Matrix Formalism of Differential Geometry

[thehangedman]

I would like to explore writing differential geometry in matrix format and was wondering if any of the experts here knows a good resource for that? I have tried Google and can't find anything definitive.

Thanks in advance!

 

[HallsofIvy]

Are you talking about representing tensors as matrices?

 

[thehangedman]

Are you talking about representing tensors as matrices?

Yes. The metric tensor would become a 4x4 hermitian matrix.

 

[haushofer]

Well, symmetric 2-tensors are represented by symmetric matrices, like the metric. Antisymmetric 2-tensors are represented by antisymmetric matrices, like the Maxwelltensor. Are there any magical things going on here?

The matrixform of pushforwards and pullbacks also are quite frequently noted in texts. I must say that I can't think of any subtleties, but maybe this topic,

https://www.physicsforums.com/showthread.php?t=353536

helps a little :)