MaplePrimes - answers and comments on Question, Pulling a connection back

Try this...

Torre — Sun, 18 Sep 2011 02:10:27 Z

If I were in this situation, I would solve the equations defining the map i: S -> M for the coordinates of S in terms of those of M. If (x, y, z) are coordintes on M and (u,v) are coordinates on S, this transformation would have equations of the form [u=u(x,y,z), v=v(x,y,z)]. Define this as a another DifferentialGeometry transformation. Use this second transformation in the PushForward command as the second argument (the first argument being the map i). This should give you vector fields on M you can differentiate.

See Example 7 in the help page and see also "PullbackVector".

Does this help?

Charlie Torre

Trying to implement a pulled back connection

kostja — Sun, 18 Sep 2011 17:37:11 Z

Hello all,

I tried to define some methods, that construct for a given Transformation i:S->M the pulled back Vectorbundle i^*TM over S, which consists of Vectorfields along i, i.e. Y=X ° i, with X Vectorfield on M.

I have defined a method, TangentBundle(j,TjN) (wrong name, isn't it?), which takes a Transformation j from one frame into another and a specifier TjN and constructs a new bundle frame:

DGsetup([vars of Domain],[vars of Range],TjN)

Then, there is the method TangenVectors(j,TjN), which constructs the Pushforward(j,FrameBaseVectors of Domain) in TjN.

And finally, there is the method PullbackConnection(i,TjN,C), which takes again, a Transformation and the above constructed bundle, and should return the pulled back connection C in TjN. If C has christoffel symbols Gamma_ab^c, than the pulled back connection should have the symbols Gamma_ab^c j^a_k, there j^a_k is the jacobian of the transformation j. Unfortunately I get an error at the end of this method, which is:

Error, (in InfoProcTable["TensorDensityType"]) improper op or subscript selector

Please help me to figure out, what is going on. I think I could figure out the failure. I'm going to test this solution.

I have attached the file to this anwser: tangentbundle.mw

Kind regards
Konstantin

Not quite what I wanted to do...

kostja — Sun, 18 Sep 2011 17:37:29 Z

Hi!

Thank you for your suggestions. I didn't looked into the PullbackVector method, because I thought I need a diffeomorphism to be able to pull back vectors (in differential geometry you, do: f^* X = f_*^(-1) X ° f).

It's also not quite what I want to do, if I would be able to find a right inverse for the imbedding map. Because in the end, I would get a vectorfiled on M, and not on S, what I'm looking for.

I have tried to construct the pulled back bundle and to pull back the connection from M to it. Maybe you can help me with it. There's a little failure in it, I can't figure out. (See my anwser on the question please.)

Kind regards
Konstantin

I misunderstood, I guess. I thought you were

Torre — Sun, 18 Sep 2011 18:42:48 Z

I misunderstood, I guess. I thought you were trying to covariantly differentiate vector fields on M.

I suspect I still don't know quite what you are after.

If you simply want to pullback a connection from M to S, you can do this with PushPullTensor applied to the connection on M. First argument should be a Transformation from N to S, with the expression of coordinates on S in terms those on M. Second argument should be the map (i) embedding S into M.

charlie

Okay, I edited the above file a little bit

kostja — Sun, 18 Sep 2011 21:51:25 Z

Okay, I edited the above file a little bit and removed some bugs. I tested the new function

SecondFundamentalForm

for S^1 in IR^2 and for S^2 in IR^3. And the results seem to be consistent.

Fell free to use it, in this case please report bugs or post even bugfixes here.

Also, feel free to ask questions about the methods.

Kind regards

Konstantin

EDIT: new version: tangentbundle.mw