kostja

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These are answers submitted by kostja

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

Hello everyone,

thank you very much for your anwsers and please excuse me, that I haven't anwsered for so long time. I'm also very sorry, that my previous posts had such a bad formating.

What Robert Israel posted is exactly what I was looking for.

I have attached an example of what I'm trying to do. Would you help me to do all the calculations under "geometric properties", that I did for the first chart, for all charts, please? Can I do that in a kind of for-loop? And how I name the charts and variables?

And I don't understand, why the color of the plot is pure red, although I added the "color=function" option.

Thanks for reading.

Kind regards

Konstantin

View 11239_mean-curvature.mw on MapleNet or Download 11239_mean-curvature.mw
View file details

Hello and thank you very much for your anwser. But this is not what I am looking for. I also have looked in the online help already. My problem is, that not all surfaces can be parameterized by just only _one_ chart. I'd like to define an altas of the surfcace...

 

I also need "access" to the coordinate functions, so I can caclulate geometric properties, like the second fundamental form of the surface.

Hello again! I

hope wasn't impolite, or why I am being ignored? A negative anwser like: "It isn't possible to plot submanifolds of R³" would also be an anwser ...

 

Kind regards

Konstantin

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