First-order invariants of differential 2-forms

TL;DR

This paper characterizes the algebra of first-order invariants of differential 2-forms on a 2n-dimensional manifold, linking it to symplectic invariants and computing their maximum number.

Contribution

It establishes an isomorphism between the algebra of invariant functions on the open subbundle and symplectic invariants, providing a new understanding of invariants of differential 2-forms.

Findings

Isomorphism between invariant function algebras and symplectic invariants

Maximum number of functionally independent invariants computed

Characterization of invariants for differential 2-forms on manifolds

Abstract

Let $M$ be a smooth manifold of dimension $2n$, and let $O_{M}$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. The algebra of $\operatorname*{Diff}M$-invariant smooth functions of first order on $O_{M}$ is proved to be isomorphic to the algebra of smooth $Sp(\Omega_{x})$-invariant functions on $\wedge^{3}T_{x}^{\ast}M$, $x$ being a fixed point in $M$, and $\Omega_{x}$ a fixed element in $(O_{M})_{x}$. The maximum number of functionally independent invariants is computed.

Equations

$$G_x^r=G_x^r(M)=\left\{j_x^r\varphi \in J^r(M,M):\varphi (x)=x,\det {\varphi }_{\ast ,x}\ne 0\right\},r\ge 1,$$

$${\delta }_x\left(j_x^2\varphi \cdot j_x^1\Omega \right)$$

$$=j_x^2\varphi \cdot {\delta }_x(j_x^1\Omega ).$$

$$w_3=\sum _{h<i<j}{y}_{hij}(w_3)(d{x}^h{)}_x\wedge (d{x}^i{)}_x\wedge (d{x}^j{)}_x,\forall w_3\in {\wedge }^3{T}_x^{\ast }M,$$

$$(df{)}_x\otimes w_2+j_x^1\Omega =j_x^1\left[\left(f-f(x)\right){\Omega }^{\prime }+\Omega \right],$$

$${\varphi }^{\ast }\Omega ={\sum }_{j<k}{\bar{F}}_{jk}d{x}^j\wedge d{x}^k,{\bar{F}}_{jk}={\sum }_{h<i}\left(F_{hi}\circ \varphi \right)\det \frac{\partial ({\varphi }^h,{\varphi }^i)}{\partial (x^j,x^k)},{\varphi }^h=x^h\circ \varphi ,$$

$$\frac{\partial {\bar{F}}_{jk}}{\partial x^l}(x)$$

$$+{\sum }_{h<i}{F}_{hi}(\varphi (x))\{\frac{{\partial }^2{\varphi }^h}{\partial x^j\partial x^l}(x)\frac{\partial {\varphi }^i}{\partial x^k}(x)+\frac{\partial {\varphi }^h}{\partial x^j}(x)\frac{{\partial }^2{\varphi }^i}{\partial x^k\partial x^l}(x)$$

$$-\frac{{\partial }^2{\varphi }^h}{\partial x^k\partial x^l}(x)\frac{\partial {\varphi }^i}{\partial x^j}(x)-\frac{\partial {\varphi }^h}{\partial x^k}(x)\frac{{\partial }^2{\varphi }^i}{\partial x^j\partial x^l}(x)\}.$$

$${\Omega }^{\prime }$$

$$=\frac{1}{2}{x}^{l}d\left(\frac{\partial {\varphi }^h}{\partial x^l}\right)\wedge \left\{F_{hk}(x)d{x}^k+F_{hj}(x)d{x}^j\right\}$$

$$=d\left\{\frac{1}{2}\left(x^l\frac{\partial {\varphi }^h}{\partial x^l}-{\varphi }^h\right)\left(F_{hk}(x)d{x}^k+F_{hj}(x)d{x}^j\right)\right\}.$$

$$\begin{array}[c]{l}J^{1}\bar{\phi}\colon J^{1}\bigl{(}\wedge^{2}T^{\ast}M\bigr{)}\to J^{1}\bigl{(}\wedge^{2}T^{\ast}M\bigr{)},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ J^{1}\bar{\phi}(j_{x}^{1}\Omega)=j_{\phi(x)}^{1}(\bar{\phi}\circ\Omega\circ\phi^{-1}),\end{array}$$

$$C^{\infty }(J_x^1{O}_M{)}^{{\mathrm{Diff}}_{x}M}=C^{\infty }(J_x^1{O}_M{)}^{G_x^2},$$

$$0={\lambda }_{ijk}-{\lambda }_{kji}+{\lambda }_{kij},1\le k<i<j\le 2n.$$

$$F_{hk}(x)\frac{{\partial }^2{\varphi }^h}{\partial x^j\partial x^l}(x)-F_{hj}(x)\frac{{\partial }^2{\varphi }^h}{\partial x^k\partial x^l}(x)={\lambda }_{jkl,}1\le j<k\le 2n,$$

