﻿What is a bundle - explained simply.


Bundles are used when you want to attach something at every point of an object, for example a fiber. The terminology works out quite nicely here because indeed in mathematics the name of the attached thing is called the “fiber”.


One thing to note is that bundles are not used for construction, but rather for analysing the intricacies of the underlying space.


Imagine you have a circle, and every point you want to attach a line. When you do that, you get a cylinder.


Now imagine you want to attach again a line at every point of the circle, but you spin it around a little bit every time. Then you get a mobius strip.


The point is that these two objects have different internal properties, and these can be explored with a bundle.


In this case the cylinder is trivial because there exists a bundle isomorphism to a product bundle (between the circle and the interval [0,2] for example). The mobius strip is locally trivial because it is locally isomorphic to a product bundle.


Both examples above are fiber bundles because all their fibers are isomorphic to each other.


Another hugely important idea of bundles is sections of the bundles, named after the cross section of a cylinder, which is indeed a valid example of a section.


The main use case for bundles is actually with vector fields, since vector fields are sections of the tangent bundle, where the tangent bundle is the bundle of all tangent vector spaces.


You can actually give bundles more structure by restricting that the map from the total space to base space is of the preserving structure (let’s say continuous and smooth). This is actually used with the tangent bundle, because we want to ensure that vector fields (sections of the tangent bundle) are smooth. This condition can be enforced by requiring that the section map (from point on base space to element in total space) is smooth. Of course this only makes sense if the total space and base space have a topology and a smooth atlas equipped, or else how can you make sense of smoothness.


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One thing we assume is that parallel transport is linear and metric compatible. This is an assumption on the status of the connection. But why do we assume that the connection is torsion free?


Cool language trick: you can say something is a necessary condition if it’s completely larger. But you can say it’s a sufficient condition if it’s completely equal. Quite cool language that I totally plan to use.


For example with my proof of the linear algebra problem with Adithya, the original proof I did of showing that blah showed that it was necessary, but not sufficient.


The other tensor based on connection we need to understand is curvature.


In fact all of these tensors based on connections are interesting because we can further restrict what our connection must be without looking at it directly, since it’s not a very obvious object, unlike tensors which are quite nice because they’re invariant under different charts.