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Terry Tao
Terry Tao
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A variant of the CayleyMonge-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 CayleyMonge-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 CayleyMonge-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 CayleyMonge-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 CayleyMonge-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 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 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 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 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 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?

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.

Source Link
Terry Tao
Terry Tao
  • 114.1k
  • 33
  • 462
  • 539

A variant of the 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 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 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 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 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.