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In article <2183@nikhefh.nikhef.nl> t16@nikhefh.nikhef.nl (Kasper Peeters)
writes:

> Can anyone tell me if it is possible to perform the following tensor-
> manipulations with Maple:

>    - automatic Einstein summation convention,
>    - convenient notation for partial derivatives (like g[a,b,c] for
>        \partial_c g_{a,b} )
>    - automatic relabeling of indices whenever a substitution is made
>         (eg.  A=B(mu)*B(mu);  C=A*A;  MAPLE:  C=B(mu)*B(mu)*B(N1)*B(N1) )


I think Robert Bryant (bryant@math.duke.edu) has used Maple to do such
calculations, although I don't know whether he's developed any software for
general use.  If you're willing to switch to Mathematica, you have two good
options: (1) there's a commercial package called MathTensor (contact
MathSolutions, Inc. at mathtensor@wri.com); or (2) I've developed a free
package called Ricci, the announcement for which is reproduced below.

==============================================================================

                         Announcing the release of

                                   RICCI

            A Mathematica package for doing tensor calculations
                         in differential geometry

                                Version 1.0


The first public release of Ricci, my Mathematica package for doing tensor
computations in differential geometry, is now available.

Ricci is designed to assist with some of the tensor calculations needed by
pure mathematics researchers in differential geometry.  It has the
following features and capabilities:

* Manipulation of tensor expressions with and without indices
* Implicit use of the Einstein summation convention
* Correct manipulation of dummy indices
* Display of results in mathematical notation, with upper and lower indices
* Automatic calculation of covariant derivatives
* Automatic application of tensor symmetries
* Riemannian metrics and curvatures
* Differential forms
* Any number of vector bundles with user-defined characteristics
* Names of indices indicate which bundles they refer to
* Complex bundles and tensors
* Conjugation indicated by barred indices
* Connections with and without torsion

Limitations: Ricci currently does not support computation of explicit
values for tensor components in coordinates, or derivatives of tensors
depending on parameters (as in geometric evolution equations or calculus of
variations), although support for these is planned for a future release.
Ricci also has no explicit support for general relativity, or for other
mathematical physics or engineering applications, and none is planned.  If
you are interested in such support, I recommend that you consider the
commercial package MathTensor, which is far more extensive than Ricci, and
provides all these capabilities and more.  MathTensor is available from
MathSolutions, Inc. (mathtensor@wri.com).

Ricci requires Mathematica version 2.0 or greater.  The source takes
approximately 270K bytes of disk storage, including about 49K bytes of
on-line documentation.  The package was developed and tested on a
DECStation 5000 running Unix, but there are no known system-dependent
features, so it should run on any system that can run Mathematica with 7
megabytes or more of available memory.

The source files for Ricci are available to the public by anonymous ftp
from the Stanford Mathematica Users Forum library (otter.stanford.edu).  To
obtain them, you need to log into a system that has an Internet connection
and supports the ftp (file transfer program) command.  If you're using
Unix, you can follow the script below.

  % ftp otter.stanford.edu
  Connected to otter.stanford.edu.
  220 otter FTP server (Version 5.20 (NeXT 1.0) Sun Nov 11, 1990) ready.
  Name (otter.stanford.edu:): anonymous
  331 Guest login ok, send ident as password.
  Password:  <---------------------------------Type your e-mail address here.
  230 Guest login ok, access restrictions apply.
  ftp> cd mma/Geometry
  250 CWD command successful.
  ftp> get Ricci.tar.Z
  200 PORT command successful.
  150 Opening ASCII mode data connection for Ricci.tar.Z (189479 bytes).
  226 Transfer complete.
  local: Ricci.tar.Z remote: Ricci.tar.Z
  190287 bytes received in 1.6 seconds (1.1e+02 Kbytes/s)
  ftp> bye
  221 Goodbye.
  % uncompress Ricci.tar.Z
  % tar -xvf Ricci.tar
  x Ricci/
  x Ricci/Bundle.m, 14905 bytes, 30 blocks
  x Ricci/Changes.doc, 2798 bytes, 6 blocks
  x Ricci/Constant.m, 4783 bytes, 10 blocks
  x Ricci/DefineRelation.m, 16903 bytes, 34 blocks
  x Ricci/Derivatives.m, 25289 bytes, 50 blocks
  x Ricci/Example.doc, 20075 bytes, 40 blocks
  x Ricci/Formatting.m, 20408 bytes, 40 blocks
  x Ricci/Index.m, 7958 bytes, 16 blocks
  x Ricci/Manual.tex, 192557 bytes, 377 blocks
  x Ricci/Products.m, 10920 bytes, 22 blocks
  x Ricci/README, 8276 bytes, 17 blocks
  x Ricci/Ricci.m, 7597 bytes, 15 blocks
  x Ricci/Ricci.tex, 1778 bytes, 4 blocks
  x Ricci/Riemann.m, 18063 bytes, 36 blocks
  x Ricci/Tensor.m, 27520 bytes, 54 blocks
  x Ricci/TensorExpressions.m, 34853 bytes, 69 blocks
  x Ricci/TensorSimplify.m, 42357 bytes, 83 blocks
  x Ricci/Usage.m, 51787 bytes, 102 blocks

The last two Unix commands (uncompress and tar) will create a directory
named Ricci, and place all the uncompressed Ricci files into it.  Look at
the file named README for more information about what's in the files and
how to get started using Ricci.

This is the first public release of Ricci.  If you use this package at all,
I would appreciate it if you would send me a message at the e-mail address
below describing your experience, and telling me whether you found the
package useful or not.  I'd especially like to hear about any bugs,
anomalous behavior, things that look like they should simplify but don't,
suggestions for improvement, things that seem to take longer than they
should, etc.  And please feel free to get in touch with me if you have
questions about the software.  If I get e-mail from you, I'll inform you
whenever I release a new production version.


Jack Lee
Department of Mathematics, GN-50
University of Washington
Seattle, WA 98195

Internet: lee@math.washington.edu
Fax:      206-543-0397


--
Jack Lee
Dept. of Mathematics
University of Washington
Seattle, WA