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MathematicalPhysicist
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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.
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.