(************************************************************************** * * * Ricci * * A Mathematica package for doing tensor calculations * * in differential geometry * * * * Version 1.34 * * * * * * By John M. Lee * * assisted by Dale Lear, John Roth, Jay Coskey, and Lee Nave * * * * Copyright (c) 1992-2000 John M. Lee * * All rights reserved * * * * Development of this software was supported in part * * by NSF grants DMS-9101832, DMS-9404107 * * * * * * The Ricci software package and its accompanying documentation are * * provided as is, without guarantee of support or maintenance. The * * copyright holder makes no express or implied warranty of any kind with * * respect to this software, including implied warranties of * * merchantability or fitness for a particular purpose, and is not liable * * for any damages resulting in any way from its use. * * Everyone is granted permission to copy, modify and redistribute this * * software package and its accompanying documentation, provided that: * * * * 1. All copies contain this notice in the main program file and in the * * supporting documentation. * * 2. All modified copies carry a prominent notice stating who made the * * last modification and the date of such modification. * * 3. No charge is made for this software or works derived from it, with * * the exception of a distribution fee to cover the cost of materials * * and/or transmission. * * * **************************************************************************) OVERVIEW Ricci is a Mathematica package for doing symbolic tensor computations that arise 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 MathTensor, Inc. (E-mail mathtensor@wolfram.com). 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 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 Before you obtain or use Ricci, please read the copyright notice above. The easiest way to get Ricci is from the Ricci World-Wide-Web site . Alternatively, you can download the Ricci files yourself by anonymous ftp from ftp.math.washington.edu, in directory pub/Ricci. You'll need to download the following files: README.txt This file Manual.ps User's Manual (420K - about 90 pages when printed) Ricci.m The source file for the Ricci package (283K) 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) USING RICCI Ricci requires Mathematica version 2.0 or greater. The source takes approximately 283K bytes of disk storage, including about 49K bytes of on-line documentation. I have tested the package on a DEC Alpha system and on a Pentium 100, where it runs reasonably fast and seems to require about 6 or 7 megabytes of memory. I don't have any idea how it will run on other systems, but I expect that Ricci will be very slow on some platforms. 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: <