Integrable GL(2) Geometry and Hydrodynamic Partial Differential Equations

INTEGRABLE GL(2) GEOMETR Y AND HYDR OD YNAMIC P AR TIAL DIFFERENTIAL EQUA TIONS ABRAHAM D. SMITH Abstract. This article is a lo cal analysis of in tegrable GL (2)-structures of degree 4. A GL (2)- structure of degree n corresp onds to a distribution of rational normal cones ov er a manifold of dimension n + 1. In tegrabilit y corresp onds to the existence of man y submanifolds that are spanned b y lines in the cones. These GL (2)-structures are imp ortant because they naturally arise from a certain family of second-order hyperb olic PDEs in three v ariables that are integrable via hydrody- namic reduction. F amiliar examples include the wa v e equation, the ﬁrst ﬂow of the dKP hierarch y , and the Bo yer–Finley equation. The main results are a structure theorem for in tegrable GL (2)-structures, a classiﬁcation for con- nected integrable GL (2)-structures, and an equiv alence b etw een local integrable GL (2)-structures and Hessian h ydro dynamic hyperb olic PDEs in three v ariables. This yields natural geometric characterizations of the wa ve equation, the ﬁrst ﬂow of the dKP hierarc hy , and sev eral others. It also provides an in trinsic, co ordinate-free infrastructure to describe a large class of h ydro dynamic integrable systems in three v ariables. Contents In tro duction 1 1. Bac kground 4 1.1. Hessian Hydro dynamic Hyp erb olic PDEs 5 2. A Preferred Connection 8 3. 2-In tegrabilit y 11 4. 3-In tegrabilit y 14 5. The Classiﬁcation 18 5.1. Symmetry Reduction 20 6. GL(2) PDEs 22 6.1. Hyp erb olic Planar PDEs 25 7. Concluding Remarks 27 App endix A. The Matrix J(T) 29 References 30 Intr oduction A fundamen tal problem in analysis is to understand why some diﬀeren tial equations (particularly h yp erb olic equations arising from w a ve-lik e equations or from diﬀeren tial geometry) are inte gr able . This problem is comp ounded b y the many comp eting notions of in tegrabilit y [35]. Informally , Date : March 26, 2022. 2000 Mathematics Subje ct Classiﬁc ation. 58A15, 37K10. Key wor ds and phr ases. GL(2)-structure, hydrodynamic reduction, hyperb olic PDE. This work is supported by an NSF All-Institutes Postdoctoral F ellowship administered b y the Mathematical Sciences Researc h Institute through its core gran t DMS-0441170. The author is hosted by the Departmen t of Mathematics and Statistics at McGill Univ ersity . 1 2 A. D. SMITH a PDE is called in tegrable if it can b e solv ed thanks to the existence of an inﬁnite hierarch y of conserv ation laws or of a family of in v ariant foliations along c haracteristics. In tegrable PDEs of all types are well-studied in t w o (1 + 1) indep enden t v ariables; how ev er, examples are increasingly rare and p o orly understo o d in higher dimensions. A particularly interesting class of in tegrable PDEs are those that can b e in tegrated by decom- p osing the equation to a set of coupled ﬁrst-order equations that represen t trav eling w av es [31]. This tec hnique is called “in tegration b y h ydro dynamic reduction,” and it has been extensively studied for v arious sp ecial classes of second-order PDEs in three or more indep enden t v ariables [7, 14, 15, 16, 17, 32, 33] Consider a second-order h yp erb olic equation of the form F ( ξ i , u, u i , u ij ) = 0 on scalar functions u ( ξ 1 , ξ 2 , ξ 3 ). In the method of hydrodynamic reduction, one hop es to integrate the PDE F = 0 b y stipulating that the