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Hi, Everyone:
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.
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.