What can be derived from the metric tensor? I am working on a  computational project about General Relativity. In this process, I want to code 'the stuff' that can be derivable from the metric tensor. So far, I have coded Riemann Tensor, Weyl Tensor, Einstein Tensors, Ricci Tensor, Ricci scalar. What are the other essential/needed quantities in the GR calculations that can be coded?
Some notes to answer the comments:

*

*It's not precisely numerical. I will not solve the Einstein field equations for a given energy stress-energy tensor etc.

*The program's purpose is to obtain possible mathematical objects that belong to GR (tensors, etc.) If the user only knows metric tensor and nothing else.

*I am using / will use python to calculate these things.  It's also kind of a relativistic tool, yes.

 A: I think you know you have a hard job ahead of you. Wishing you success. In order to know the things that physicists need in this area, I would recommend you first see these two packages: xAct (in Wolfram Mathematica) and GRtensor (in Maple). These are really great. Here I summarize some of the most important things (with appropriate links) that come to my mind:

*

*Many essential operations in GR including Covariant derivative, Lie derivative, etc (you can find these in any GR textbook's index part)


*Einstein's field equations in flat, de Sitter and anti-de Sitter backgrounds


*Geodesic deviation equation and related topics


*Hawking radiation (related to the surface gravity)


*A code for computing Unruh temperature for accelerating Observer.


*Frame dragging


*Bekenstein-Hawking entropy formula


*Wald entropy (for higher derivative gravity models)


*A code for testing Killing vectors


*Perhaps some codes for computing Komar mass or ADM mass.


*Kretschmann scalar


*Christoffel symbols
...
And, a number of important metrics/coordinates in GR and Cosmology such as


*Schwarzschild spacetime metric


*Kruskal–Szekeres coordinates


*Eddington–Finkelstein coordinates


*Kerr spacetime metric


*Boyer-Lindquist coordinates (for Kerr spacetime)


*Kerr-Newman spacetime metric


*Friedmann–Lemaître–Robertson–Walker metric
.
.
.
For this purpose, I think the "Catalogue of Spacetimes" by  Mueller & Grave can help you a lot about this. In this book we read:

The aim of the catalogue is to give a quick reference for students
who need some basic facts of the most well-known spacetimes in GR.

In addition, I highly recommend you to search in this site (nLab).
Finally, according to your edit, I have three suggestions:
i) It would be nice if you could define the most commonly used metrics by default in your program (see items 13-19, please).
ii) It would be nice if you consider the possibility of defining metrics in dimensions higher or lower than 4 dimensions (they are very important from different aspects). This feature will increase the usefulness and also the number of users of your program. Some examples of lower/higher dimensional black hole spacetimes are


*BTZ black hole metric (in 3 dimensions)


*Charged rotating BTZ black hole metric (in 3 dimensions)


*Higher dimensional Kerr black hole
...
iii) It would be nice if you provide a graphical environment (if possible) for your program, in which users can obtain a desired property for their metric by a click. In fact, sometimes it is not easy for undergraduate or even some of graduate students to work with those packages (GRtensor/xAct) and such a program will be useful. For example, see this handy calculator for Hawking radiation of static black holes, which is useful and easy to work (by Viktor Toth, a member here).
Good luck
A: *

*Metric $ds^2$ in Cartesian/Spherical/... coordinates

*Inverse of the metric

*Angle between $d^{(1)}x^{\alpha}$ and $d^{(2)}x^{\alpha}$

*Christoffel symbols

*Geodesic equations

*Geodesic equations in Newtonian limit

*Components of generalized momentum

*Riemann tensor

*Ricci tensor

*Traceless Ricci tensor

*Ricci scalar

*Einstein tensor

*Weyl tensor

*Some of the identities (e.g Bianchi) and properties