solution function u (and its deriv atives) may b e written as u ( R 1 , . . . , R k ) for an a priori unkno wn n umber k of functions R 1 ( ξ ) , . . . , R k ( ξ ) whose deriv ativ es admit the “comm uting” relations (1) ∂ ∂ ξ 2 R i = ρ i 2 ( R ) ∂ ∂ ξ 1 R i , ∂ ∂ ξ 3 R i = ρ i 3 ( R ) ∂ ∂ ξ 1 R i whic h also imply (2) 1 ρ i 2 − ρ j 2 ∂ ρ i 2 ∂ R j = 1 ρ i 3 − ρ j 3 ∂ ρ i 3 ∂ R j , ∀ i 6 = j. Equations (1) and (2) can be solv ed b y other metho ds [31]. The reduction of F = 0 to Equation (2) is called a k -parameter h ydro dynamic reduction. The equation F = 0 is called in tegrable via k - parameter h ydro dynamic reduction if it admits an inﬁnite family of k -parameter h ydro dynamic reductions and this family is itself parametrized b y k functions of one v ariable. The functions R i ( ξ 1 , ξ 2 , ξ 3 ) are called Riemann inv arian ts, and their level sets deﬁne the foliations along c har- acteristics in the informal deﬁnition ab ov e. See the works cited ab ov e, particularly [14], [15] and [17], for examples and detailed exp osition of this tec hnique. Man y examples of these so-called h ydro dynamic equations are kno wn, such as the w a v e equation u 22 = u 13 , the disp ersionless Kadom tsev–P etviashvili (dKP) equation u 22 = ( u 1 − uu 3 ) 3 , the ﬁrst ﬂo w of the dKP hierarc hy u 22 = u 13 − 1 2 ( u 33 ) 2 , the Bo yer–Finley equation u 22 + u 33 = e u 11 , and some well-kno wn integrable hierarchies. T ests for h ydro dynamic in tegrability also exist, but so far there is no in trinsic, co ordinate-free theory that classiﬁes and generates all of these equations. F or the case of second-order hydrodynamic equations in three v ariables that inv olve only the Hessian of the solution function, recen t work b y F eraponto v, et al. , reveals a relationship betw een the integrabilit y of the PDEs, the symplectic con tact symmetries of the equations, and the equa- tions’ underlying degree-4 GL (2)-structures [14]. Integrable GL (2)-structures are not well-kno wn in the literature; they arise in the searc h for exotic aﬃne holonomies in diﬀeren tial geometry [5] and a similar conformal geometry app ears useful in the equiv alence problem for ODEs [24], but the application to PDEs is very recen t [14, 27]. The purp ose of this article is to fully analyze the local geometry of integrable GL (2)-structures of degree 4, with an ey e to wards understanding PDEs that are in tegrable via h ydro dynamic reduction. The main result is a natural classiﬁcation of lo cal integrable GL (2)-structures and the asso ciated PDEs. This is achiev ed using geometry ` a la Cartan , namely exterior diﬀeren tial systems, moving frames, and the metho d of equiv alence. Here is a summary of the con tents of this article, including abbreviated and non-te chnic al v er- sions of the main theorems: Section 1 deﬁnes GL (2)-structures and their 2-in tegrability and 3-in tegrabilit y . The property of k -integrabilit y in v olv es the existence of “man y” foliations b y certain submanifolds of dimension k . GL(2) PDES 3 Section 1 also introduces the motiv ating observ ation by F erap onto v et al. that h yp erb olic PDEs in three v ariables that are integrable by means of h ydro dynamic reductions lead to GL (2)-structures that are b oth 2-in tegrable and 3-in tegrable (abbreviated 2,3-in tegrable). Section 2 con tains a description of the necessary S L (2) represen tation theory and an explicit solution to the ﬁrst-order equiv alence problem for GL (2)-structures. Corollary 2.2 deﬁnes a global preferred connection for GL (2)-structures that has essential torsion in the direct sum of R 3 , R 7 , R 9 , and R 11 , each of whic h is an irreducible S L (2) mo dule. In Section 3, the exterior diﬀeren tial system describing 2-integrabilit y for a GL (2)-structure is analyzed. The most important ob ject in this article, the irreducible R 9 -v alued torsion, is ﬁrst emphasized in Theorem 3.2. Theorem 3.2. A GL (2) -structur e is 2-inte gr able if and only if its torsion takes values only in R 9 . Section 4 provides Theorem 4.3, a complete lo cal description of GL (2)-structures that are 2,3- in tegrable. The immediate consequence of this theorem is that the v alue of the torsion at a single p oin t completely determines the lo cal structure of a 2,3-in tegrable GL (2)-structure. Theorem 4.3. A GL (2) -structur e is 2,3-inte gr able if and only if its structur e e quations ar e of the form d ω = − θ ∧ ω + T ( ω ∧ ω ) d θ = − θ ∧ θ + T 2 ( ω ∧ ω ) d T = J ( T )( ω , θ ) (3) for an essential torsion T in R 9 and a 9 × 9 matrix J ( T ) dep ending on T , as in App endix A. Section 5 presents a classiﬁcation of connected 2,3-in tegrable GL (2)-structures. This classiﬁca- tion is provided b y a natural stratiﬁcation of R 9 in to equiv alence classes that are deﬁned by the matrix J ( T ). These equiv alence classes are explicitly iden tiﬁed in Theorem 5.3. Theorem 5.3. Every c onne cte d 2,3-inte gr able GL (2) -structur e b elongs to exactly one of 55 e quiv- alenc e classes, which ar e given by a str atiﬁc ation of R 9 into the r o ot factorization typ es of r e al binary o ctic p olynomials. In Section 6, the structure equations from Section 4 are applied to repro duce the in tegrable PDEs, yielding Theorem 6.3. Theorem 6.3. Every lo c al 2,3-inte gr able GL (2) -structur e arises fr om a Hessian hydr o dynamic hyp erb olic PDE. Therefore, the classiﬁcation in Theorem 5.3 is also a classiﬁcation of Hessian h ydro dynamic h yp erb olic PDEs. Some previously known examples of suc h PDEs are identiﬁed in the new clas- siﬁcation, and sev eral new in tegrable PDEs are constructed. Also, a relationship b etw een Hessian h ydro dynamic h yp erb olic PDEs in three v ariables and hyperb olic PDEs in t wo v ariables (Monge– Amp ` ere, Goursat, or generic type) is observ ed. This w ork would ha ve been impossible to complete without the uncann y sp eed and accuracy of computer algebra soft ware, and there is no sense in ret yping dozens of pages of formulas and iden tities. Th us, a reader lo oking for explicit computational details ma y b e disapp ointed b y this article (in which case I direct the reader to [27]). Ho wev er, most pro ofs rely only on basic concepts from diﬀerential geometry , linear algebra, and representation theory , and a reader comfortable with the standard metho ds of exterior diﬀeren tial systems (and their v arious soft ware implemen tations) can readily supply the computational details. A Maple ﬁle containing the structure equations app earing in Theorem 4.3 and the Maurer–Cartan form appearing in Theorem 6.3 will remain 4 A. D. SMITH a v ailable at [26] as long as p ossible. This ﬁle should also be bundled with this article on The conten t of this ﬁle allows rapid repro duction of all results following Theorem 4.3. Please note that the terms “h yp erb olic” and “integrable” are heavily used in sligh tly diﬀerent con texts throughout this article. “Hyp erb olic” refers