Neumann boundary conditions on S^1/Z_2

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Discussion Overview

The discussion revolves around the application of Neumann boundary conditions on the orbifold S^1/Z_2, particularly focusing on the implications of the Z_2 action and the nature of singularities in this context. Participants explore the mathematical properties of functions defined on this space and the conditions that arise from the boundary behavior.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant describes the action of Z_2 on S^1 and the resulting orbifold structure, suggesting that functions on S^1 must be invariant under this action, leading to Neumann boundary conditions at the boundaries.
  • Another participant corrects the initial claim about singularities, noting that there are actually two singularities at x=0 and x=1, which may necessitate the use of two coordinate patches.
  • A participant questions the approach, asking if the modding out is being done on the unit disc rather than the circle.
  • Another clarifies that the modding out is being performed on the circle by reflection about the y-axis, indicating a potential misunderstanding in the previous post.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the singularities and the appropriate boundary conditions. There is no consensus on the necessity of Neumann conditions or the implications of the singularities, indicating ongoing debate and exploration of the topic.

Contextual Notes

The discussion highlights potential limitations in understanding the boundary conditions due to the presence of singularities and the specific nature of the Z_2 action. The implications of these factors on the choice between Neumann and Dirichlet conditions remain unresolved.

AntideSitter
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Hello everybody,

I've been puzzling over something (quite simple I assume).

Take S^1. Now consider the action of a Z_2 which takes x to -x, where x is a natural coordinate on the cylinder ( -1< x <1). Now we mod out by this action. The new space is an orbifold: smooth except at x=0. It is diffeomorphic to the line internal. We can take 0< x <1 as a coordinate on S^1/Z_2.

Take the set of functions on S^1, call it F_s. To find the set of functions on S^1/Z_2, we should restrict F_s to the the functions which are invariant under the Z_2 action. I.e. they should be symmetric under the reflection. These functions all have f '(0)=f '(1) = 0. These are Neumann boundary conditions on the boundary of S^1/Z_2.

Firstly, correct any errors I have made here. My question is: why must there be Neumann boundary conditions on functions on the line interval? Is this a generic property of manifolds with boundaries? Why are Dirichlet conditions not possible here? Is the singularity important in this example?

Comments welcome. Thank you!
 
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Just to correct myself: there are two singularities on S_1/Z_2 , at x=0,1. Strictly speaking we should use two coordinate patches. So there are singularities at each end of the line interval, just where my boundary conditions are.
 
I am not sure what you are doing. Are you modding the unit disc out by reflection along the x axis?
 
I'm modding out the circle (e.g. x^2 + y^2 =1) by reflection about the y-axis.
 

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