SageManifolds

What is SageManifolds?

SageManifolds is a sub-project of SageMath and provides a comprehensive symbolic and numeric tensor calculus on manifolds. The code is distributed under an open-source license and hence freely available. All functionalities are interfaced via the Python programming language which makes it a perfect tool for research and teaching purposes. The capabilities range from computations with tensor fields, curvatures and Lie derivatives to vector bundles, bundle connections and, most recently, characteristic classes.

Vector Bundles

In 2019/2020, I contributed vector bundles to the SageManifolds project by generalizing the preexisting implementation of tensor fields. All computations are done in local frames. A short introduction is planned in a future post.

Characteristic Classes

The primary goal of my master’s thesis was to implement characteristic classes on arbitrary vector bundles. This has been achieved in early 2020. Since version 9.0 of SageMath, it is possible to compute the characteristic form of a given characteristic class and a bundle connection. The implementation bases on the Chern–Weil homomorphism and includes the computation of multiplicative as well as additive characteristic classes and the Euler class. An introductory post on this topic is coming soon.

Currently Working on

  • Remove code redundancies caused by the new implementation of sections on vector bundles
  • Add new vector bundle features such as vector-valued forms, bundle automorphisms, bundle metrics
  • Transgression forms of characteristic classes
  • Alternative algorithm for characteristic forms involving multiplicative/additive sequences (possible speed-up)

Interested?

If you have questions regarding this project, don’t hesitate to contact me via email. You are also invited to contribute.