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Orinetability questions.

  1. Jul 28, 2008 #1
    I have a few question, I hope you can help me on some of them.

    1.Show that if A (subset of R^n) is a submanifold with dimension, n, with boundary then dA (the boundary of A) is orientable.
    2. Show that a torus in R^3 is orientable.
    3.Show that a mobius band isn't orientable.
    4. Let M,N be two connected oriented manifolds. Let f:M->N be a diffeomorphism.
    Show that [tex]df_x:T_x M\rightarrow T_f(x) N[/tex] either preserves or reverses orientation for all x in M simultaneously.

    Here is what I thought of:
    1)I think that the standard orientation on R^n is induced to the boundary of A.

    4) I need to prove that the determinant of df_x is always positive or negative, now from the definition of orientation on M and N, we have two diffeomorphism [tex]\psi , \phi[/tex]
    such that for every x in M there's a neighbourhood U, such that: psi is a local diffeomorphism of U onto an open set V of R^N, and for every z in U [tex]d\psi_z : T_z M\rightarrow R^m[/tex] keeps the orientation, the same for N.
    Now if f can be broken into two diffeomorphisms one from [tex]T_x M \rightarrow R^m[/tex]
    the other from [tex] T_f(x) N \rightarrow R^m[/tex], then the determinant of df_x would be equal the product of two determinants which both of them have a plus sign cause they keep the orientation.

    2. a torus is [tex]S^1 x S^1[/tex] where S^1 is a circle, intuitively I understand why it's orintebale but how to prove it rigourosly?
    I mean I think I need to show that if I induce the standrad orientation of R^3 onto the torus, it keeps orientation, not sure.

    3. the same for 2, just inducing the standard orientation and to show the determinat changes signs from some point.
  2. jcsd
  3. Jul 28, 2008 #2
    for 4, I think it should be phrased differently, like this:
    Suppose that f:M->N is a diffeomorphism of connected oriented manifolds with boundary, show that if [tex]df_x[/tex] for some point x in M, preserves orientation, then f preserves orientation globally.
    (the first statement of problem was from my lecturer).

    well I think this follows quite immediately if we look at first order approximation of f, like this:
    [tex]f(x+x0)=df_x_0 x +o(||x||)[/tex]
    then if we take df_x at some point x in M (other than x0), then its determinant will have the same sign as of [tex]df_x_0[/tex] cause the other orders of the taylor expansion are o(1) and thus don't affect its sign.

    I feel that it's mambo jambo, can anyone help me on this?

    thanks in advance.
  4. Aug 2, 2008 #3
    1. Using the standard orientation to prove that all closed n-1 dimensional manifolds embedded in [tex]R^n[/tex] are orientable? Of course you must then prove that said boundaries are of this form.

    2. First prove that [tex]S^1[/tex] is orientable.
    Can you show that if [tex]M,N[/tex] are smooth manifolds then [tex]M\times N[/tex] has a canonical differentiable structure? What if the original manifolds were orientable? (This is a standard theorem).

    3. I'll have a think :)

    4. The restatement of the problem makes sense. Play around with the following..
    Take a curve c passing through p, then [tex]df_p[/tex] maps the vector [tex](\phi\circ c)'(0)[/tex] to [tex](\psi\circ f\circ c)'(0)=(\psi\circ f\circ\phi^{-1}\circ\phi\circ c)'(0)=d(\psi\circ f\circ\phi^{-1})_{\phi(p)}\circ(\phi\circ c)'(0)[/tex] (where we're using appropriate charts). Because we're dealing with a diffeomorphism, [tex]d(\psi\circ f\circ\phi^{-1})[/tex] is invertible, so nonzero, and varies continuously through members of GL(n,R) as we go from point to point.
  5. Aug 4, 2008 #4
    For part 1, orientability is a topological invariant, right? If this is so, wouldn't it suffice to prove the result for very simple shapes such as a rectangular n-box with rectangular holes cut out of it?

    Part 2 looks like a straightforward application of part 1.

    Part 3. For a mobius band, one could take a triangulization with only a few triangles, and then use counting arguments on the edges and faces to show that at least 2 faces next to each other must be oriented oppositely. The orientation of a face may be expressed by directing its edges in a loop, and for it to be orientable is equivalent to having each edge that shares a face be directed oppositely, unlike gears. The outer edge will all have to go the same direction, which will wreak havok with any attempts to antialign the inner edges.

    I don't have enough background to understand what part 4 is asking.
    Last edited: Aug 4, 2008
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