- #1
Jianphys17
- 66
- 2
Hi, i would be very interested to start learning hyperbolic geometry, what would be the necessary prerequisites to begin it's study? 
Prerequisites for non Euclidean geometry
- Context: Undergrad
- Thread starter Jianphys17
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Discussion Overview
The discussion revolves around the necessary prerequisites for studying hyperbolic geometry, exploring the foundational knowledge required to approach this topic effectively. It includes considerations of related fields such as differential geometry and linear algebra.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant expresses interest in learning hyperbolic geometry and seeks guidance on prerequisites.
- Another participant suggests that familiarity with vector calculus and basic linear algebra is sufficient for starting the study of hyperbolic geometry.
- A different viewpoint is presented, indicating that knowledge of differential geometry of surfaces is also important, as referenced in a Wikipedia article.
- One participant elaborates that while vector calculus and linear algebra prepare one for differential geometry, understanding hyperbolic geometry ideally requires a background in the differential geometry of curves and surfaces in 3-space.
- Another participant asserts that learning about differential geometry of surfaces in 3D space is necessary before studying hyperbolic geometry.
- In contrast, a participant argues that differential hyperbolic geometry is a subset of hyperbolic geometry and not the only aspect to consider, suggesting it is not strictly necessary.
- One participant acknowledges differing viewpoints on the necessity of differential geometry, stating that while it may not be essential, it could enhance understanding of hyperbolic geometry.
Areas of Agreement / Disagreement
Participants express differing opinions on the necessity of differential geometry as a prerequisite for studying hyperbolic geometry. Some argue it is essential, while others believe it is not strictly required, leading to an unresolved discussion on the topic.
Contextual Notes
Participants reference various levels of understanding in mathematics, indicating that the prerequisites may depend on the specific aspects of hyperbolic geometry one wishes to study.
- #2
zinq
- 399
- 119
A good familiarity with vector calculus and basic linear algebra should suffice.
- #3
Jianphys17
- 66
- 2
Ah, but reading the article on wikipedia https://en.wikipedia.org/wiki/Hyperbolic_geometry believed is also required knowledge of differential geometry of surfaces...
- #4
zinq
- 399
- 119
The reason I didn't mention hyperbolic geometry specifically is that vector calculus and linear algebra prepare one for a basis course on the differential geometry of curves and surfaces in 3-space. This is just what one would need to understand hyperbolic geometry in the right context (even though a complete hyperbolic plane of constant Gaussian curvature = -1 cannot be embedded in 3-space if it is of differentiability class C2. — Hilbert's theorem).
Hyperbolic geometry is ideally learned as an advanced topic in a course on the differential geometry of curves and surfaces in 3-space. So, the prerequisites are for such a course.
Hyperbolic geometry is ideally learned as an advanced topic in a course on the differential geometry of curves and surfaces in 3-space. So, the prerequisites are for such a course.
- #5
Jianphys17
- 66
- 2
Hence so it would be necessary to learn before some notion of differential geometry of surfaces in 3d-space!
- #6
- 22,170
- 3,332
Jianphys17 said:
No, it is not necessary. Differential hyperbolic geometry is a subset of hyperbolic geometry, but definitely not the only part.
- #7
Jianphys17
- 66
- 2
Thank you for helping me solve this doubt! 
- #8
zinq
- 399
- 119
It is possible that micromass and I have different viewpoints about this issue. I won't say that differential geometry is absolutely necessary if all you want to learn is hyperbolic geometry.
But in my opinion it would greatly enhance your understanding, and would be an extremely wise thing to learn either before or during the time you learn about hyperbolic geometry.
But in my opinion it would greatly enhance your understanding, and would be an extremely wise thing to learn either before or during the time you learn about hyperbolic geometry.
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