Ricci

Ricci is a Mathematica package for doing symbolic tensor computations that arise in differential geometry. It has the following features and capabilities:

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• 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

Disclaimer: Be warned that I make no claims that this is a professional-quality software package. I have tried to make it as general and error-free as possible, and I think it is reasonably robust. However, I do most of the work on this package in my "spare time", with only very limited programming assistance, so I don't have time to check everything. I will try to fix any bugs that you encounter.

If you use this package at all, I would appreciate it if you would send me a message at lee@math.washington.edu 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. If I get e-mail from you, I'll put you on my mailing list to be informed whenever I release a new production version.

How to get Ricci

• Manual.pdf: User's Manual (380K - about 90 pages when printed)
• Ricci.m: the source file for the latest version of Ricci (291K)
• Example.txt: an example of Ricci usage (ASCII, 20K).
• Changes.txt: A list of all significant changes to Ricci since the first beta release (ASCII, 6K).
• Ricci.tex: TeX macros needed for Ricci's TeXForm output (ASCII, 2K)

Once you have downloaded the files, put the source file Ricci.m in the directory in which you place Mathematica input files. If you plan to use TeXForm output from Ricci, put Ricci.tex in the directory in which TeX looks for its input files.

Using Ricci

To use Ricci, put the Ricci.m source file into a directory of your own that is accessible to Mathematica. (You may need to change the value of Mathematica's $Path variable in your initialization file--see the documentation for the version of Mathematica that you're using.) Then load the Ricci package by typing the following Mathematica command:

<<Ricci.m

Once you've loaded Ricci into Mathematica, you can type ?name for information about any Ricci function or command.

Changes in Version 1.53

• Fixed protection error caused by new system symbol PermutationOrder.

Here's a log of changes in prior versions of Ricci.

Plans for future releases

• WYSIWIG input and output under Mathematica 3.0 or higher.
• Tensors depending on parameters, such as g[t] or u[s,t], together with support for computing derivatives of tensor expressions with respect to the parameters. This will be useful for studying evolution equations in geometry, and for computing variational equations of geometric functionals.
• Tensors whose rank is a symbolic constant, such as n-forms or k-forms on an n-manifold. Along with this, I will implement Riemannian volume elements (what physicists call the Levi-Civita tensor or epsilon tensor) and the Hodge star operator. The limitation will be that you cannot insert indices into a tensor expression unless its rank is an explicit nonnegative integer.
• Computation of explicit values for the components of tensor expressions in terms of local coordinates.
• Kähler metrics, and the holomorphic and antiholomorphic parts of the exterior derivative (Cauchy-Riemann operators).
• Vector-valued differential forms, including wedge products and covariant exterior derivatives.

Phone: (206) 543-1735
Fax: (206) 543-0397
E-mail: lee@math.washington.edu