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Symmetric connections, corrected version November 9, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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My post yesterday on the torsion tensor and symmetry had a serious error.  For some reason I thought that connections can be pulled back.  I am correcting the latter part of that post (where I used that erroneous claim) here. I decided not to repeat the (as far as I know) correct earlier part.

Proposition 1 Let {s} be a surface in {M}, and let {\nabla} be a symmetric connection on {M}. Then\displaystyle \frac{D}{\partial x} \frac{\partial}{\partial y} s = \frac{D}{\partial y} \frac{\partial}{\partial x} s.\ \ \ \ \ (1)  (more…)

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The torsion tensor and symmetric connections November 8, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Today I will discuss the torsion tensor of a Koszul connection. It measures the deviation from being symmetric in a sense defined below.

Torsion

Given a Koszul connection {\nabla} on the smooth manifold {M}, define the torsion tensor {T} by

\displaystyle T(X,Y) := \nabla_X Y - \nabla_Y X - [X,Y].  (more…)