Conformal Symplectic Invariant Curvatures

• Conformal symplectic invariant curvatures are differential invariants under the CSp group that classify geometric structures in both surface and curve settings.
• They are derived via Cartan reduction, Schwarzian-type constructions, and Lie–Tresse methods to yield complete local invariants with practical applications.
• These invariants facilitate the classification of hyperbolic surfaces in LG(2,4), analysis of Jacobi curves, and the study of differential invariant algebras in symplectic geometry.

Conformal symplectic invariant curvatures are differential invariants preserved by the conformal symplectic group, typically denoted $CSp$, and extracted from geometric data in symplectic or contact settings. In the sources considered here, they appear in three closely related forms: as scalar curvatures classifying hyperbolic surfaces in $LG(2,4)$, as operator-valued and Cartan-type invariants classifying Jacobi curves in a general Lagrangian Grassmannian, and as generating differential invariants for linear symplectic and conformal symplectic actions on functions and curves (The, 2010, Bautista et al., 19 Sep 2025, Jensen et al., 2020). Across these settings, the common theme is that Cartan reduction, Schwarzian-type constructions, and Lie–Tresse methods produce complete local invariants modulo $CSp$, often together with invariant derivations or an invariant parameter.

1. Ambient conformal symplectic geometry

The conformal symplectic group is the natural symmetry group whenever the underlying Lagrangian geometry depends only on a symplectic form up to scale. In the general symplectic setting $(W,\omega)$, it is given by

$$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$

In the linear model $V=\mathbb R^{2n}(x_i,y_i)$ with

$\omega=\sum_{i=1}^n dx_i\wedge dy_i,$

one also writes

$\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$

with infinitesimal generators consisting of the Hamiltonian generators spanning $\mathfrak{sp}(2n)$ together with the Euler vector field

$$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$

which commutes with the Hamiltonian generators and integrates to the scaling factor (Jensen et al., 2020).

Within this general framework, Lagrangian Grassmannians play a distinguished role. For a $LG(2,4)$0-dimensional symplectic vector space $LG(2,4)$1, the Lagrangian Grassmannian is

$LG(2,4)$2

A particularly rigid case occurs for $LG(2,4)$3. Among all real Lagrangian–Grassmannians $LG(2,4)$4, only $LG(2,4)$5 admits a distinguished Lorentzian conformal structure and is therefore identified with the indefinite Möbius space $LG(2,4)$6 (The, 2010). A common misconception is that every real Lagrangian Grassmannian carries an analogous distinguished conformal structure; the cited result excludes this and singles out $LG(2,4)$7 as exceptional.

This exceptional geometry is the basis for the surface theory developed in $LG(2,4)$8, while the general definition of $LG(2,4)$9 controls the curve theory of Jacobi curves and the differential invariant theory of linear actions.

2. Cartan reduction for hyperbolic surfaces in $CSp$0

For a hyperbolic, equivalently timelike, surface $CSp$1, the conformal symplectic invariant curvatures arise from Cartan’s method of moving frames. One identifies

$CSp$2

and chooses a frame

$CSp$3

adapted so that $CSp$4, $CSp$5 span $CSp$6, $CSp$7 is chosen so that $CSp$8 gives the central tangent sphere, and $CSp$9 completes the hyperbolic frame (The, 2010).

In the chart

$(W,\omega)$0

the pullback of the Maurer–Cartan form $(W,\omega)$1 satisfies

$(W,\omega)$2

The first normalization, or 1-adaptation, imposes $(W,\omega)$3, and then

$(W,\omega)$4

form a coframe on $(W,\omega)$5, with $(W,\omega)$6. After eliminating the mixed second fundamental form component, the 2-adapted frame satisfies

$(W,\omega)$7

In the generic 3-adapted frame one further arranges

$(W,\omega)$8

and the residual group is at most $(W,\omega)$9 (The, 2010).

At that stage one has

$$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$0

together with the remaining connection forms. The resulting exterior-derivative relations reduce to \begin{align*} d\theta_1&=-\alpha_1\wedge\theta_1,\qquad d\theta_2=-\alpha_2\wedge\theta_2,\ d\alpha_1&=(\lambda_{11}\lambda_{22}-b_{21})\,\theta_2\wedge\theta_1,\qquad d\alpha_2=-(\lambda_{11}\lambda_{22}-b_{12})\,\theta_1\wedge\theta_2,\ d\omega^4{}_0&=\beta_1\wedge\theta_1+\beta_2\wedge\theta_2, \end{align*} where $$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$1, $$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$2, and $$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$3 are additional relative invariants.

These structure equations provide the local differential system from which the scalar conformal symplectic curvatures are extracted.

