> > \Gamma^m_{nb,a}-\Gamma^m_{na,b}+\Gamma^m_{sa}\Gamma^s_{nb}
> > -\Gamma^m_{sb}\Gamma^s{na}.
>
> I think this is the convention for the Riemann tensor. This
> formula can be defined as $R^m_{nba}$ or $R^m_{nab}$
Or any other index ordering. I checked several books last night, and just
about every author uses a different convention. Maxima's convention,
R_{nba}^m, is somewhat unusual, but probably not unique. (Far more confusing
was the fact that ctensor and itensor used to employ different conventions,
but I corrected that years ago.)
> Ok, I didn't read so many books and I wonder if I can find all
> these books in our library:-)
Well, here are two more just to show that Maxima is also in respectable
company: Einstein (in The Meaning of Relativity) and Schr?dinger (in
Space-Time Structure) both contract with the third covariant index.
> Since there is freedom to choose which covariant index to contract
> with, I think it need to be point out explicitly which one ctensor
> has chosen in the document.
Well, this is why in the documentation, we have ric[i,j] under the
definition of the function ricci (compare with the definition of the Riemann
tensor a few entries later, and note the position of the i,j indices.)
Having said that, I certainly recognize that, even after substantial
improvements, the documentation doesn't tell the whole story, this is why I
wrote that supplementary paper which is on arXiv. I also recommend checking
out the tensor demos (do a demo(tensor) in a fresh copy of Maxima),
especially the complete examples. (A new demo, which will be in the
soon-to-be-released version, demonstrates how Maxima can be used to obtain
the Friedmann equations of cosmology, for instance.)
> I learn general relativity mainly by reading the books written by
> Schutz and Weinberg.
I'm not familiar with Schutz, but Weinberg's book is one of my favorites.
> Thank you for your patience.
Thank you for yours, and especially for your interest in putting the Maxima
tensor packages to good use!
Viktor