$$\begin{array}[c]{rl}\tfrac{\partial^{2}\phi^{2k^{\prime}-1}}{\partial x^{2j^{\prime}}\partial x^{l}}(x)-\tfrac{\partial^{2}\phi^{2j^{\prime}-1}}{\partial x^{2k^{\prime}}\partial x^{l}}(x)=&\lambda_{2j^{\prime},2k^{\prime},l},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ -\tfrac{\partial^{2}\phi^{2k^{\prime}}}{\partial x^{2j^{\prime}}\partial x^{l}}(x)-\tfrac{\partial^{2}\phi^{2j^{\prime}-1}}{\partial x^{2k^{\prime}-1}\partial x^{l}}(x)=&\lambda_{2j^{\prime},2k^{\prime}-1,l},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ \tfrac{\partial^{2}\phi^{2k^{\prime}-1}}{\partial x^{2j^{\prime}-1}\partial x^{l}}(x)+\tfrac{\partial^{2}\phi^{2j^{\prime}}}{\partial x^{2k^{\prime}}\partial x^{l}}(x)=&\lambda_{2j^{\prime}-1,2k^{\prime},l},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ -\tfrac{\partial^{2}\phi^{2k^{\prime}}}{\partial x^{2j^{\prime}-1}\partial x^{l}}(x)+\tfrac{\partial^{2}\phi^{2j^{\prime}}}{\partial x^{2k^{\prime}-1}\partial x^{l}}(x)=&\lambda_{2j^{\prime}-1,2k^{\prime}-1,l},\end{array}$$

$${\left({\bar{\delta }}_x\right)}^{\ast }:C^{\infty }((O_M{)}_x\times {\wedge }^3{T}_x^{\ast }M)\stackrel{\cong }{⟶}{C}^{\infty }(J_x^1{O}_M{)}^{G_x^{21}}.$$

$${\left({\bar{\delta }}_x\right)}^{\ast }:C^{\infty }((O_M{)}_x\times {\wedge }^3{T}_x^{\ast }M{)}^{G_x^1}\stackrel{\cong }{⟶}{\left(C^{\infty }(J_x^1{O}_M{)}^{G_x^{21}}\right)}^{G_x^1}=C^{\infty }(J_x^1{O}_M{)}^{G_x^2}.$$

$$\begin{array}[c]{l}\alpha_{\Omega_{x}}^{1}\colon\wedge^{3}T_{x}^{\ast}M\to(O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M,\\ \alpha_{\Omega_{x}}^{1}(\theta_{x})=(\Omega_{x},\theta_{x}),\end{array}$$

$$\left(f\circ {\alpha }_{{\Omega }_x}^1\right)\left(A\cdot {\theta }_x\right)$$

$$=\left(f\circ {\alpha }_{{\Omega }_x}^1\right)\left({\theta }_x\right).$$

$${\left({\alpha }_{{\Omega }_x}^1\right)}^{\ast }:C^{\infty }\left((O_M{)}_x\times {\wedge }^3{T}_x^{\ast }M\right)\to C^{\infty }\left({\wedge }^3{T}_x^{\ast }M\right)$$

$${\left({\alpha }_{{\Omega }_x}^1\right)}^{\ast \prime }:C^{\infty }{\left((O_M{)}_x\times {\wedge }^3{T}_x^{\ast }M\right)}^{G_x^1}\to C^{\infty }{\left({\wedge }^3{T}_x^{\ast }M\right)}^{Sp({\Omega }_x)}.$$

$$f\left({\Omega }_x,{\theta }_x\right)=0,\forall {\theta }_x\in {\wedge }^3{T}_x^{\ast }M.$$

$$\theta ={\sum }_{1\le a<b<c\le 2n}{y}_{abc}(\theta )\left(v^a\wedge v^b\wedge v^c\right)\in {\wedge }^3{V}^{\ast }.$$

$$A\cdot \left(v^a\wedge v^b\wedge v^c\right)$$

$$=\left(v^a\circ A^{-1}\right)\wedge \left(v^b\circ A^{-1}\right)\wedge \left(v^c\circ A^{-1}\right)$$

$$=\left({\lambda }_h^a{v}^h\right)\wedge \left({\lambda }_i^b{v}^i\right)\wedge \left({\lambda }_j^c{v}^j\right)$$

$$\displaystyle=\sum\nolimits_{1\leq h<i<j\leq 2n}\left|\begin{array}[c]{ccc}\lambda_{ah}&\lambda_{bh}&\lambda_{ch}\\ \lambda_{ai}&\lambda_{bi}&\lambda_{ci}\\ \lambda_{aj}&\lambda_{bj}&\lambda_{cj}\end{array}\right|v^{h}\wedge v^{i}\wedge v^{j},$$

