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](https://terrytao.wordpress.com/2014/03/28/the-cayley-salmon-theorem-via-classical-differential-geometry/); 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 variant of the Monge-Cayley-Salmon theorem?
Setting aside the assumption that $\phi$ be a polynomial mapping for the moment (however, see below for a construction of a large family of polynomial solutions), if one makes the 'nondegeneracy' assumptions


1. $\mathrm{dim}\ \mathrm{span}\bigl( \phi\_s(s,t), \phi\_t(s,t)\bigr) =2 $,
2. $\mathrm{dim}\ \mathrm{span}\bigl( \phi\_s(s,t), \phi\_t(s,t), \phi\_{ss}(s,t), \phi\_{st}(s,t), \phi\_{tt}(s,t) \bigr) = 3$ for all $(s,t)\in[0,1]^2$, and
3. the subspace $W(s,t) =\mathrm{span}\bigl( \phi\_s(s,t), \phi\_t(s,t), \phi\_{ss}(s,t), \phi\_{st}(s,t), \phi\_{tt}(s,t) \bigr)\subset\mathbb{R}^4$ is not constant, in the sense that $W:[0,1]^2\to \mathrm{Gr}\_3(\mathbb{R}^4)\simeq\mathbb{RP}^3$ has nonvanishing differential,


then one can show that the surface $\phi\bigl([0,1]^2\bigr)\subset\mathbb{R}^4$ is ruled (and does not lie in an affine $3$-space). 


Such surfaces locally depend on three arbitrary functions of one variable in Cartan's sense. One way of describing them is this: Let $\Lambda$ be the space of (affine) lines in $\mathbb{R}^4$, a manifold of dimension $6$. Consider the ($9$-dimensional) bundle $\pi:F\to\Lambda$ whose fiber over $\lambda\in\Lambda$ is the flag variety of the $3$-dimensional vector space $\mathbb{R}^4/\lambda'$, where $\lambda'\subset\mathbb{R}^4$ is the linear subspace parallel to $\lambda$. Then there exists a smooth $4$-plane field $D\subset TF$ such that, if $\gamma\subset F$ is a generic curve tangent to $D$, then regarding $\gamma$ as a $1$-parameter family of affine lines via $\pi(\gamma)\subset \Lambda$, the union of these lines is a surface satisfying the above assumptions. (Here 'generic' means that the tangents to $\gamma$ do not lie in triple of hyperplanes in $D$.)


It is not hard to write down polynomial solutions: For example, if $f:\mathbb{R}\to\mathbb{R}^4$ is a polynomial curve that satisfies $f'(s)\wedge f''(s)\wedge f'''(s) \wedge f''''(s)\not = 0$, then the mapping $\phi:\mathbb{R}^2\to\mathbb{R}^4$ given by 
$$
\phi(s,t) = f(s) + f'(s)\, t
$$
(which parametrizes the 'tangential development' of the curve $f$) satisfies these conditions when $t\not=0$. (By replacing $t$ by $t{+}1$, say, one could arrange that $\phi$ be an immersion on all of $[0,1]^2$.)


The following analysis is a more-or-less standard approach to verifying the above description using the so-called 'moving frame'. (I'm sure that the result itself
is classical in some sense, though I don't know offhand where to look
in the literature to find it.)


The three conditions listed above are actually independent of the choice of $st$-coordinates on the surface in $\mathbb{R}^4$ and so can be regarded as conditions on a surface $S\subset\mathbb{R}^4$ that, for local analysis purposes, can be taken to be smoothly embedded.


Let $B\_0(S)$ denote the space of quintuples $(p;v\_1,v\_2,v\_3,v\_4)$ where $p$ lies in $S$ and the quadruple $(v\_1,v\_2,v\_3,v\_4)$ is a basis of $\mathbb{R}^4$ such that $(v\_1,v\_2)$ is a basis of $T\_pS$ while $(v\_1,v\_2,v\_3)$ is a basis of the $3$-dimensional subspace $W\_pS\subset \mathbb{R}^4$. Then $B\_0$ is a smooth submanifold of the product $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ that has dimension $2 + 4 + 3 + 4 = 13$. 
 
