Differential Geometry

Lecture Notes

Author

Affiliation

Dr. Silvio Fanzon

University of Hull

Welcome

These are the Lecture Notes of Differential Geometry 661955 for T1 2023/24 at the University of Hull. We will study curves and surfaces in \(\mathbb{R}^3\). I will follow these lecture notes during the course. 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/67594

and on the Course Webpage hosted on my website

silviofanzon.com/blog/2023/Differential-Geometry

Digital Notes

Features of these digital notes:

• Navigation bar is available on the left
• A search function is available on the top-right
• You can toggle dark mode on the top-right
• A pdf version of these notes is available to download on the top-right

Readings

Main textbooks:

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

Other interesting readings are the books by do Carmo (do Carmo 2017) and Abate, Tovena (Abate, Marco and Tovena, Francesca 2011). 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 course. I will show basic Python code to plot curves and surfaces. This part of the course is not required for the final examination. 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 course using the excellent online 3D grapher tool CalcPlot3D. To understand how it works, please refer to the help manual or to the short video introduction. Another nice tool is Desmos.

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 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.