$$\begin{array}[c]{rc}U^{\ast}=&\sum\nolimits_{1\leq h<i<j\leq 2n}\left(\sum\nolimits_{1\leq a<b<c\leq 2n}U_{hij}^{abc}y_{abc}\right)\tfrac{\partial}{\partial y_{hij}},\vskip 3.0pt plus 1.0pt minus 1.0pt\\ U=&\lx@intercol(u_{ij})_{i,j=1}^{2n}\in\mathfrak{sp}(2n,\mathbb{R}),\hfil\lx@intercol\end{array}$$

$$U_{hij}^{abc}=-\left|\begin{array}[c]{ccc}u_{ha}&\delta_{hb}&\delta_{hc}\\ u_{ia}&\delta_{ib}&\delta_{ic}\\ u_{ja}&\delta_{jb}&\delta_{jc}\end{array}\right|-\left|\begin{array}[c]{ccc}\delta_{ha}&u_{hb}&\delta_{hc}\\ \delta_{ia}&u_{ib}&\delta_{ic}\\ \delta_{ja}&u_{jb}&\delta_{jc}\end{array}\right|-\left|\begin{array}[c]{ccc}\delta_{ha}&\delta_{hb}&u_{hc}\\ \delta_{ia}&\delta_{ib}&u_{ic}\\ \delta_{ja}&\delta_{jb}&u_{jc}\end{array}\right|.$$

$$I\Bigl{(}\sum\nolimits_{\begin{subarray}{c}1\leq a<b<c\leq 2n\\ 1\leq h<i<j\leq 2n\end{subarray}}~{}y_{abc}\left|\begin{array}[c]{ccc}\lambda_{ah}(t)&\lambda_{bh}(t)&\lambda_{ch}(t)\\ \lambda_{ai}(t)&\lambda_{bi}(t)&\lambda_{ci}(t)\\ \lambda_{aj}(t)&\lambda_{bj}(t)&\lambda_{cj}(t)\end{array}\right|v^{h}\wedge v^{i}\wedge v^{j}\Bigr{)}=I(\theta),$$

$$0={\sum }_{1\le a<b<c\le 6,1\le h<i<j\le 6}{U}_{hij}^{abc}{y}_{abc}\frac{\partial I}{\partial y_{hij}}(\theta ),$$

$$N_{2n}=\left\{\begin{array}[c]{rl}0,&1\leq n\leq 2,\\ 2,&n=3,\\ \frac{n(4n^{2}-12n-1)}{3},&n\geq 4.\end{array}\right.$$

Full text

Mathematics Subject Classification 2010: Primary: 53A55 ; Secondary: 22E15, 53D05, 58A10, 58A20

*Key words and phrases:*Differential invariant function, differential $2$-form, jet bundle, linear representation, symplectic group.

1 Reduction to symplectic group

Let $M$ be a $C^{\infty}$ manifold and let

[TABLE]

be the Lie group of $r$-jets of diffeomorphisms at $x\in M$. If $r\geq s$, then $G_{x}^{rs}$ denotes the kernel of the natural projection $G_{x}^{r}\to G_{x}^{s}$.

In particular, for every $r\geq 2$, $G_{x}^{r,r-1}$ is isomorphic to the vectorial group $S^{r}T_{x}^{\ast}(M)\otimes T_{x}(M)$, as $J^{r}(M,M)\to J^{r-1}(M,M)$ is an affine bundle modelled over $S^{r}T^{\ast}(M)\otimes_{J^{r-1}(M,M)}T(M)$ and, therefore for every $j_{x}^{r}\phi\in G_{x}^{r,r-1}$ there exists a unique $t\in S^{r}T_{x}^{\ast}(M)\otimes T_{x}(M)$ such that $t+j_{x}^{r}(1_{M})=j_{x}^{r}\phi$. Hence we can identify $j_{x}^{r}\phi$ to $t$.

Theorem 1**.**

Let $M$ be a $C^{\infty}$ manifold of dimension $2n$, and let $p\colon O_{M}\to M$ be the dense open subbundle in $\wedge^{2}T^{\ast}M$ of $2$-covectors of maximal rank. Given a point $x\in M$, the map $\delta_{x}\colon J_{x}^{1}O_{M}\to\wedge^{3}T_{x}^{\ast}M$, $\delta_{x}(j_{x}^{1}\Omega)=(d\Omega)_{x}$, is a $G_{x}^{2}$-equivariant $G_{x}^{21}$-invariant epimorphism.

Proof.