3. Fundamental scalar curvatures for surfaces

For hyperbolic surfaces in $$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$4, the two quantities $$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$5 and $$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$6 are the analogues of principal curvatures, since they occur in the second fundamental form through

$$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$7

Because of the residual $$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$8-ambiguity of the 3-frame, these quantities are not themselves absolute invariants: one finds $$CSp(W)=\{\,c\in GL(W)\;|\;c^*\omega=\lambda_c\,\omega\,,\;\lambda_c\neq0\}.$$9 or a sign flip. The genuine absolute invariants are the symmetric combinations $V=\mathbb R^{2n}(x_i,y_i)$0 defined by

$V=\mathbb R^{2n}(x_i,y_i)$1

A further invariant arises from

$V=\mathbb R^{2n}(x_i,y_i)$2

whose coefficient $V=\mathbb R^{2n}(x_i,y_i)$3 is the conformal torsion (The, 2010).

These invariants may be recovered directly from the adapted frame: $V=\mathbb R^{2n}(x_i,y_i)$4 Under residual scalings $V=\mathbb R^{2n}(x_i,y_i)$5, $V=\mathbb R^{2n}(x_i,y_i)$6, the quantities $V=\mathbb R^{2n}(x_i,y_i)$7 are unchanged, while $V=\mathbb R^{2n}(x_i,y_i)$8 may pick up a sign if the orientation flips. No smooth scale can force $V=\mathbb R^{2n}(x_i,y_i)$9 to vanish unless they identically vanish, so $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$0 descend to the quotient by $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$1 and are true scalar invariants (The, 2010).

Geometrically, $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$2 and $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$3 measure the second-order bending of the surface in the two null directions $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$4 and $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$5. Their vanishing signals that the corresponding family of null lines is totally geodesic in the conformal symplectic sense. The torsion $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$6 measures the third-order twist of the central sphere congruence and detects whether the conjugate surface $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$7 is well-defined, a curve, or a point.

In this surface-theoretic setting, every hyperbolic surface in $\omega=\sum_{i=1}^n dx_i\wedge dy_i,$8 carries the three fundamental conformal symplectic curvatures

$\omega=\sum_{i=1}^n dx_i\wedge dy_i,$9

and these completely classify the surface locally (The, 2010).

4. Classification, PDE interpretation, and the Lorentzian Dupin cyclide

The conformal symplectic classification of hyperbolic surfaces in $\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$0 is simultaneously a contact-invariant classification of second-order scalar hyperbolic PDE in the plane. The same geometric framework yields a simple argument for the invariance of the general hyperbolic Monge–Ampère equation and the relative invariants that characterize it. For hyperbolic PDE of non-Monge–Ampère type, there exists a geometrically associated conjugate PDE (The, 2010).

The sign of the product $\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$1 separates the two generic classes:

• 2-elliptic if $\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$2,
• 2-hyperbolic if $\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$3.

This division is intrinsic to the conformal symplectic geometry of the surface. A plausible implication is that the principal-curvature analogy is not merely formal: the pair $\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$4 encodes the generic second-order regime, while $\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$5 controls higher-order coupling through the central sphere congruence.

The simplest non-trivial CSI 2-generic examples are the Lorentzian Dupin cyclides. In these examples,

$\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$6

so both curvature families admit 1-parameter sphere envelopes. A model is given in $\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$7-coordinates by

$\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$8

for which an explicit 3-adapted frame gives

$\CSp(2n)=\bigl\{(A,\lambda)\in GL(2n)\times\mathbb R^+\;\big|\;A^*\omega=\lambda\,\omega\bigr\}\cong \Sp(2n)\rtimes\mathbb R^+,$9

No further normalization is possible, and the remaining parameter $\mathfrak{sp}(2n)$0 runs through all nonzero values up to sign. The same source identifies this as the first known example of a Dupin cyclide in a Lorentzian space (The, 2010).

A possible misunderstanding is that vanishing $\mathfrak{sp}(2n)$1 and $\mathfrak{sp}(2n)$2 should force complete flatness. In the Dupin cyclide example this is false: the nonzero constant torsion shows that higher-order conformal symplectic geometry remains nontrivial.

5. Jacobi curves and absolute conformal curvatures

A different but closely related theory arises for Jacobi curves, which are regular curves in the Lagrangian Grassmannian of a symplectic vector space. A curve

$\mathfrak{sp}(2n)$3

is called non-singular if $\mathfrak{sp}(2n)$4 never vanishes, regular if $\mathfrak{sp}(2n)$5 is a non-degenerate quadratic form on $\mathfrak{sp}(2n)$6, and monotone if $\mathfrak{sp}(2n)$7 is definite for all $\mathfrak{sp}(2n)$8. In the cited terminology, “Jacobi curve” means a regular curve, and admissibility adds the requirement that a certain Ricci endomorphism be everywhere invertible (Bautista et al., 19 Sep 2025).