It is useful to define $\mathbb{R}^4$-valued functions $x,e\_i: \mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})\to\mathbb{R}^4$ such that $x(p;v\_1,v\_2,v\_3,v\_4) = p$ while $e\_i(p;v\_1,v\_2,v\_3,v\_4) = v\_i$. Then there
exist unique (linearly independent) $1$-forms $\omega^i$ and $\theta^i\_j$ 
on $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ such that (assuming the
usual summation convention on repeated indices) the following *structure equations* hold:
$$
\mathrm{d}x = e\_i \omega^i
\qquad\text{and}\qquad
\mathrm{d}e\_i = e\_j\,\theta^j\_i\,.\tag1
$$ 


$$
\mathrm{d}\omega^i = -\theta^i\_j\wedge\omega^j
\qquad\text{and}\qquad
\mathrm{d}\theta^i\_j = -\theta^i\_k\wedge\theta^k\_j\,.\tag2
$$


Now, pull back these functions and $1$-forms to $B\_0(S)$ (but, as is customary, not notate the pullback). The definition of $B\_0(S)$ and the assumptions on $S$ imply that
$$
\mathrm{d}x\wedge e\_1\wedge e\_2 = \mathrm{d}e\_1\wedge e\_1\wedge e\_2\wedge e\_3
= \mathrm{d}e\_2\wedge e\_1\wedge e\_2\wedge e\_3 = 0
$$
while the two expressions 
$$
\bigl(\,\mathrm{d}e\_1\wedge e\_1\wedge e\_2,\ \mathrm{d}e\_2\wedge e\_1\wedge e\_2\bigr)
\qquad\text{and}\qquad
\mathrm{d}e\_3\wedge e\_1\wedge e\_2\wedge e\_3
$$
are nowhere vanishing. Using the above structure equations, these imply the relations
$$
\omega^3 = \omega^4 = \theta^4\_1 = \theta^4\_2 = 0,\tag3
$$
while, because of the three assumptions, $\omega^1\wedge\omega^2$ is nonvanishing, the pair $(\theta^3\_1,\theta^3\_2)$ do not simultaneously vanish,
and $\theta^4\_3$ is nonvanishing.


Meanwhile, the structure equations yield
$$
0 = \mathrm{d}\omega^3 
= -\theta^3\_1\wedge\omega^1 -\theta^3\_2\wedge\omega^2,
$$
so there must exist functions $h\_{ij}=h\_{ji}$ for $1\le i,j,\le 2$, not all simultaneously vanishing, such that $\theta^3\_i = h\_{ij}\omega^j$. The quadratic form $h = h\_{ij}\,\omega^i\omega^j$ is then nonvanishing and well-defined up to multiples on the surface $S$. Moreover, for $i = 1$ or $2$, 
$$
0 = \mathrm{d}\theta^4\_i 
= -\theta^4\_k\wedge\theta^k\_i = -\theta^4\_3\wedge\theta^3\_i\,.
$$
Thus, since $\theta^4\_3$ is nonvanishing, it follows that $\theta^3\_1$ and $\theta^3\_2$ are multiples of $\theta^4\_3$. In particular, $\theta^3\_1\wedge\theta^3\_2$ vanishes identically, so $h\_{11}h\_{22}-{h\_{12}}^2$ vanishes identically. Thus, the quadratic form $h$ has rank $1$.


Let $B\_1(S)\subset B\_2(S)$ denote the submanifold defined by $h\_{11} = h\_{12}=0$. It is a smooth submanifold of $B\_0(S)$ of dimension $11$, and when all the forms and functions are pulled back to $B\_1(S)$, we have $\theta^3\_1 = 0$ while $\theta^3\_2 = h\_{22}\,\omega^2$. In particular, it now follows that $\theta^4\_3$ is also a multiple of $\omega^2$, say $\theta^4\_3 = f\,\omega^2$ for some $f$ (which is nonvanishing).


Moreover,
$$
0 = \mathrm{d}\theta^3\_1 = -\theta^3\_k\wedge\theta^k\_1 
= -\theta^3\_2\wedge \theta^2\_1 = -h\_{22}\,\omega^2\wedge\theta^2\_1, 
$$ 
so it follows (since $h\_{22}$ is nonvanishing) that $\theta^2\_1 = g\,\omega^2$ for some function $g$ on $B\_1(S)$. 


Now,
$$
\mathrm{d}\omega^2 = -\theta^2\_j\wedge\omega^j 
= -g\,\omega^2\wedge\omega^1 - \theta^2\_2\wedge\omega^2
= -(\theta^2\_2 - g\,\omega^1)\wedge\omega^2.
$$ 
Thus, $\omega^2$ is an integrable $1$-form, and, because it is semi-basic
for the submersion $x:B\_1(S)\to S\subset\mathbb{R}^4$, it follows that $\omega^2$ is a multiple of the $x$-pullback of a (nonvanishing) $1$-form on $S$. Thus, $S$ is foliated by (connected) curves whose $x$-preimages in $B\_1(S)$ are codimension $1$ integral submanifolds of $\omega^2$.


