A maybe trivial question about fiber bundles (I'm not an expert, and I didn't find quickly an answer looking here and there). Suppose you are given fiber bundle $p\colon E\to M$, where $E,M$ are smooth manifolds and $p$ is smoothly locally trivial. Also suppose that the fiber $F$ of the bundle is smoothly contractible. Is that true that $p$ admits a smooth section?

The answer is: Yes (at least for finite dimensional manifolds). In fact you only need that the fiber is contractible not smoothly contractible. Take any continuous section $s_0 \colon B \to E$. cover $B$ by open sets $U_i$ such that the bundle is trivializable over each $U_i$, also make sure that the closure of each $U_i$ is compact and that the cover is locally finite. Furthermore, give $E$ any complete Riemannian structure. This provides each fiber $E_x$ with an induced Riemannian structure, which is also complete. We use this to define the obvious supremum distance between any two sections of $E$ over any subspace of $B$. We may also construct a continuous map $r \colon E \to \mathbb{R}_+$ such that the ball of radius $r(x)$ and center $x$ in each fiber is geodesically convex. The construction now goes in 2 steps: 1) local construction: for each $i$ we may find a smooth section $s \colon U_i \to E$ such that the supremum distance defined above is smaller than $r$ on all the points of $S_0$ restricted to $U_i$. This is easy and follows from smooth approximation of any function from $U_i \to F$ defining a section $U_i \to U_i \times F$. 2) global construction: use a partion of unity to get a global construction. This is now possible because we were carefull enough to create the smooth local sections such that they lie in a geodesically convex neighborhood. While finishing this Andrew Stacey posted a similar answer, but it seemed a waste not to poste this also. Especially since we are focussing on different details. 


A proof of the existence of this section appears in Steenrod "The Topology of fibre bundles". The first edition is about 1950, at your knowledge, are there older proofs? 


Yes. This follows from smooth paracompactness of $M$. Basically, you choose local trivialisations over a suitable cover and then on the intersections you smoothly interpolate between the different choices. By choosing your cover carefully, at each stage you are interpolating between a finite number of choices so the interpolation amounts to smoothly filling in a simplex, which by assumption you can do. (In slightly more detail, but only slightly, once you've filled in the simplex, you use a partition of unity to specify where in the simplex you should be. Pictorially, if you imagine a line at $y = 0$ and one at $y = 1$, then between $x=0$ and $x=1$ you must get from one line to the other. "Filling in the simplex" corresponds to filling in the rectangle $(0,0)  (1,0)  (1,1)  (0,1)$. A partition of unity in this case is just a smooth function $[0,1] \to [0,1]$ which is $0$ near $0$ and $1$ near $1$. The graph of this defines the interpolation between the two lines and thus the way to get from the line $y = 0$ to the one at $y= 1$.) 

