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O.K, please let me see if I got it right:
Let M be an orientable m-manifold with non-empty boundary B.
Let p be a point in B , and let {del/delX^1,...,del/delX^(m-1) }_p
be a basis for T_pB for every p in a boundary component .
Let N be a unit normal field on B . Now, this is the induced orientation (is it?):
We consider the collection {N_p, del/delX^1,...,delX^(m-1)}_p
(with N_p normal to M at p.)
AS IF p were a point in M, and not in the boundary B, (e.g., we can
smooth out the boundary so that it disappears, or we can cap
a disk or something, so that one boundary component disappears).
Then, if this basis {N_p, del/delX^1,...,delX^(m-1)}_p for p in M
of T_pM is oriented in agreement with the given orientation of M, (Jacobian of
chart overlap is positive, etc. ) then the boundary is positively oriented,
otherwise it is negatively oriented.
Is this it?
Thanks.
Let M be an orientable m-manifold with non-empty boundary B.
Let p be a point in B , and let {del/delX^1,...,del/delX^(m-1) }_p
be a basis for T_pB for every p in a boundary component .
Let N be a unit normal field on B . Now, this is the induced orientation (is it?):
We consider the collection {N_p, del/delX^1,...,delX^(m-1)}_p
(with N_p normal to M at p.)
AS IF p were a point in M, and not in the boundary B, (e.g., we can
smooth out the boundary so that it disappears, or we can cap
a disk or something, so that one boundary component disappears).
Then, if this basis {N_p, del/delX^1,...,delX^(m-1)}_p for p in M
of T_pM is oriented in agreement with the given orientation of M, (Jacobian of
chart overlap is positive, etc. ) then the boundary is positively oriented,
otherwise it is negatively oriented.
Is this it?
Thanks.