Flow Chart of material to learn differential geometry

[jhoffs1]

I am a 3rd year mechanical engineering student at LSU, but my true interest lies in theoretical physics and mathematics (specifically general relativity and differential geometry). I've taken calculus 1,2,3, linear algebra, ordinary differential equations, number theory, discrete math, and statistics (at a local university while I was in high school). My question is, where would I go next after tensor calculus if I want to fully grasp differential geometry/relativity in a full sense (not watered down)? I bought "introduction to tensor calculus and continuum mechanics" and I am almost finished with the tensor analysis (not too difficult) and am trying to find where I need to go next. Sent from my iPhone using Physics Forums

 

[jhoffs1]

Flow chart for general relativity

What classes should be taken/self-taught to fully understand and work with GR? I have taken calc 1,2,3 linear algebra, statistics, number theory, discrete math, and am just finishing "introduction to tensor calculus and continuum mechanics", but where should I go next? Straight to differential geometry?Sent from my iPhone using Physics Forums

 

[WannabeNewton]

You already know enough to delve into a myriad of GR texts.

 

[lavinia]

if you are interested in modern physics I suggest learning Differential Geometry on Manifolds. General Relativity is one entry point to this but pure mathematical texts can be helpful.

 

[Matrix0]

As a fellow scientist in the field of differential geometry, I understand your passion for theoretical physics and mathematics. It is great to see that you have already taken a variety of courses in mathematics and have a solid foundation in calculus, linear algebra, and differential equations.

To fully grasp differential geometry and its application in general relativity, I would suggest delving into more advanced topics such as differential topology, Riemannian geometry, and differential manifolds. These areas of study will help you understand the geometric concepts and structures that underlie general relativity.

Some specific topics to focus on would be the curvature of space-time, the Einstein field equations, and the geodesic equation. It may also be helpful to study the mathematical theories and frameworks that have been developed to describe and explain these concepts, such as the Cartan formalism and the theory of connections.

In addition to theoretical study, I would also recommend exploring practical applications of differential geometry in fields such as cosmology, astrophysics, and quantum gravity. This will give you a deeper understanding of how this mathematical framework is used to describe and model the physical world.

Overall, I would encourage you to continue learning and exploring in this field, as there is always more to discover and understand. Best of luck on your journey towards a full grasp of differential geometry and its applications in general relativity!