How to understand fig. 15.16, page 346 of Penrose's R2R

  • A
  • Thread starter math771
  • Start date
In summary, fig. 15.16 on page 346 of Penrose's R2R is a diagram or illustration that visually represents a concept or data related to the topic being discussed in the text. It is important to carefully read and understand the accompanying text and any labels or legends in order to interpret the figure correctly. This figure is important because it can help clarify complex concepts or provide evidence for scientific claims. If you are having trouble understanding the figure, it is recommended to review the text and seek clarification from a teacher or mentor. If you wish to use the figure in your own work, make sure to properly cite it and obtain permission from the copyright holder.
  • #1
math771
204
0

Hi there! I can understand your confusion about this example of a `strained' line bundle. It definitely seems counterintuitive that a vertical displacement would result in a helical path. However, I think the key to understanding this is to remember that a `connexion' on a bundle allows for a non-trivial parallel transport between fibers. In other words, the connection allows the fiber to rotate as it moves along the base manifold.

In this case, the failed attempt to trace out a horizontal section is due to the fact that the fiber is rotating as it moves along the base manifold. So even though you start at point b and move vertically, the fiber is also rotating, causing the path to wind upwards. This is why the rotation is double the initial displacement after completing a 360 degree traversal.

I hope this helps to clear up some of your confusion. It's definitely a tricky concept to wrap your head around, but just remember that the connection allows for non-trivial parallel transport and that should help make sense of the helical path. Happy learning!
 
Physics news on Phys.org
  • #2
I am very confused by this otherwise intriguing example of a `strained' line bundle. RP claims it is an example of a `connexion' on the bundle.
The figure depicts the bundle above the S1 fibre base manifold, & a failed attempt to trace out a horizontal section. A black & a white arrow indicate CW & CCW paths to complete a 360 degree traversal. Yet the `attempt' traces out a Helical Path ! This is what's giving me fits ! How can a mere vertical displacement (above pt. b) result in a helical path, causing the path to wind upwards by double the initial displacement after the rotation is complete ?
Thanx !
 
  • #3
Jim said:
I am very confused by this otherwise intriguing example of a `strained' line bundle. RP claims it is an example of a `connexion' on the bundle.
The figure depicts the bundle above the S1 fibre base manifold, & a failed attempt to trace out a horizontal section. A black & a white arrow indicate CW & CCW paths to complete a 360 degree traversal. Yet the `attempt' traces out a Helical Path ! This is what's giving me fits ! How can a mere vertical displacement (above pt. b) result in a helical path, causing the path to wind upwards by double the initial displacement after the rotation is complete ?
Thanx !
Can you describe the example in words?
 

1. What does fig. 15.16 on page 346 of Penrose's R2R represent?

Fig. 15.16 on page 346 of Penrose's R2R is a diagram or illustration that visually represents a concept or data related to the topic being discussed in the text. In this case, it could be a diagram of a scientific process, a graph of data, or an illustration of a complex concept.

2. How can I interpret fig. 15.16 on page 346 of Penrose's R2R?

Interpreting fig. 15.16 on page 346 of Penrose's R2R requires a thorough understanding of the topic being discussed and the information presented in the figure. It is important to carefully read the accompanying text and refer to any labels, legends, or scales included in the figure to understand its meaning.

3. Why is fig. 15.16 on page 346 of Penrose's R2R important?

Fig. 15.16 on page 346 of Penrose's R2R is important because it visually represents information or data that is relevant to the topic being discussed. It may help to clarify complex concepts, provide evidence for a scientific claim, or aid in understanding a scientific process.

4. What should I do if I don't understand fig. 15.16 on page 346 of Penrose's R2R?

If you are having trouble understanding fig. 15.16 on page 346 of Penrose's R2R, it is important to first review the accompanying text and any labels or legends included in the figure. You can also refer to other sources or ask for clarification from a teacher or mentor.

5. Can I use fig. 15.16 on page 346 of Penrose's R2R in my own work?

If you wish to use fig. 15.16 on page 346 of Penrose's R2R in your own work, it is important to properly cite the figure and obtain permission from the copyright holder if necessary. Additionally, make sure to fully understand the figure and its relevance to your work before using it.

Similar threads

A How to derive Quantum Mechanics in curved physical space?
    • Quantum Physics
Replies
6
Views
928
I Arbitrariness of connection and arrow on sphere
    • Differential Geometry
Replies
14
Views
3K
I Understanding Parallel Transport
    • Differential Geometry
Replies
9
Views
5K
Horizontal Lift vs Parallel Transport in Principal Bundle & Riemannian Geometry
    • Differential Geometry
Replies
4
Views
6K
How arbitrary are connections?
    • Differential Geometry
Replies
4
Views
3K
Connections in differential geometry
    • Differential Geometry
Replies
13
Views
5K
Baez and Schreiber: Higher Gauge Theory
    • Beyond the Standard Models
Replies
2
Views
3K
LQG: Gauge Theory, Holonomies, Ashtekar's GR, Loop Representation
    • Beyond the Standard Models
Replies
2
Views
3K
Requesting suggestions for languages, libraries, and architectures for parallel (and sometimes non parallel) numerical and scientific computations
    • Programming and Computer Science
Replies
8
Views
2K
Prerequisite mathematics for string theory and loop quantum gravity
    • Beyond the Standard Models
Replies
21
Views
25K

Hot Threads

Recent Insights

Back
Top