Symmetric connections, corrected version November 9, 2009
Posted by Akhil Mathew in differential geometry, MaBloWriMo.Tags: connections, corrections, symmetric connections, torsion tensor
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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 be a surface in , and let be a symmetric connection on . Then (more…)
The torsion tensor and symmetric connections November 8, 2009
Posted by Akhil Mathew in differential geometry, MaBloWriMo.Tags: connections, symmetric connections, torsion tensor
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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 on the smooth manifold , define the torsion tensor by