Neumann boundary conditions on S^1/Z_2

  1. 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!
     
  2. jcsd
  3. 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.
     
  4. lavinia

    lavinia 2,056
    Science Advisor

    I am not sure what you are doing. Are you modding the unit disc out by reflection along the x axis?
     
  5. I'm modding out the circle (e.g. x^2 + y^2 =1) by reflection about the y-axis.
     
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