The $G_{x}^{2}$-equivariance of $\delta_{x}$ is a consequence of the following well-known property: $(d(\phi^{\ast}\Omega))_{x}=\phi^{\ast}((d\Omega)_{\phi(x)})$. In fact, for every $\phi\in\operatorname*{Diff}_{x}M$ one obtains

[TABLE]

Next, we show that $\delta_{x}$ is surjective. Let $w_{3}\in\wedge^{3}T_{x}^{\ast}M$. Let $(x^{i})_{i=1}^{2n}$ be a coordinate system centred at $x$. If $w_{3}=\sum_{h<i<j}\lambda_{hij}(dx^{h})_{x}\wedge(dx^{i})_{x}\wedge(dx^{j})_{x}$, then $w_{3}=\delta_{x}(j_{x}^{1}\Omega)$, where $\Omega=\sum_{h<i<j}\lambda_{hij}x^{h}dx^{i}\wedge dx^{j}$. If $(y_{hij})_{1\leq h<i<j\leq 2n}$ is the coordinate system on $\wedge^{3}T_{x}^{\ast}M$ given by

[TABLE]

then the equations for $\delta_{x}$ are $y_{abc}\circ\delta_{x}=y_{bc,a}-y_{ac,b}+y_{ab,c}$, $a<b<c$.

This proves that $\delta_{x}$ is the restriction of a linear mapping. Moreover, the projection $p^{10}\colon J^{1}(\wedge^{2}T^{\ast}M)\to J^{0}(\wedge^{2}T^{\ast}M)=\wedge^{2}T^{\ast}M$ is an affine bundle modelled over $T^{\ast}M\otimes\wedge^{2}T^{\ast}M$, where the sum operation is defined as follows:

[TABLE]

$\Omega^{\prime}$ being any $2$-form such that $\Omega_{x}^{\prime}=w_{2}\in\wedge_{x}^{2}T^{\ast}M$.

If $j_{x}^{2}\phi\in G_{x}^{21}$, then $(\phi^{\ast}\Omega)_{x}=\Omega_{x}$, i.e, $p^{10}(j_{x}^{1}\Omega)=p^{10}(j_{x}^{1}(\phi^{\ast}\Omega))\,$; hence there exists a unique $\tau=\sum_{j<k}\tau_{jkl}(dx^{l})_{x}\otimes(dx^{j})_{x}\wedge(dx^{k})_{x}\in T_{x}^{\ast}M\otimes\wedge^{2}T_{x}^{\ast}M$ such that $j_{x}^{1}(\phi^{\ast}\Omega)=\tau+j_{x}^{1}\Omega$. If $\Omega=\sum_{h<i}F_{hi}dx^{h}\wedge dx^{i}$, then

[TABLE]

and taking derivatives for $\bar{F}_{jk}$ and evaluating at $x$, one obtains

[TABLE]

As $j_{x}^{2}\phi\in G_{x}^{21}$, it follows: $\phi(x)=x$, $\frac{\partial\phi^{h}}{\partial x^{i}}(x)=\delta_{i}^{h}$, and from the previous formula one thus deduces $\tau_{jkl}=F_{hk}(x)\frac{\partial^{2}\phi^{h}}{\partial x^{j}\partial x^{l}}(x)-F_{hj}(x)\frac{\partial^{2}\phi^{h}}{\partial x^{k}\partial x^{l}}(x)$. As $\tau_{jkl}$ is alternate on $j,k$, one can write $\tau=\frac{1}{2}\tau_{jkl}(dx^{l})_{x}\otimes(dx^{j})_{x}\wedge(dx^{k})_{x}$, and recalling that the coordinates are centred at $x$, taking the formula (1) into account, it follows: $j_{x}^{1}\Omega^{\prime}=\tau$, where $\Omega^{\prime}$ is the $2$-form given by

[TABLE]

Hence, $\delta_{x}\left(j_{x}^{1}(\phi^{\ast}\Omega)\right)=\delta_{x}\left(j_{x}^{1}\left(\Omega^{\prime}+\Omega\right)\right)=\delta_{x}\left(j_{x}^{1}\Omega\right)$. ∎

Let $M$ be an arbitrary $C^{\infty}$-manifold and let $\bar{\phi}\colon\wedge^{2}T^{\ast}M\to\wedge^{2}T^{\ast}M$ be the natural lift of a diffeomorphism $\phi\in\operatorname*{Diff}M$; i.e., $\bar{\phi}(w)=(\phi^{-1})^{\ast}w$ for every $2$-covector $w\in\wedge^{2}T^{\ast}M$. If $\Omega$ is a $2$-form on $M$, then $\bar{\phi}\circ\Omega\circ\phi^{-1}=(\phi^{-1})^{\ast}\Omega$. Let

[TABLE]

be the $1$-jet prolongation of $\bar{\phi}$. A subset $S\subseteq J^{1}(\wedge^{2}T^{\ast}M)$ is said to be natural if $\left(J^{1}\bar{\phi}\right)(S)\subseteq S$ for every $\phi\in\operatorname*{Diff}M$. Let $S\subseteq J^{1}(\wedge^{2}T^{\ast}M)$ be a natural embedded submanifold. A smooth function $I\colon S\to\mathbb{R}$ is said to be an invariant of first order under diffeomorphisms or even $\operatorname*{Diff}M$-invariant if $I\circ J^{1}\bar{\phi}=I$, $\forall\phi\in\operatorname*{Diff}M$. If we set $I(\Omega)=I\circ j^{1}\Omega$, for a given $2$-form $\Omega$ on $M$, then the previous invariance condition reads as $I\left((\phi^{-1})^{\ast}\Omega)(\phi(x)\right)=I(\Omega)(x)$, for all $x\in M$,$\;\phi\in\operatorname*{Diff}M$, thus leading us to the naive definition of an invariant, as being a function depending on the coefficients of $\Omega$ and its partial derivatives up to first order, which remains unchanged under arbitrary changes of coordinates.

