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Topic:
Differential Forms and Orientation. Case for curves, and Codimension>1 .
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Bacle H
Posts:
283
Registered:
4/8/12


Differential Forms and Orientation. Case for curves, and Codimension>1 .
Posted:
Nov 8, 2012 9:35 PM


Hi, All:
One way of defining an orientation form for an codimension1 ,
orientable nmanifold N embedded in R^{n+1} , in which
the gradient ( of the parametrized image ) is nonzero, is to
consider the nowherezero normal vector n(x), and to define the
form w(v)_x : =  n(x) v1 , v2 ,...,v_n1 (##)
Where {vi}_i=1,..,n1 is an orthogonal basis for T_x N , written
as column vectors, and n(x) is the vector normal to N at x .
Then the vectors in (##) are pairwise orthogonal, and so are
Linearlyindependent.
*QUESTION* : How do we define a form for a curve of codimension1,
and, in general, for orientable manifolds of codimension larger
than 1 ?
Thanks.



