Difference between revisions of "A Portal Special Presentation- Geometric Unity: A First Look"
A Portal Special Presentation- Geometric Unity: A First Look (view source)
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<p>[01:12:17] We now define the chimeric bundle, right? And the chimeric bundle is this direct sum of the vertical tangent bundle along the fibers with the pullback, which we're going to call the horizontal bundle from the base space. | <p>[01:12:17] We now define the chimeric bundle, right? And the chimeric bundle is this direct sum of the vertical tangent bundle along the fibers with the pullback, which we're going to call the horizontal bundle from the base space. | ||
So the chimeric bundle is going to be the vertical tangent space of 10-dimensions to $$U | So the chimeric bundle is going to be the vertical tangent space of 10-dimensions to $$T^10_V(U) \oplus T^4_H(U)$$ the four-dimensional cotangent space, which we're going to call horizontal to $$U$$. And the great thing about the chimeric bundle is that it has an a priori metric. It's got a metric on the four, a metric on the 10, and we can always decide that the two of them are naturally perpendicular to each other. | ||
<p>[01:13:00] Furthermore, it is almost canonically isomorphic to the tangent bundle or the cotangent bundle, because we either have 4 out of 14, or 10 out of 14 dimensions, on the nose. So the question is, what are we missing? And the answer is that we're missing exactly the data of a connection. So this bundle, chimeric $$C$$. We have $$C$$ equal to the tangent bundle of $$U$$ up to a choice of a connection $$\theta$$. | <p>[01:13:00] Furthermore, it is almost canonically isomorphic to the tangent bundle or the cotangent bundle, because we either have 4 out of 14, or 10 out of 14 dimensions, on the nose. So the question is, what are we missing? And the answer is that we're missing exactly the data of a connection. So this bundle, chimeric $$C$$. We have $$C$$ equal to the tangent bundle of $$U$$ up to a choice of a connection $$\theta$$. | ||