Using xCoba with a product manifold

[Leo Stein]

Hello xActers,

  Has anybody tried to use xCoba with a product manifold? I was attracted to the idea of using a product manifold since I thought it would save some computation time, when one knows this property of the manifold a priori. However, I don't see how to get it to work. Maybe I'm approaching it wrong.

  Let's say I have two manifolds M1 and M2, with metrics g1 and g2. Then I define a product manifold M with product metric g. This is all fine for simplifying abstract expressions such as ChristoffelCD[a1,-b2,-c2] in terms of the g1 and g2. But then I want to do some component calculations. Since the abstract expressions like ChristoffelCD[a1,-b2,-c2] are decomposed in terms of g1 and g2, it is natural to define charts B1 and B2 on M1 and M2 in order to assign component values to g1 and g2 (after all, g1 and g2 have indices in TangentM1 and TangentM2).

  However, there's no such machinery as "DefProductChart" to create a chart on M out of the two charts B1 and B2. So, this works for calculations which have every index either in M1 or M2, but there can be no mixing of indices, since there haven't been derivatives of one set of basis vectors defined with respect to the others.

  So, do I have the wrong approach, or is this an unimplemented feature?

  If the above was not clear, I can attach an example notebook.

Cheers

Leo

[JMM]

Hi Leo,

This is indeed a partially implemented feature. "Partially" means it
is implemented for bases, but not for charts. There is a command
SplitBasis which will tells xCoba to break the basis of a product
manifold into bases of the submanifolds. Note also that the basic
function CNumbersOf admits a second argument, precisely to specify to
which of the vbundles you are referring to.

I imagine that commands like TraceBasisDummy, ComponentArray might not
be enough to work in this more general case. Consider the use of the
more powerful low-level commands TraceDummy and SplitIndex, both in
xTensor.

Cheers,
Jose.