Theorem 2**.**

Let $M$ be a smooth connected manifold of dimension $2n$. The ring of invariants of first order on $O_{M}$ is isomorphic to $C^{\infty}\left(\wedge^{3}T_{x}^{\ast}M\right)^{Sp(\Omega_{x})}$, where $\Omega_{x}$ is a fixed element in $O_{M}$.

Proof.

As $M$ is connected, the group $\operatorname*{Diff}M$ acts transitively on $M$. Therefore, it suffices to fix a point $x\in M$ and to compute $\operatorname*{Diff}_{x}M$-invariant functions in $C^{\infty}(J_{x}^{1}O_{M})$. From the very definitions it follows:

[TABLE]

and by virtue of Theorem 1 the map $\bar{\delta}_{x}\colon J_{x}^{1}O_{M}\to(O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M$, defined as follows: $\bar{\delta}_{x}(j_{x}^{1}\Omega)=(\Omega_{x},(d\Omega)_{x})$, is $G_{x}^{21}$-invariant and surjective; hence the induced homomorphism $\left(\bar{\delta}_{x}\right)^{\ast}\colon C^{\infty}((O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M)\to C^{\infty}(J_{x}^{1}O_{M})^{G_{x}^{21}}$ is injective. Next, we shall prove that $\left(\bar{\delta}_{x}\right)^{\ast}$ is also surjective, by showing that every $I\in C^{\infty}(J_{x}^{1}O_{M})^{G_{x}^{21}}$ takes constant value on the fibres of $\bar{\delta}_{x}$, as in this case $I$ induces $\bar{I}\in C^{\infty}((O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M)$ such that $I=\bar{I}\circ\bar{\delta}_{x}=\left(\bar{\delta}_{x}\right)^{\ast}(\bar{I})$. Actually, this is a consequence of the fact that the fibres of $\bar{\delta}_{x}$ coincide with the orbits of $G_{x}^{21}$ on $J_{x}^{1}O_{M}$. To prove this, we first observe that every $j_{x}^{1}\bar{\Omega}\in\left(\bar{\delta}_{x}\right)^{-1}(\bar{\delta}_{x}(j_{x}^{1}\Omega))$ can be written as $j_{x}^{1}\bar{\Omega}=j_{x}^{1}(\tilde{\Omega}+\Omega)$ with $\tilde{\Omega}_{x}=0$, $(d\tilde{\Omega})_{x}=0$, and the proof reduces to show the existence of a $2$-form $\Omega^{\prime}$ given by the formula (2) such that $j_{x}^{1}\Omega^{\prime}=j_{x}^{1}\tilde{\Omega}$, since we have seen that $j_{x}^{1}(\phi^{\ast}\Omega)=j_{x}^{1}\left(\Omega^{\prime}+\Omega\right)$, for some $j_{x}^{2}\phi\in G_{x}^{21}$. The rank of $\Omega$ being $2n$, there exists a coordinate system $(x^{i})_{i=1}^{2n}$ centred at $x$ such that $\Omega_{x}=\sum_{i=1}^{n}(dx^{2i-1})_{x}\wedge(dx^{2i})_{x}$, or equivalently, $F_{2i-1,2i}(x)=1$, and $F_{jk}(x)=0$, $1\leq j<k\leq 2n$ otherwise. We have $\tilde{\Omega}=\sum\nolimits_{i<j}\lambda_{ijk}x^{k}dx^{i}\wedge dx^{j}+$ terms of order $\geq 2$, because $\tilde{\Omega}$ vanishes at $x$, and by imposing $\tilde{\Omega}$ to be closed at $x$, we obtain

[TABLE]

Then, the equation $j_{x}^{1}\Omega^{\prime}=j_{x}^{1}\tilde{\Omega}$ is equivalent to the system

[TABLE]

or equivalently, for $1\leq j^{\prime}<k^{\prime}\leq n$,

[TABLE]

as follows by taking derivatives with respect to $x^{l}$, $1\leq l\leq 2n$, and evaluating at $x$, in the coefficient of $dx^{j}\wedge dx^{k}$ in the right-hand side of the first equation in (2). Furthermore, as a computation shows, the equations (3) are seen to be the compatibility conditions of the system (4), thus concluding that an element $j_{x}^{2}\phi\in G_{x}^{21}$ satisfying such a system really exists.