The central invariant in this setting is the Ricci endomorphism $\mathfrak{sp}(2n)$9, constructed from the derivative curve $$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$0. If in a symplectic splitting one writes

$$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$1

then the derivative curve is

$$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$2

The Ricci endomorphism is then

$$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$3

and in local coordinates its matrix is the matrix Schwarzian

$$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$4

By construction $$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$5 is $$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$6-invariant (Bautista et al., 19 Sep 2025).

Under a re-parametrization $$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$7, one has

$$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$8

where $$\zeta=\sum_{i=1}^n\bigl(x_i\partial_{x_i}+y_i\partial_{y_i}\bigr),$$9 is the scalar Schwarzian. Taking the trace gives the scalar Ricci curvature $LG(2,4)$00, and there is a unique projective parameter $LG(2,4)$01 for which $LG(2,4)$02. In that gauge, the geometric arc element is

$LG(2,4)$03

which is both $LG(2,4)$04- and re-parametrization-invariant. The absolute curvature operator is then

$LG(2,4)$05

and its eigenvalues are true absolute conformal symplectic invariants.

In geometric arc-parameter one diagonalizes $LG(2,4)$06 to obtain $LG(2,4)$07 and a skew-symmetric matrix $LG(2,4)$08, and the reduced Cartan matrix takes the form

$LG(2,4)$09

The pair $LG(2,4)$10 is a complete set of absolute conformal symplectic curvatures, and the assignment

$LG(2,4)$11

is one-to-one up to $LG(2,4)$12-action. Conversely, given smooth $LG(2,4)$13, diagonal $LG(2,4)$14, and an arc element $LG(2,4)$15 satisfying the stated compatibility condition, the corresponding linear ODE system reconstructs a unique Jacobi curve with those invariants (Bautista et al., 19 Sep 2025).

The flat case is characterized by $LG(2,4)$16, equivalently $LG(2,4)$17. Then the curve closes up at infinity into a cycle, and the theory identifies cycles precisely with flat Jacobi curves.

6. Differential invariant algebras and low-dimensional models

A complementary viewpoint comes from the equivalence problem for symplectic and conformal symplectic group actions on submanifolds and functions of symplectic and contact linear spaces. In this setting, differential invariants are computed via the Lie–Tresse theorem, and the outcome is a finite generating set of scalar invariants and invariant derivations, together with commutator relations and the corresponding syzygies (Jensen et al., 2020).

For functions $LG(2,4)$18 on the symplectic plane in the case $LG(2,4)$19, one begins with the $LG(2,4)$20-invariants $LG(2,4)$21 on second jets and the invariant derivations

$LG(2,4)$22

Passing from $LG(2,4)$23 to $LG(2,4)$24 amounts to retaining only objects of weight $LG(2,4)$25 under the homothety generated by $LG(2,4)$26. The cited computation shows that the only new scalar $LG(2,4)$27-invariant of pure second order is $LG(2,4)$28, regarded there as the conformal-symplectic curvature of the graph of $LG(2,4)$29, and that the invariant derivations reduce to weight-zero derivations $LG(2,4)$30 and $LG(2,4)$31. All higher invariants are obtained by repeated application of $LG(2,4)$32 and $LG(2,4)$33 to $LG(2,4)$34. Their commutator has the form

$LG(2,4)$35

with explicit rational functions $LG(2,4)$36 and $LG(2,4)$37 in jet variables (Jensen et al., 2020).

For curves in the contactification

$LG(2,4)$38

with group $LG(2,4)$39, one starts from the $LG(2,4)$40-invariants

$LG(2,4)$41

Weight considerations then produce the unique weight-zero third-order invariant

$LG(2,4)$42

together with the weight-zero invariant derivation

$LG(2,4)$43

All higher invariants are $LG(2,4)$44, and there are no further syzygies. In the language of the source, $LG(2,4)$45 plays the role of the conformal-symplectic curvature of the curve (Jensen et al., 2020).

The cited examples illustrate constant-curvature models. For $LG(2,4)$46, one obtains a constant $LG(2,4)$47. For the curve $LG(2,4)$48, $LG(2,4)$49, one obtains a constant $LG(2,4)$50. In both the function case and the curve case, the Lie–Tresse count confirms that the listed basic curvature together with the invariant derivations form a complete generating system for all local differential invariants of the conformal symplectic action (Jensen et al., 2020).

Taken together, these low-dimensional models show that conformal symplectic invariant curvatures are not confined to one geometric category. They govern surface theory in $LG(2,4)$51, classification of Jacobi curves in arbitrary even dimension, and differential invariant algebras for linear and contact actions. A plausible implication is that the common mechanism is the same in each case: a reduction to weight-zero or projectively normalized data, followed by extraction of a complete invariant system under $LG(2,4)$52.