Nontrivial homogeneous Vector bundle

Solution 1:

One example is the Möbius strip, which admits a homogeneous $E(2)$ (or $SE(2)$) structure. To see this note that the set of affine lines in $\mathbb{R}^2$ (A manifold I'll call $M$) is diffeomorphic to the Möbuis strip.

One way to see this is using the projection $M\to\mathbb{RP}^1\cong S^1$ given by mapping each line to the unique parallel line which passes through the origin. Each fiber $\pi^{-1}(l)$ of this projection can be identified with the orthogonal compliment of $l$., and the resulting bundle has typical fiber $\mathbb{R}$ and is nontrivial. (If you're familiar, this construction is equivalent to the tautological line bundle of $\mathbb{RP}^1$.)

This set of lines admits a transitive $E(2)$ action, so we have $M\cong E(2)/G$, where $G$ is the stabilizer of the $x$-axis.