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analytic science forum beginner
Joined: 14 May 2005
Posts: 18

Posted: Tue Jul 11, 2006 4:46 pm Post subject:
Intrinsic definitions of pseudovectors/&c.



Hey,
Just a question: tangent vectors/cotangent vectors can be defined
either intrinsically as equivalence classess of differentiable
paths/scalar fields, or as quantities that reside in the charts with
various transformation rules; I only know pseudovectors/spinors/etc.
through transformation rules; is there any similarly pleasant
geometrical definition of these quantities? 

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Bossavit science forum beginner
Joined: 17 Jul 2005
Posts: 10

Posted: Tue Jul 18, 2006 11:41 am Post subject:
Re: Intrinsic definitions of pseudovectors/&c.



Lavelle:
Quote:  Just a question: tangent vectors/cotangent vectors can
be defined intrinsically as equivalence classes
(...) is there any similarly pleasant
geometrical definition of [pseudovectors/spinors/etc.]

You might find useful to refer to Burke's
"Applied Differential Geometry" (one among many
other sources): a pseudovector, or as Burke
says, a "twisted vector", in dimension n,
is indeed an equivalence class, made of two
equivalent pairs {v, Or} of type VECTOR x nCOVECTOR,
with Or (the ncovector) non null. The equivalence relation is {v,
Or} equiv. {v, Or'} iff Or and
Or' have opposite signs (ncovectors make a
1dimensional space, so this makes sense).
The geometrical picture is quite clear: a
pseudovector is a vector bearing, piggyback, an
orientation (a local one) of ambient space.
The opposite vector, burdened with the
opposite orientation, is another representative
of the same pseudovector (aka axial vector, or
twisted vector). As Burke shows convincingly,
vectors can also be identified with equivalence
classes of pairs, of type
{twisted vector, (orienting) ncovector},
so there is a nice symmetry between "straight"
objects, such as vectors, and twisted ones.
To the extent that tensors can be defined
intrinsically (as, for instance, multilinear
maps), this applies to them as well.
I have a litle more on that in
http://www.icm.edu.pl/edukacja/mat/axial.php 

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