Ruled surface of constant K gauss curvature Ruled surfaces have negative /zero K ;  so what are some examples, with  parametrization, of a ruled surface with constant negative K ?
EDIT1:
For standard ruled surface we need to integrate linked Reference Equn (14.11):
$$ K= \dfrac{-M ^2}{EG-F^2}=-1$$
K of Ruled surface
 A: Every ruled surface may be parametrized in the form
$$
x(s, t) = c(t) + su(t),
$$
with $u$ a unit-vector-valued function and $c$ a unit-speed curve with $c' \cdot u = 0$. (That is, parametrize the rulings at unit speed, and take $c$ to be a unit-speed curve orthogonal to the rulings.) A short calculation (e.g.,  B. O'Neill, Elementary Differential Geometry, revised second edition, Exercise 5.4.12, p. 233, compare wikipedia) shows the Gaussian curvature is a function of $t$ divided by a function of $s$ and $t$.
Particularly, the Gaussian curvature is not constant (depends on $s$) unless the numerator vanishes identically. That is, there exists no ruled surface of constant negative Gaussian curvature in three-dimensional Euclidean space, even locally.
A: Since every complete, simply connected surface of constant negative curvature is diffeomorphic to the plane, and in fact isometric to the hyperbolic plane, your first example of such as thing is two-dimensional hyperbolic space. There are lots of different models and associated charts (i.e. parametrizations) readily available online.
If you drop the simply connected condition, then the ruled surface must be a quotient of hyperbolic space by a subgroup of the isometry group. If you want the rulings to be of infinite length, then this leaves you with the pseudosphere, which is homeomorphic to a cylinder. Otherwise you can look at other quotients of hyperbolic space.
As far as incomplete surfaces go, I'm not so sure.
