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In R.W. Sharpe's Differential Geometry, a flat fibre bundle is defined as a bundle whose transition functions are constant. I don't understand the difference between this and a trivial bundle. Because, surely if the transition functions are constant you can just perform a rescaling so that they all become the identity, making the bundle trivial?

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2 Answers 2

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In my opinion, you should always try to think of the easiest counter example. In this case it is the Mobius bundle.

Let $S^1$ be the circle with standard open cover $\{U,V, W\}$, so all the intersections are connected. Define a real vector bundle by the transition functions $g_{UV}=1$, $g_{VW}=1$ and $g_{WU}=-1$. This is the Mobius bundle. The transition functions are defined to be constant, but the bundle is non-trivial.

More generally, we should address your confusion regarding rescaling. It is not true that two bundles whose transition functions are related by a scalar multiple, or a matrix-valued function of sorts, are isomorphic. What is true is that two bundles whose transition functions are homotopic are isomorphic. But the maps $1:W\cap U\to\mathbb{R}^\times$ and $-1:W\cap U\to\mathbb{R}^\times$ are not homotopic, and that is a very important detail to consider. They are homotopic as maps into $\mathbb{R}$, but we have to "remove" the origin when we're talking about transition functions.

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Let us consider an example. We are on a circle. Draw it so that there is a North pole $N$, and a South pole $S$ for a one dimensional ant. The ant has two charts of its world, the one around $N$ is colored in (the color) navy, the one around $S$ is colored in (the color) sienna. Each chart goes show below the "ecuatorial region", so the two charts intersect in two disjoint arcs of the world, call them East and West.

The ant wants to make a trip. It walks, and can understand somehow an up and a down, because it knows something about a vector bundle of the world. This vector bundle has fibers $\Bbb R$ in each point in navy, and also in each point in sienna.

However, the many fibers are pasted together in an unexpected manner for the ant. The Western common piece uses the constant $+1$ to identify the fibers over the same points. The Eastern common piece uses the constant $-1$ to identify the fibers over the same points.

This bundle is not trivial! The ant needs to make two loops on the world to be "up again". (Möbius band like situation.)

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