@ny2292000 

I am not sure what input you are using.  It could be the `dtheta` `dphi` is not being interpreted as you wish.  Does this input work?

DGsetup([r, theta, phi, w], M)

g1 := evalDG((dw &t dw) - w^2*(dr &t dr/(-kr^2 + 1) + r*sin(theta)^2*(dtheta &t dtheta) + r*(dphi &t dphi))/R0^2)

I have also encountered this kind of output in the North American release.  I am not sure what triggers it since it doesn't happen consistently.  (I have gone back to a previous version to avoid this problem.)

I do not know if Atlas is capable of this, but if you can't get it to work you might try the DifferentialGeometry package included with Maple.  Maple 14 is pretty far back, and I can't recall exactly what the state of DifferentialGeometry was back then.  But if you look for help under "DGsetup" it should show you how to define an "anholonomic frame" via its structure relations.  Then one can define the metric in terms of the coframe, then compute curvature via the commad CurvatureTensor, and so forth. 

@mskalsi 

The error comes because you did not specify an ordering for all of the variables in the jet space.  I believe that, for this example, you can just leave out the "variableorder" option altogether and it should work ok. 

@mskalsi 

1.  To convert the differential 1-form to a vector field (actually a vector density) in the space of (t, x) variables you should use the contravariant form of the permutation symbol. See the help page for DifferentialGeometry:-Tensor[ PermutationSymbol]. This should explain the minus sign.

2. For three independent variables you can proceed in exactly the same way.  You use the permutation symbol to convert the 2-form to a vector (density). 

How exactly is the tensor being represented in Maple?

@PerKirkegaard 

Sorry, I do not know.

https://en.wikipedia.org/wiki/Scalar_curvature

@Agharabo 

Yes, the version of DifferentialGeometry you mention includes different typesetting routines for output than what is included in the Maple release.  You are welcome to use that version, but you do not have to do so.

With the commands you wrote,

DGsetup([t, x, u], M);

v := evalDG(D_t):
w := evalDG(D_t*t+D_u):

LieBracket(v, w);

the correct result is D_t (i.e., v).  Maybe you meant w := evalDG(x*D_t + D_u)?

To get various representations of the commutator Lie algebra you can use commands like

LD := LieAlgebraData([v,w], alg);

DGsetup(LD);

MultiplicationTable();

@trace 

I answered this in your thread entitled "Anholonomic Frame".

You might be interested in the command CovariantlyConstantTensors, within the Tensor sub-package of DifferentialGeometry.

The field equations of the Einstein-Hilbert action are given by the vanishing of the Einstein tensor.  Within the DifferentialGeometry package you can compute the Einstein tensor of a metric g using the command EinsteinTensor(g).

Is this what you are asking?  Or are you asking for the Einstein equations with a prescribed matter field? Then you must also use the energy-momentum tensor, which is obtained using the EnergyMomentumTensor command in DifferentialGeometry. 

Or do you mean something still different?

Aside from the help pages, there is no manual or booklet for DifferentialGeometry, as far as I know.  There is a website intended to address the need for more information on how to best use this package.  You can find it here (http://digitalcommons.usu.edu/dg/).  The website is under active development so it is steadily becoming more extensive. 

Glad to hear it!

Was it in fact so(6) + so(6) ?