Are there more distinctive names of "null curves" with certain additional properties (absence of "chord curves")?

Question

In this answer (to the question "In general relativity, are light-like curves light-like geodesics?", PSE/q/76170) a particular example of a curve is discussed whose "tangent is everywhere null" and which is therefore called a "null curve". I'll restate the example curve explicitly as

$$\nu : \mathbb R \rightarrow \cal M,$$

together with a coordinate function

$$\mathbf r : \cal M \rightarrow \mathbb R^{1,2}; \qquad \mathbf r := \{~t,~x,~y~\}$$

such that

$$\mathbf r \circ \nu[~\lambda~] := \{~t_{\nu}[~\lambda~],~x_{\nu}[~\lambda~],~y_{\nu}[~\lambda~]~\} = \{~\lambda,~\text{Cos}[~\lambda~],~\text{Sin}[~\lambda~]~\}.$$

The calculation of the corresponding "tangent" value proceeds (roughly) via

$$\begin{align} \left( \frac{d}{d\lambda}[~t_{\nu}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~x_{\nu}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~y_{\nu}[~\lambda~]~] \right)^2 & = & \\ 1 - \left( -\text{Sin}[~\lambda~] \right)^2 - \left( \text{Cos}[~\lambda~] \right)^2 & = 0\end{align}$$ for all values $\lambda$ in the domain of "null curve" $\nu$.

Now, interestingly, for any two distinct values $\lambda := a$ and $\lambda := b$ from the domain of "null curve" $\nu$, there exist curves (which in the following are suggestively called "chord curves") $$\kappa_{ab} : [~a, ~b~] \subset \mathbb R \rightarrow \cal M,$$ such that $$\kappa_{ab}[~a~] = \nu[~a~], \qquad \kappa_{ab}[~b~] = \nu[~b~]$$ and: the "tangent of $\kappa_{ab}$" is everywhere positive.

As one concrete case consider $$\mathbf r \circ \kappa[~\lambda~] := \{~t_{\kappa}[~\lambda~],~x_{\kappa}[~\lambda~],~y_{\kappa}[~\lambda~]~\} = $$ $$\{~\small{\lambda,~\text{Cos}[~a~] + \left( \frac{\lambda - a}{b - a} \right) \left( \text{Cos}[~b~] - \text{Cos}[~a~] \right),~\text{Sin}[~a~] + \left( \frac{\lambda - a}{b - a} \right) \left( \text{Sin}[~b~] - \text{Sin}[~a~] \right)~}\},$$

with the corresponding "tangent" value

$$\left( \frac{d}{d\lambda}[~t_{\kappa}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~x_{\kappa}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~y_{\kappa}[~\lambda~]~] \right)^2 = $$ $$\small{1 - \left( \frac{\text{Cos}[~b~] - \text{Cos}[~a~]}{b - a} \right)^2 - \left( \frac{\text{Sin}[~b~] - \text{Sin}[~a~]}{b - a} \right)^2 = 1 - 4 \left( \frac{\text{Sin}[~(b - a)/2~]}{b - a} \right)^2 \gt 0.}$$

(As a sidenote: similarly, for given distinct values $\lambda := a$ and $\lambda := b$ from the domain of "null curve" $\nu$, one may ask for "chord curves" whose "tangent" value should be everywhere negative; but I have not been able to construct a corresponding concrete example. In the following, it is not necessary to distinguish "positive chord curves" from "negative" ones, if cases of the latter exist at all. Relevant is only that the "tangent" value of a "chord curve" exists everywhere and is not "null" anywhere.)

My question is:
Is there some specific name (i.e. more distinctive than "just another null curve") referring to "null curves" which do not have any "chord curves" at all? (Are such special cases of "null curves" perhaps called "null geodesics"?)

Answer 1

As in your post, let $\gamma : [a,b] \rightarrow \mathcal{M}$ be a smooth curve onto a Lorentzian manifold.

You might be interested in this math.SE post, indicating that all points in more than two Lorentzian dimensions can be joined by a spacelike curve, so space-like chord curves are utterly uninteresting, as they can be constructed from taking a small tube around $\gamma$ from $\gamma(a)$ to $\gamma(b)$ and letting the spacelike path wind around that tube.

The existence of time-like chord curves implies that $\gamma$ is not a maximum of the length functional, and that therefore $\gamma$ is not a length-maximizing geodesic. I am not yet sure if it implies that $\gamma$ is no geodesic at all, since geodesics on Lorentzian manifolds are weird. However, if time-like chord curves exist between every $\gamma(t)$ and $\gamma'(t)$ for every $t,t'\in [a,b]$, then $\gamma$ is no geodesic, since Lorentzian time-like geodesics are locally length-maximizing.

Thus, being a null geodesic indeed implies that, locally, there are no time-like chord curves.