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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.

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# Using Forms to Define Orientation of Curves.

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