Manifolds

This python package is designed to work with sage. Attempting to run it in plain python will not work!

All the libraries in this package are designed to work in local coordinates within a single coordinate chart, with the focus on explicit computations. The three base object for all of this are: Chart, Tensor, and Form.

Chart Objects

All computations which use this package require a base Chart object. In additions to the base Chart class, there are several subclasses that are more convenient to use:

• RiemChart - A subclass of Chart for charts on Riemannian manifolds.
• ComplexChart - A subclass of Chart for charts with complex coordinates, which are split into holomorphic and antiholomorphic.
• KahlerChart - A subclass of both ComplexChart and RiemChart classes to model Kähler manifolds, built from a Kähler potential function.
• GenChart - A meta-class which is a subclass of RiemChart, meant for manifolds with a metric and a torsion-free connection, which need not be compatible with the metric. There are 2 implementations of this provided in the package: DensChart and StatChart.

Note that for many computations either RiemChart or a subclass of it is required.

Tensor Objects

The Tensor class is used to model arbitrary tensors on any chart, one of the few computations that can be performed in general is tracing the tensor's upper and lower indices. If the tensor lives on a RiemChart, one can also raise/lower indices and compute the covariant derivative.

Form Objects

The Form class is used to model arbitrary forms on any chart. Each form is stored using a dictionary in the .vals property, which is structured as {frozenset({i1,i2,...,ik}): f_{i1,i2,...,ik}} to represent f_{i1,i2,...,ik} dx^{i1} \wedge dx^{i2} \wedge ... \wedge dx^{ik}.