Differential Geometry

Lecture Notes

Author

Affiliation

Dr. Silvio Fanzon

University of Hull

Welcome

These are the Lecture Notes of Differential Geometry 661955 for 2024/25 at the University of Hull. I will use this material during lectures. If you have any question or find any typo, please email me at

S.Fanzon@hull.ac.uk

Up to date information about the course, Tutorials and Homework will be published on the University of Hull Canvas Website

canvas.hull.ac.uk/courses/73612

Digital Notes

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Readings

We will study curves and surfaces in \(\mathbb{R}^3\), as well as some general topology. The main textbooks are:

• Pressley (Pressley 2010) for differential geometry,
• Manetti (Manetti 2023) for general topology.

Other good readings are the books:

• do Carmo (do Carmo 2017), a classic and really nice textbook
• Abate, Tovena (Abate, Marco and Tovena, Francesca 2011), for a more in depth analysis

I will assume some knowledge from Analysis and Linear Algebra. A good place to revise these topics are the books by Zorich (Zorich 2015, 2016).

Visualization

It is important to visualize the geometrical objects and concepts we are going to talk about in this module. Chapter 5 contains a basic Python tutorial to plot curves and surfaces. This part of the notes is not examinable.

If you want to have fun plotting with Pyhton, I recommend installation through Anaconda or Miniconda. The actual coding can then be done through Jupyter Notebook. Good references for scientific Python programming are (Johansson 2019; Kong, Qingkai, Siauw, Timmy, and Bayen, Alexandre 2020).

If you do not want to mess around with Python, you can still visualize pretty much everything we will do in this module using

• Desmos
• CalcPlot3D

Important

You are not expected to purchase any of the above books. These lecture notes will cover 100% of the topics you are expected to known in order to excel in the Homework and Final Exam.

Abate, Marco, and Tovena, Francesca. 2011. Curves and Surfaces. Springer.

do Carmo, Manfredo P. 2017. Differential Geometry of Curves and Surfaces. Second Edition. Dover Books on Mathematics.

Johansson, Robert. 2019. Numerical Python. Scientific Computing and Data Science Applications with Numpy, SciPy and Matplotlib. Second Edition. Apress.

Kong, Qingkai, Siauw, Timmy, and Bayen, Alexandre. 2020. Python Programming and Numerical Methods. Academic Press.

Manetti, Marco. 2023. Topology. Second Edition. Springer.

Pressley, Andrew. 2010. Elementary Differential Geometry. Second Edition. Springer.

Zorich, Vladimir A. 2015. Mathematical Analysis I. Second Edition. Springer.

———. 2016. Mathematical Analysis II. Second Edition. Springer.