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My question is simply: how does one think about sections of nontrivial vector bundles on a smooth manifold, for example? The canonical example I think of is a vector field, i.e. a section of the tangent bundle, or even a differential form (section of the cotangent bundle; slightly harder to visualize as literally as tangent vectors, but still somewhat concrete). I know that the Mobius strip gives a nontrivial bundle over the circle, which is perhaps the only one I can see clearly because of the low dimension, and even then the sections seem not so intuitive. However, when it comes to an arbitrary vector bundle, I have no idea how to imagine the sections.

My motivation is that I'm trying to learn algebraic geometry and sheaf theory, which is explained to me as trying to generalize vector bundles (by just declaring the sections on an open set), but I'm having trouble how to think of these if they're not just trying to emulate vector fields/differential forms (or of course, "smooth functions" on a space in the case of the trivial bundle, which I like as motivation for the structure sheaf on a scheme). Any examples from (beginner) algebraic geometry would be appreciated too.

Thanks.

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Only very few mathematicians understand a concept without a lot of examples. So learn more examples!

I think a great class of examples form the Grassmannians (and are somehow all that you really need to understand, by some universality magic). These are the spaces of $n$-dimensional subspaces of $\mathbb{R}^k$. The bundle just attaches to a point (i.e. an $n$ plane) the plane itself.

You should spend time with the $n=1$ case: the projective spaces. For low dimensions you can draw the sections. The complex analogous are somehow easier to understand. These are $n$ dimensional complex subspaces of some $\mathbb{C}^n$ for $n$ large. Over complex projective space $CP^n$ there is the tautological bundle $L\rightarrow CP^n$, defined as above. Homogenous polynomials in $n+1$ variables are sections of the dual of this line bundle $L$. It is a good exercise to write this down with details.

Finally I suggest to look at vector bundles of rank $r$ over the sphere $S^n$. The sphere is built up from two hemispheres. As these hemispheres are contractible, the restriction of any vector bundle over the sphere is trivial when restricted to one of the hemispheres. The only information of the vector bundle is contained in how these trivializations are glued together. This is a map $S^{n-1}\rightarrow GL(r)$ and closely correspond to the homotopy groups of $GL(r)$. A great resource for this is Hatcher's vector bundles and $K$-theory.

ps: To see the sections of the mobius strip clearly, see the mobius strip as $[0,1]\times \mathbb{R}$ under the identification $(0,x)\sim (1,-x)$. Then a section of this bundle is just a map $f:[0,1]\rightarrow \mathbb{R}$ with $f(0)=-f(1)$.

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