to h yp erplanes having maximal intersection with certain pro jective v arieties (for example, in Lemma 1.2), to PDEs in both t wo and three indep enden t v ariables with appropriate leading sym b ol (for example, the wa ve equation), and to exterior diﬀerential systems with appropriate tableaux (for example, in the pro of of Theorem 3.1). The term “integrable” refers b oth to PDEs that admit inﬁnitely man y exact solutions and to GL (2)- structures that admit man y secant submanifolds (submanifolds which one migh t b e tempted to also call “hyperb olic”). Of course, all of these concepts are intimately related, so the standard o v erloading of these deﬁnitions is reasonable. I o we man y thanks to Robert Bry an t and Niky Kamran for their wisdom, guidance, and support, to Dennis The and Eugene F erap on to v for their man y helpful commen ts, and to Jeanne Clelland, whose “Cartan” pack age for Maple made the computations b earable. 1. Back ground Let V n denote the vector space of degree n homogeneous p olynomials in x and y with real co eﬃcien ts. Iden tify V n with R n +1 using the terms from the binomial theorem to pro duce a basis; for example, V 4 → R 5 b y (4) v − 4 x 4 + v − 2 4 x 3 y + v 0 6 x 2 y 2 + v 2 4 xy 3 + v 4 y 4 7→ ( v − 4 , v − 2 , v 0 , v 2 , v 4 ) ∈ R 5 . Let M denote a 5-dimensional smo oth manifold, and let F denote the V 4 -v alued coframe bundle o v er M , so the ﬁb er F p is the set of all isomorphisms T p M → V 4 . The coframe bundle F is a principal right GL ( V 4 ) bundle. Recall that V n is the unique irreducible represen tation of sl (2) of dimension n +1. The action of sl (2) on V n is generated by (5) X = y ∂ ∂ x , Y = − x ∂ ∂ y , and H = x ∂ ∂ x − y ∂ ∂ y . Deﬁnition 1.1. A GL (2) -structur e B → M is a r e duction of F with ﬁb er gr oup GL (2) ⊂ GL ( V n ) inﬁnitesimal ly gener ate d by X , Y , H , and the sc aling action I . Note that throughout this article, the sym b ols GL (2) and gl (2) are used to denote the abstract Lie group and its Lie algebra as w ell as an y particular irreducible represen tations GL (2) ⊂ GL ( V n ) and gl (2) ⊂ gl ( V n ). Since S L (2) has a unique irreducible represen tation V n in eac h dimension n +1, it is alwa ys clear from context whic h representation of GL (2) is b eing considered. Generally , a GL (2)-structure is said to hav e “degree n ” if the base space M has dimension n +1. The degree 3 case is thoroughly studied in [5], and v arious observ ations are made for all n in [27]. This article considers only the degree 4 case. The P GL (2) action generated b y X , Y , and H is the symmetry group of a rational normal curve in P ( V 4 ). A rational normal curv e is a curv e of degree n in P ( R n +1 ). All such curv es are P GL ( n + 1)-equiv alent to { [ g n : g n − 1 h : · · · : g h n − 1 : g n ] } [20]. The de-pro jectivization of the rational normal curv e is the rational normal cone, which has a GL (2) symmetry group and is usually describ ed as { ( g n , g n − 1 h, . . . , g h n − 1 , h n ) } ⊂ R n +1 . Th us, the geometric conten t of Deﬁnition 1.1 is contained in the following lemma. Lemma 1.1. A GL (2) -structur e B → M 5 is e quivalent to a distribution of r ational normal c ones C ⊂ T M . F or any b ∈ B p , b ( C p ) = { ( g x + hy ) 4 : g , h ∈ R } ⊂ V 4 . Deﬁnition 1.2 (In tegrabilit y) . L et B → M b e a GL (2) -structur e. A k -dimensional line ar subsp ac e E ⊂ T p M is k-se c ant if E ∩ C p is k distinct lines. Equivalently, E