Therefore, a $G_{x}^{2}$-equivariant isomorphism of algebras holds:

[TABLE]

Moreover, as $G_{x}^{2}$ is the semidirect product $G_{x}^{2}=G_{x}^{21}\rtimes G_{x}^{1}$, taking invariants with respect to $G_{x}^{1}\cong GL(2n,\mathbb{R})$, we finally obtain an isomorphism

[TABLE]

Once an element $\Omega_{x}\in(O_{M})_{x}$ has been fixed, the following injective map is defined:

[TABLE]

such that $L_{A}\circ\alpha_{\Omega_{x}}^{1}=\alpha_{A\cdot\Omega_{x}}^{1}\circ L_{A}$, $\forall A\in G_{x}^{1}$; in particular, if $A\in Sp(\Omega_{x})$, we have $L_{A}\circ\alpha_{\Omega_{x}}^{1}=\alpha_{\Omega_{x}}^{1}\circ L_{A}$. Hence the map $\alpha_{\Omega_{x}}^{1}$ is $Sp(\Omega_{x})$-equivariant. If $f\in C^{\infty}((O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M)^{G_{x}^{1}}$, then $f\circ\alpha_{\Omega_{x}}^{1}\in C^{\infty}(\wedge^{3}T_{x}^{\ast}M)^{Sp(\Omega_{x})}$. In fact, if $A\in Sp(\Omega_{x})$, then for every $\theta_{x}\in\wedge^{3}T_{x}^{\ast}M$ we have

[TABLE]

Therefore, the $Sp(\Omega_{x})$-equivariant ring-homomorphism

[TABLE]

induced by $\alpha_{\Omega_{x}}^{1}$ maps $C^{\infty}((O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M)^{G_{x}^{1}}$ into $C^{\infty}(\wedge^{3}T_{x}^{\ast}M)^{Sp(\Omega_{x})}$, and the restriction of $\left(\alpha_{\Omega_{x}}^{1}\right)^{\ast}$ to $C^{\infty}((O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M)^{G_{x}^{1}}$ is denoted by

[TABLE]

We prove that $\left(\alpha_{\Omega_{x}}^{1}\right)^{\ast\prime}$ is injective. Actually, if $\left(\alpha_{\Omega_{x}}^{1}\right)^{\ast\prime}(f)=0$, then

[TABLE]

As $f$ is invariant under the action of $G_{x}^{1}$, we also have $f\left(A\cdot\Omega_{x},A\cdot\theta_{x}\right)=0$, $\forall A\in G_{x}^{1}$, $\forall\theta_{x}\in\wedge^{3}T_{x}^{\ast}M$; as $G_{x}^{1}$ operates transitively on $(O_{M})_{x}$, it follows: $f=0$, because given an arbitrary point $\left(\Omega_{x}^{\prime},\theta_{x}^{\prime}\right)\in(O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M$ there exists $A\in G_{x}^{1}$ such that $A\cdot\Omega_{x}=\Omega_{x}^{\prime}$ and by taking $\theta_{x}=A^{-1}\cdot\theta_{x}^{\prime}$, we conclude $f\left(\Omega_{x}^{\prime},\theta_{x}^{\prime}\right)=0$.

The map $\left(\alpha_{\Omega_{x}}^{1}\right)^{\ast\prime}$ is also surjective: For every $g\in C^{\infty}\left(\wedge^{3}T_{x}^{\ast}M\right)^{Sp(\Omega_{x})}$ we define $f\colon(O_{M})_{x}\times\wedge^{3}T_{x}^{\ast}M\to\mathbb{R}$ as follows: $f\left(\Omega_{x}^{\prime},\theta_{x}\right)=g(A^{-1}\cdot\theta_{x})$, $A\in G_{x}^{1}$ being any transformation verifying $\Omega_{x}^{\prime}=A\cdot\Omega_{x}$. The definition is correct, since if $B$ also verifies the equation $\Omega_{x}^{\prime}=B\cdot\Omega_{x}$, then $A^{-1}B\in Sp(\Omega_{x})$ and $g$ being invariant under the action of $Sp(\Omega_{x})$, we have $g(B^{-1}\cdot\theta_{x})=g((A^{-1}B)^{-1}\cdot A^{-1}\cdot\theta_{x})=g(A^{-1}\cdot\theta_{x})$. Furthermore, $f$ is $G_{x}^{1}$-invariant, as $f(A^{\prime}\cdot\Omega_{x}^{\prime},A^{\prime}\cdot\theta_{x})=g((A^{\prime}A)^{-1}\cdot A^{\prime}\cdot\theta_{x})=f\left(\Omega_{x}^{\prime},\theta_{x}\right)$, $\forall A^{\prime}\in G_{x}^{1}$, thus concluding. ∎

2 The number of invariants

2.1 Infinitesimal invariants

From now onwards, $V$ denotes a real vector space of dimension $2n$ and $\Omega\in\wedge^{2}V^{\ast}$ denotes a non-degenerate skew-symmetric bilinear form on $V$.

