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A way of defining an orientation form when given a codimension-1 ,

orientable n-manifold N embedded in R^{n+1} , in which

the gradient ( of the parametrized image ) is non-zero (I think n(x)

being nonzero is equivalent to N being orientable), is to

consider the nowhere-zero normal vector n(x), and to define the

form w(v)_x : = | n(x) v1 , v2 ,...,v_n-1| (**)

Where {vi}_i=1,..,n-1 is an orthogonal basis for T_x N , written

as column vectors, and n(x) is the vector normal to N at x , so that we write:

| n_1(x) v_11 v_21..... v_n1|

| n_2(x) v_12 v_22......v_n2|

................................

................................

|n_n(x) v_1n v_2n.......v_nn|

For vi= (vi1, vi2,....,vin )

Then the vectors in (##) are pairwise orthogonal, and so are

Linearly-independent.

*QUESTION* : How do we define a form for a curve of codimension-1,

and, in general, for orientable manifolds of codimension larger-

than 1 ? I have seen the expression t(x).v , meaning <t(x),v> ,for the curve. But the

tangent space of a curve is 1-dimensional, so, how is this a dot product?

Also, for codimension larger than one: do we use some sort of tensor contraction?

Thanks.

Thanks.

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# Orientation Forms in Different Codimension.

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