## Symmetric connections, corrected versionNovember 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…)

## The torsion tensor and symmetric connectionsNovember 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…)