Let $(v_{i})_{i=1}^{2n}$ be a basis for $V$ with dual basis $(v^{i})_{i=1}^{2n}$. We define coordinate functions $y_{abc}$, $1\leq a<b<c\leq 2n$, on $\wedge^{3}V^{\ast}$ by setting

[TABLE]

If $A\in GL(V)$, then for $1\leq a<b<c\leq 2n$ we have

[TABLE]

where $(\lambda_{ij})_{i,j=1}^{2n}$ is the matrix of $(A^{-1})^{T}$ in the basis $(v_{i})_{i=1}^{2n}$ and the superscript $T$ means transpose. In what follows we assume $\Omega=\sum_{i=1}^{n}v^{i}\wedge v^{n+i}$.

A function $I\colon\wedge^{3}V^{\ast}\to\mathbb{R}$ is $Sp(\Omega)$-invariant if $I\left(\Lambda\cdot\theta\right)=I(\theta)$, $\forall\theta\in\wedge^{3}V^{\ast}$, $\forall\Lambda\in Sp(\Omega)$.

Lemma 3**.**

A smooth function $I\colon\wedge^{3}V^{\ast}\to\mathbb{R}$ is $Sp(\Omega)$-invariant if and only if $I$ is a first integral of the distribution spanned by the following vector fields:

[TABLE]

where the functions $U_{hij}^{abc}$ are given by the formulas

[TABLE]

Proof.

If $I$ is invariant, then, in particular, we have $I\left(\exp(tU)\cdot\theta\right)=I(\theta)$, $\forall t\in\mathbb{R}$, $U=(u_{ij})_{i,j=1}^{2n}\in\mathfrak{sp}(\Omega)$. If $\Lambda(t)=\exp(-tU^{T})$, then

[TABLE]

and taking derivatives at $t=0$, it follows:

[TABLE]

$U_{hij}^{abc}$ being as in the statement. The converse follows from the fact that the symplectic group is connected and hence, every symplectic transformation is a product of exponentials of matrices in the symplectic algebra. ∎

Theorem 4**.**

The distribution $\mathcal{D}\subset T\mathfrak{(}\wedge^{3}V^{\ast})$ whose fibre $\mathcal{D}_{\theta}$ over $\theta\in\wedge^{3}V^{\ast}$ is the subspace $(U^{\ast})_{\theta}$, $U\in\mathfrak{sp}(2n,\mathbb{R})$, is involutive and of locally constant rank on a dense open subset $\mathcal{O}\subset\wedge^{3}V^{\ast}$.

The number $N_{2n}$ of functionally independent $Sp(\Omega)$-invariant functions defined on $\mathcal{O}$ is equal to $N_{2n}=\tbinom{2n}{3}-\operatorname*{rank}\mathcal{D}|_{\mathcal{O}}$.

Proof.

