A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions
A Portal Special Presentation- Geometric Unity: A First Look (edit)
Revision as of 17:08, 25 April 2020
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===== The Levi-Civita Connection ===== | ===== The Levi-Civita Connection ===== | ||
<p>[01:20:01] Okay. What is it that we get for the Levi-Civita connection? Well, not much. One thing we get is that normally the space of connections is an affine space | <p>[01:20:01] Okay. What is it that we get for the Levi-Civita connection? Well, not much. One thing we get is that normally the space of connections is an affine space. Not a vector space, but an affine space. Almost a vector space, a vector space up to a choice of origin. But with the Levi-Civita connection, rather than having an infinite plane with an ability to take differences, but no real ability to have a group structure. | ||
<p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection $$A$$ has a torsion tensor, $$T_A$$, which is equal to the connection, $$A$$ minus the Levi-Civita connection ($$\nabla^{LC}$$). So we get a tensor that we don't usually have. Gauge potentials are not usually well-defined. They're are only defined up to a choice of gauge. | <p>[01:20:35] You pick out one point, which then becomes the origin. That means that any connection $$A$$ has a torsion tensor, $$T_A$$, which is equal to the connection, $$A$$ minus the Levi-Civita connection ($$\nabla^{LC}$$). So we get a tensor that we don't usually have. Gauge potentials are not usually well-defined. They're are only defined up to a choice of gauge. | ||