Exploring Various Tools of Differential Topology on a Smooth Manifold using SageMath

Affiliation

UM-DAE Centre for Excellence in Basic Sciences, Mumbai

Author

• Vishvas RANJAN
  UM-DAE Centre for Excellence in Basic Sciences, Mumbai

Supervisors

• Prof. Dr. Ajit KUMAR
  Institute of Chemical Technology, Mumbai

Project Overview

This repository contains a SageMath‐based exploration of differential topology concepts on the 2‐sphere (S^2). We start by recalling the definitions of:

• Topological (n)‑manifolds
• Coordinate charts & transition maps
• Atlases & smooth manifolds

and then move on to:

• Tangent spaces and tangent vectors
• The differential of a smooth map
• Tangent bundles
• Vector fields, local & global frames

Finally, we apply all of the above to the 2‐sphere:

1. Define stereographic charts (north and south) and the transition map between them.
2. Compute and visualize tangent spaces on each chart.
3. Illustrate how the sphere “looks locally” via plotting routines in SageMath.
4. Represent and analyze vector fields on (S^2), including global frames where possible.

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Features

• Chart Definitions – Explicit SageMath code for the two standard stereographic charts on (S^2).
• Transition Maps – Symbolic computation of the transition functions between north and south charts.
• Tangent Bundle – Construction and visualization of tangent vectors and bundles.
• Plotting & Visualization – 2D and 3D plots to illustrate coordinate patches and vector fields.
• Notebook‐Driven – Everything is packaged as Jupyter/SageMath notebooks for easy exploration.

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Thanks for checking out our work!