Re: An affine connection ain't an affine connection
- From: "Shmuel (Seymour J.) Metz" <spamtrap@xxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Wed, 21 Feb 2007 18:28:46 -0500
In <1170840059.874571.42250@xxxxxxxxxxxxxxxxxxxxxxxxxxx>, on
02/07/2007
at 01:20 AM, startdancingonfive@xxxxxxxxx said:
Parallel transport of a vector along a geodesic is defined by holding
the angle of the vector constant along the geodesic.
Perhaps in a really old book. That's certainly not a contemporary
definition.
Curvature is defined by parallel transporting a vector along a
(small) loop and calculating the angle between the original and the
transported vector.
C 'angle' 'difference'
So, parallell transport and curvature depends heavily on the fact
that we (me and the trouts?) are able to calculate angles.
No,
Without a metric it makes no sense to talk about curvature.
Incorrect.
Also parallel transport is meaningless in the absense of a metric.
Likewise incorrect.
Instead of curvature we have torsion.
No; curvature is curvature and torsion is torsion; they're both
relevant.
There are two kinds of connections:
Metric connection. Has curvature. No such thing as torsion.
The fact that the torsion happens to be zero doesn't mean that the
torsion tensor doesn't exist.
In S^3 the great circles are geodesics of both the natural metric
connection and the (Lie group) affine connection.
What Lie group? There's no group structure in S^3.
What'ya say,
Read a book on Differential Geometry. I'd expect that Spivak covers
connections.
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