Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.



Elliptic curves and connected components
Yes. It is not hard to find an example: Take
$$E \colon y^2 = x^3 - 12 x - 1\,.$$
Then $E(\mathbb Q) \cong \mathbb Z$ and $P = (5, 8)$ is a generator (according
to Magma). Since $P$ is on the component of the identity, all rational points are on that component.