Every pair of vector fields $U^{\prime\ast},U^{\prime\prime\ast}$ belonging to $\mathcal{D}$ on an open subset $O\subseteq\wedge^{3}V^{\ast}$ can be written as $U^{\prime\ast}=\sum_{h=1}^{n(2n+1)}f^{h}(U_{h})^{\ast}$, $U^{\prime\prime\ast}=\sum_{i=1}^{n(2n+1)}g^{i}(U_{i})^{\ast}$, with $f^{h},g^{i}\in C^{\infty}(O)$, where $(U_{1},\ldots,U_{n(2n+1)})$ is a basis of $\mathfrak{sp}(2n,\mathbb{R)}$. As $[U_{h}^{\ast},U_{i}^{\ast}]=-[U_{h},U_{i}]^{\ast}$, it follows that $[U^{\prime\ast},U^{\prime\prime\ast}]$ can be written as a linear combination of $(U_{h})^{\ast}$, $(U_{i})^{\ast}$, and $[U_{h}^{\ast},U_{i}^{\ast}]=-[U_{h},U_{i}]^{\ast}=-c_{hi}^{j}(U_{j})^{\ast}$, where $c_{hi}^{j}$ are the structure constants of $\mathfrak{sp}(2n,\mathbb{R})$ on this basis. This proves that $\mathcal{D}$ is involutive. Moreover, we first recall that the dimension of the vector spaces $\{\mathcal{D}_{\theta}:\theta\in\wedge^{3}V^{\ast}\}$ is uniformly bounded by $\dim(\wedge^{3}V^{\ast})=\tbinom{2n}{3}$. Let $\mathcal{O}\subset\wedge^{3}V^{\ast}$ be the subset defined as follows: A point $\theta\in\wedge^{3}V^{\ast}$ belongs to $\mathcal{O}$ if and only if $\theta$ admits an open neighbourhood $N$ such that $d=\dim\mathcal{D}_{\theta}=\max_{\theta^{\prime}\in N}\left(\dim\mathcal{D}_{\theta^{\prime}}\right)$. We claim that $\mathcal{O}$ is an open subset. Actually, there exists an open neighbourhood $N^{\prime}$ of $\theta$ such that the dimension of the fibres of $\mathcal{D}$ over the points $\theta^{\prime}\in N^{\prime}$ is at least $d$, as if $\left.(X_{i})^{\ast}\right|_{\theta}$, $1\leq i\leq d$, is a basis for $\mathcal{D}$ at $\xi$, for certain $X_{i}\in\mathfrak{sp}(2n,\mathbb{R})$, $1\leq i\leq d$, then the vector fields $(X_{i})^{\ast}$, $1\leq i\leq d$, are linearly independent at each point of a neighbourhood of $\theta$. From the definition of $\mathcal{O}$ we thus conclude that if $\theta\in\mathcal{O}$, then we have $\dim\mathcal{D}_{\theta^{\prime}}=d$ for every $\theta^{\prime}\in N\cap N^{\prime}$; hence $N\cap N^{\prime}\subseteq\mathcal{O}$. The same argument proves that the rank of $\mathcal{D}$ is locally constant over $\mathcal{O}$. Next, we prove that $\mathcal{O}$ is dense. Let $N$ be an open neighbourhood of an arbitrarily chosen point $\theta\in\wedge^{3}V^{\ast}$ and let $\theta^{\prime}$ be a point in $N$ such that the rank of $\mathcal{D}|_{N}$ takes its greatest value at $\theta^{\prime}$. By proceeding as above, we deduce that $\theta^{\prime}$ belongs to $\mathcal{O}$. Finally, the formula for the number of invariants in the statement now follows from Frobenius’ theorem. ∎

Remark 5**.**

We have $N_{2}=0$, as $\wedge^{3}V^{\ast}=\{0\}$ if $\dim V=2$, and $N_{4}=0$, as $Sp(2n)$ acts transitively on $\wedge^{3}V^{\ast}\backslash\{0\}$ if $\dim V=2n=4$. Furthermore, as a consequence of the results obtained in [1], it follows that the generic rank of $\mathcal{D}$ for $\dim V=2n=6$ is $18$; hence $N_{6}=2$.

2.2 $N_{2n}$ computed

Theorem 6**.**

We have

[TABLE]

Proof.

The formula in the statement for $1\leq n\leq 3$ follows from Remark 5. Hence we can assume $n\geq 4$. For every $3$-covector $\theta\in\wedge^{3}V^{\ast}$, let us define

[TABLE]

It suffices to prove that there exists a dense open subset $\mathcal{O}^{\prime}\subset\wedge^{3}V^{\ast}$ such that $\mu_{\theta}$ is an immersion if $\theta\in\mathcal{O}^{\prime}$, since $\operatorname*{im}\left[(\mu_{\theta})_{\ast}\colon T_{I}Sp(\Omega)\to T_{\theta}(\wedge^{3}V^{\ast})\right]=\mathcal{D}_{\theta}$.

By expanding on (6) it follows:

[TABLE]

where $Y_{hij}=\tfrac{\partial}{\partial y_{hij}}$, $1\leq h<i<j\leq 2n$. Given indices $1\leq\alpha<\beta<\gamma\leq 2n$, the coefficient of $Y_{\alpha\beta\gamma}$ in (7) is

[TABLE]

As the matrix $U=(u_{ij})_{i,j=1}^{2n}$ is symplectic, the following symmetries hold:

[TABLE]

A vector $U_{\theta}^{\ast}$ belongs to $\ker(\mu_{\theta})_{\ast}$ if and only if $C_{\alpha\beta\gamma}=0$ for every system of indices $1\leq\alpha<\beta<\gamma\leq 2n$. We thus obtain a homogeneous linear system $S_{2n}$ of $\tbinom{2n}{3}$ linear equations in the $n(2n+1)$ unknowns $u_{ij}$, $i,j=1,\ldots,n$; $u_{i,n+j}$, $u_{n+i,j}$, $1\leq i\leq j\leq n$, and we have $\tbinom{2n}{3}>n(2n+1)$ for every $n\geq 4$. Evaluating $S_{2n}$ at the $3$-covector $\theta^{0}$ of coordinates $y_{abc}(\theta^{0})=a+b+c$, $1\leq a<b<c\leq 2n$, as a numerical calculation shows, the only solution to $S_{2n}(\theta^{0})$ is given by $u_{ij}=0$, $i,j=1,\ldots,n$. We can thus conclude by simply applying the formula (4) for $N_{2n}$ in Theorem 4. ∎