I claim that these curves in $S$ are, in fact, lines in $\mathbb{R}^4$. To see this, note that, when one pulls back to a leaf of $\omega^2=0$ in $B\_1(S)$, one has $\theta^2\_1 = \theta^3\_1=\theta^4\_1= 0$ as well, so one has
$$
\mathrm{d}x = e\_1\,\omega^1\qquad\text{and}\qquad \mathrm{d}e\_1 = e\_1\,\theta^1\_1\,.
$$
In particular, the direction of $e\_1$ is fixed on this leaf, and this is the tangent direction of the mapping $x$ restricted to this leaf. Hence the $x$-image of this leaf is an open interval in a line in $\mathbb{R}^4$. Thus, the surface is ruled, as claimed.
There still remains the question of how one could 'generate' these surfaces, at least locally. I claim that, since these surfaces are, in fact, $1$-parameter families of lines, one should think of them as curves in the space of lines that satisfy some differential equations. Here is how one can think of this system of equations:
Let $\mathcal{I}$ denote the Pfaffian system of rank $5$ that is generated by the five linearly independent $1$-forms
$$
\omega^3,\ \omega^4,\ \theta^4\_1\,,\ \theta^4\_2\,,\ \theta^3\_1 
$$
(these are the $1$-forms that vanish when pulled back to $B\_1(S)$ when $S\subset\mathbb{R}^4$ is a surface satisfying our hypotheses). By the structure equations, 
$$
\left.
\begin{aligned}
\mathrm{d}\omega^4 &\equiv 0\\
\mathrm{d}\theta^4\_1 &\equiv 0\\
\mathrm{d}\omega^3 &\equiv -\theta^3\_2\wedge\omega^2\\
\mathrm{d}\theta^3\_1 &\equiv -\theta^3\_2\wedge\theta^2\_1\\
\mathrm{d}\theta^4\_2 &\equiv \phantom{-}\theta^3\_2\wedge\theta^4\_3\\
\end{aligned}\ \right\} \mathrm{modulo}\ \mathcal{I}
$$
It follows that $\mathcal{J}$, the Cartan system of $\mathcal{I}$, 
has rank $9$ and is spanned by the nine $1$-forms
$$
\omega^3,\ \omega^4,\ \theta^4\_1\,,\ \theta^4\_2\,,\ \theta^3\_1\,,\ 
\omega^2,\ \theta^2\_1\,,\ \theta^3\_2\,,\ \theta^4\_3
$$
In fact, $\mathcal{J}$ is easily seen to be the bundle of $1$-forms 
on $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ that is semibasic for the projection to the $9$-manifold $F$, where the projection sends $(p; v\_i)$
to the element of $F$ described by
$$
\bigl (\lambda(p,v\_1),\ [v\_1\wedge v\_2], [v\_1\wedge v\_2\wedge v\_3]\ \bigr)
$$
and where $\lambda(p,v\_1)$ is the line through $p$ in the direction $v\_1$.


In particular $\mathcal{I}$ is the pullback of a well-defined Pfaffian system of rank $5$ on $F$, whose annihilator is the $4$-plane field $D\subset TF$. Moreover, it is easy to show that the three rank $6$ Pfaffian systems generated by adjoining any one of $\omega^2$, $\theta^3\_2$, or $\theta^4\_3$ to $\mathcal{I}$ are themselves pullbacks of rank $6$ Pfaffian systems on $F$ whose annihilators in $TF$ are each $3$-plane subbundles of $D$. 
By its very construction, the projection of $B\_1(S)$ into $F$ is a curve that is tangent to $D$ and not tangent to any of these three $3$-plane subbundles of $D$.


Conversely, if $\gamma\subset F$ is any integral curve of $D$ that is not tangent to any of the three $3$-plane subbundles of $D$, its preimage in $\mathbb{R}^4\times\mathrm{GL}(4,\mathbb{R})$ is a submanifold of the form $B\_1(S)$ for a surface $S$ satisfying our conditions, in fact, the surface swept out by the union of the lines represented by $\lambda(\gamma)$, where $\lambda:F\to \Lambda$ is the obvious map to the lines.


It is a standard fact that the curves tangent to a 4-plane field in a smooth manifold are locally described (up to reparametrization) by prescribing $3$ functions of one variable.










