

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With another parallel derivative,

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Complex bases have FRiemann and AChristoffel tensors, as explained in xTensorDoc, section 6.8.

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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

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Derivatives of complex bases are also replaced by the corresponding Christoffels

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Both basis indices must belong to the same vbundle:

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 Created by Mathematica  (May 16, 2008)