Why you should take this online course

Whether or not you already know the basics of differential geometry, this course takes you step by step from the most elementary facts about curves and surfaces in three-space to fundamental ideas on Riemannian geometry. 

After completing the course, you will have a good understanding of concepts such as Riemannian manifolds, curvature, geodesics, covariant derivative, connections, etc. In short, you will learn pretty much everything you need to be able to read, understand, and discuss with your friends and colleagues the contemporary literature dealing with the nature and structure of spacetime. 

Learn from an expert

Oswaldo Zapata PhD has more than 20 years of experience helping physics students and professionals understand the mathematics of physics. 

He received his Bachelor's degree from Universidad Central de Venezuela, then he went to study particles physics in Paris (MSc) and string theory in Rome (PhD). Afterwards, he carried out research at the Albert Einstein Institute for Gravitational Physics in Golm (Germany), the International Center for Theoretical Physics in Trieste (Italy), the Kavli Institute for Theoretical Physics in Beijing (China), and the London School of Economics (UK).


The best quality content

With this online course you get access to:

  • The book written by Oswaldo Zapata PhD to complement the course, "Modern Geometry for Physicists 2. From Curves and Surfaces to Riemannian Manifolds" (approx. 130 pages in pdf format).
  • 25 hours of pre-recorded video lessons.
  • The personal notes Oswaldo Zapata PhD uses to teach the course (approx. 100 pages).
  • 30 minutes of one-on-one consultation with the author.
  • Personalized follow-up emails directly from Oswaldo Zapata PhD in order to check your progress as well as emails containing supplementary material (new videos, many exercises with solutions, discussion of advanced topics, answers to students' questions, etc.).


If you think you need extra help with any of the concepts treated in the course, you can also book private one-on-one classes with the author. Do not forget that the goal of this online course is to ensure that by the time you complete it, you master each and every topic covered in class. So, do not hesitate to get in touch!

With Oswaldo Zapata's expertise and your passion for physics, you will finally learn the arcane but simple and beautiful language in which Einstein's general theory of relativity is written.

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Contents

1 Curves and Surfaces

1.1 Curves

1.2 Surfaces

1.3 Differential Calculus on Surfaces

1.4 Integral Calculus on Surfaces

2 Dynamical Systems

2.1 Introduction

2.2 Linear Systems in the Plane

2.3 Nonlinear Systems in the Plane

2.4 Examples of Dynamical Systems on Surfaces

2.5 Covariant Derivatives and Geodesics

3 The Geometry of Surfaces

3.1 The Metric and the First Fundamental Form

3.2 The Second Fundamental Form

3.3 Connections

3.4 Curvatures

3.5 Covariant Derivative of Tensors

3.6 Regular Surfaces

4 Geodesics

4.1 The Global Approach

4.2 The Exponential Map and Geodesic Coordinates

4.3 Minimizing Geodesics

4.4 Completeness and the Theorem of Hopf-Rinow

4.5 Jacobi Fields

4.6 Gauss-Bonnet

5 Riemannian Geometry

5.1 The Metric

5.2 The Intrinsic Geometry of Riemannian Manifolds

5.3 Variations of the Energy

5.4 The Topology of Riemannian Manifolds

5.5 Analysis on Riemannian Manifolds