Neumann boundary conditions on S^1/Z_2

by AntideSitter
Tags: boundary, conditions, neumann, s1 or z2
AntideSitter is offline
Apr30-10, 10:37 PM
P: 7
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!
Phys.Org News Partner Science news on
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered
AntideSitter is offline
Apr30-10, 10:48 PM
P: 7
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.
lavinia is offline
May1-10, 08:37 AM
Sci Advisor
P: 1,716
I am not sure what you are doing. Are you modding the unit disc out by reflection along the x axis?

AntideSitter is offline
May1-10, 04:32 PM
P: 7

Neumann boundary conditions on S^1/Z_2

I'm modding out the circle (e.g. x^2 + y^2 =1) by reflection about the y-axis.

Register to reply

Related Discussions
how to define neumann boundary in matlab? Differential Equations 0
Find Flux Density On One Side of Dielectric Boundary Given Boundary Conditions Advanced Physics Homework 0
Second Order Numerical Integration w/ Neumann Boundary Conditions Differential Equations 1
What's the difference between initial conditions and boundary conditions? Differential Equations 9
Boundary-value problems (Neumann condition) Differential Equations 2