Distribution on a manifold

Question

I have a serious question about the existence of a distribution.can anybody help me? Clearly I don't know whether any smooth manifold has a distribution? If not, why?

Answer 1

A $k$-dimensional distribution on a smooth manifold $M$ is a subbundle $D$ of $TM$ of rank $k$. So every $n$-dimensional smooth manifold has a $0$-dimensional distribution (the image of the zero section) and an $n$-dimensional distribution, namely $D = TM$.

Depending on the manifold, these may be the only distributions. For example, a closed surface admits a $1$-dimensional distribution if and only if it has Euler characteristic zero. So $S^2$ for example does not admit any other distributions. More generally, the only distributions on $S^{2n}$ are the ones mentioned in the aforementioned paragraph. Both of these claims can be proved by using the multiplicativity of the Euler class.