- #1
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Hi, All:
There is a standard method to construct a nowhere-zero form to show embedded
(in R^n ) manifolds are orientable ( well, actually, we know they're orientable and
we then construct the form).
Say M is embedded in R^n, with codimension -1. Then we can construct a nowhere-
zero top form by selecting the vector N(x) normal to the manifold ( say, using the Riemann metric inherited from R^n), and choosing an orthonormal frame {v1,v2,..,v(n-1)} for M . Then the form:
w:=Det | N(x) v1 v2...vn-1 | ,
where we write the vectors as columns, is nowhere-zero, since any two vectors are
perpendicular, and so the collection is linearly-independent.
**Now** how do we construct a form when:
i) M is embedded in R^n, and the codimension is larger than 1.
ii) For a curve, say a smooth curve.
iii) Can this be done/does it make sense when M is not embedded?
Thanks.
There is a standard method to construct a nowhere-zero form to show embedded
(in R^n ) manifolds are orientable ( well, actually, we know they're orientable and
we then construct the form).
Say M is embedded in R^n, with codimension -1. Then we can construct a nowhere-
zero top form by selecting the vector N(x) normal to the manifold ( say, using the Riemann metric inherited from R^n), and choosing an orthonormal frame {v1,v2,..,v(n-1)} for M . Then the form:
w:=Det | N(x) v1 v2...vn-1 | ,
where we write the vectors as columns, is nowhere-zero, since any two vectors are
perpendicular, and so the collection is linearly-independent.
**Now** how do we construct a form when:
i) M is embedded in R^n, and the codimension is larger than 1.
ii) For a curve, say a smooth curve.
iii) Can this be done/does it make sense when M is not embedded?
Thanks.