A variant of the Monge-Cayley-Salmon theorem? Suppose one has a smooth non-degenerate curve $\gamma: [0,1] \to {\bf R}^n$ into Euclidean space (thus $\gamma'$ never vanishes), with the property that the velocity $\gamma'(t)$ and acceleration $\gamma''(t)$ are always parallel, thus
$$ \mathrm{dim}\ \mathrm{span}( \gamma'(t), \gamma''(t) ) \leq 1$$
for all $t \in [0,1]$.  To avoid technicalities let us assume that $\gamma$ is a polynomial.  Then it is easy to see that $\gamma$ actually traverses a straight line (there is no curvature).
More generally, if we have
$$ \mathrm{dim}\ \mathrm{span}( \gamma'(t), \dots, \gamma^{(k)}(t) ) \leq k-1$$
for some fixed $k$ and all $t$, then one can show that the curve $\gamma([0,1])$ is trapped inside some (affine) $k-1$-dimensional subspace of ${\bf R}^n$.  (For instance, the $k=2$ case corresponds to the case when the curve has no torsion and is thus a plane curve.)  One can prove this for instance by showing that the $k$-form $\gamma'(t) \wedge \dots \wedge \gamma^{(k-1)}(t)$ has a direction that is fixed in $t$ (assuming it does not degenerate to zero, in which case one can instead induct).
A variant of this is the Monge-Cayley-Salmon theorem: if $\phi: [0,1]^2 \to {\bf R}^3$ is a polynomial immersed surface in three dimensions (thus $D\phi$ has maximal rank everywhere), and for each point $(s,t) \in (0,1)^2$ there is a smooth non-degenerate curve $\gamma = \gamma_{s,t}: [0,1] \to \phi([0,1]^2)$ on the surface passing through $\phi(s,t)$ at time zero, thus $\gamma(0) = \phi(s,t)$ and $\gamma'(0) \neq 0$, which is linear to third order, in that
$$ \mathrm{dim}\ \mathrm{span}( \gamma'(0), \gamma''(0), \gamma'''(0) ) \leq 1$$
(or equivalently, $\phi(s,t)$ is a flecnode of the surface, in that there is a tangent line that matches the surface to third order).  Then the Monge-Cayley-Salmon theorem asserts that $\phi([0,1]^2)$ is a ruled surface.  (I discuss this theorem in this blog post; the previous results about curves are used in the proof of the Monge-Cayley-Salmon theorem.)
In all of the above results, the hypothesis is about the local geometry of the surface (a condition on some finite Taylor expansion of $\phi$, or equivalently some finite jet of the surface), but the conclusion constrains the global geometry of the surface (in particular locating linear spaces that globally relate to the surface).
My (somewhat ill-posed) question concerns the following variant of the Monge-Cayley-Salmon situation: suppose that $\phi: [0,1]^2 \to {\bf R}^4$ is a polynomial immersed surface, and suppose that one has the dimension condition
$$ \mathrm{dim}\ \mathrm{span}( \phi_s(s,t), \phi_t(s,t), \phi_{ss}(s,t), \phi_{st}(s,t), \phi_{tt}(s,t) ) \leq 3$$
on the first and second partial derivatives of $\phi$ for all $(s,t) \in [0,1]^2$ (thus the Taylor expansion to second order of $\phi([0,1]^2)$ around any point is always at most three-dimensional rather than four).  Does this place a strong geometric constraint on the surface $\phi([0,1]^2)$, such as being ruled, or being trapped in a three-dimensional subspace of ${\bf R}^4$?  I am a bit vague on what type of conclusion I want here, but it should somehow control the "global" geometry of the surface in a manner similar to the previous examples.  The requirement that $\phi$ be a polynomial might be unnecessary, but that is what actually occurs for the application I have in mind.
One can also pose this question for higher dimensional varieties in higher dimensional Euclidean spaces, but the above situation of two-dimensional surfaces in ${\bf R}^4$ seems to be the first non-trivial case that is not directly treated by the previous assertions about curves.
 A: Just some suggestions:
I am concerned you are talking about parameterization (as would be the natural thing in an engineering problem) rather than the intrinsic geometry of the curve itself (as a geometer should).


*

*in your first case $v'(t) \propto v(t)$

*in  your second case $ \sum a_k(t)v^{(k)}(t) = 0$ for some time-dependent functions $a_k(t)$.  

*in your third case, it seems too much to have a third-order tangent everywhere but I'm not sure.  who knows?  You have certainly ruled it out.
I don't know how the everywhere existence of a particular catastrophe leads to a global restriction.  There is a Gauss map from curves to projective space, or from surfaces to Grassmanian $\mathrm{Gr}(2,n)$ which encodes how the tangent plane changes as you move around the surface.

Whenever you write down an equation or do an approximations, polynomials have been the choice for describing these kind of relations.  Therefore, algebraic geometry -- and algebraic varieties are always the main source of examples.
e.g. a Lie group is a variety
For me it's the fact that your theorem comes from a textbook, suggests the main geometers of the time knew your result under a different name.
I am researching the name of the map $C \to SO(k)$ from your curve to the first $k$ derivative at that point.  There must be an analogue of the Gauss map there.
Just a bit from Wikipedia:

In algebraic geometry, ruled surfaces were originally defined as projective surfaces in projective space containing a straight line through any given point. This immediately implies that there is a projective line on the surface through any given point, and this condition is now often used as the definition of a ruled surface: ruled surfaces are defined to be abstract projective surfaces satisfying this condition that there is a projective line through any point. This is equivalent to saying that they are birational to the product of a curve and a projective line. Sometimes a ruled surface is defined to be one satisfying the stronger condition that it has a fibration over a curve with fibers that are projective lines. This excludes the projective plane, which has a projective line though every point but cannot be written as such a fibration.

Instance of Monge-Cayley-Salmon