is k-se c ant if E is sp anne d by GL(2) PDES 5 k ve ctors in C p . A k -dimensional submanifold N ⊂ M is k-se c ant if T p N is a k-se c ant subsp ac e of T p M for every p ∈ N . A GL (2) -structur e is k-inte gr able if, for every k-se c ant line ar subsp ac e E ∈ Gr k ( T M ) , ther e exists a k -se c ant submanifold N with E = T p N . Lo cally , 1-integrabilit y is uninteresting, as it is simply the existence of a lo cal ﬂo w for a v ector ﬁeld. Since M has dimension 5, b eing 4-secant is an op en condition on E ∈ Gr 4 ( T M ); ho wev er, the condition is closed for k =2 (bi-secan t) and k =3 (tri-secant). A GL (2)-structure that is b oth 2-in tegrable and 3-integrable is called “2,3-integrable,” or simply “integrable.” Notice that all the deﬁnitions are pro jectively inv arian t. Indeed, one could equiv alen tly study the principal righ t P GL (2) bundle ov er M deﬁned b y the symmetries of rational normal curves in P T M . When considering the de-pro jectivized GL (2) geometry , one o ccasionally encounters an additional Z / 2 Z symmetry by ± I . F or the purp oses of this article, it is easier to pro ceed using aﬃne geometry and deal with this symmetry when it arises. The geometric conten t of this article is a complete local description and classiﬁcation of GL (2)- structures of degree 4 that are 2,3-integrable. Such GL (2)-structures are particularly interesting b ecause of their deep connection to in tegrable PDEs. The remainder of this section outlines ho w F erap onto v, Hadjikos, and Kh usnutdino v a obtain 2,3-in tegrable GL (2)-structures from certain h ydro dynamic PDEs in [14]. 1.1. Hessian Hydro dynamic Hyp erb olic PDEs. The space of 1-jets ov er R 3 → R is J 1 = { ( ξ i , z , p i ) } where 1 ≤ i, j ≤ 3. The space of 2-jets is J 2 = { ( ξ j , z , p i , U ij ) } where 1 ≤ i, j ≤ 3 and U ij = U j i . The con tact system on J 2 is the diﬀerential ideal J 2 generated by (6) { d z − p i d ξ i , d p i − U ij d ξ j } . A second-order PDE on u : R 3 → R is equiv alent to the lev el set F − 1 (0) of some smo oth function F : J 2 → R on whic h the pro jection J 2 → J 1 is a submersion, and a solution is a function u : R 3 → R whose jet-graph is a subset of F − 1 (0). Recall that a section of J 2 is the jet-graph of a function if and only if it is an integral of the con tact system. A con tact transformation on J 2 is a diﬀeomorphism ψ : J 2 → J 2 suc h that ψ ∗ ( J 2 ) = J 2 , and one studies PDEs geometrically b y examining the prop erties of F − 1 (0) that are in v ariant under con tact transformations. By a theorem of B¨ ac klund [25, Theorem 4.32], every con tact transformation on J 2 is the prolon- gation of a contact transformation on J 1 . Therefore, for any con tact transformation ψ , there exist functions A ij , B ij , C ij , D ij , a i , m i , ˆ a i , ˆ m i , and c , (written as four matrices, four column vectors, and a scalar) suc h that (7) ψ ∗   d ξ d z d p   =   B ˆ m C m t c a t A ˆ a D     d ξ d z d p   . The assumption that ψ is a contact transformation implies that the symplectic form σ = d p 1 ∧ d ξ 1 + d p 2 ∧ d ξ 2 + d p 3 ∧ d ξ 3 is preserv ed up to scale, so ψ ∗ ( σ ) = λσ , λ 6 = 0. One can compute ˜ U ij = ψ ∗ ( U ij ) by noting that ψ ∗ (d p i − U ij d ξ j ) ≡ 0 mo dulo J 2 , so in matrix notation ψ ∗ (d p ) − ˜ U ψ ∗ (d ξ ) = ( A d ξ + ˆ a d z + D d p ) − ˜ U ( B d ξ + ˆ m d z + C d p ) ≡ ( A + ˆ ap t + D U )d ξ − ˜ U ( B + ˆ mp t + C U )d ξ . (8) In particular, under the non-degeneracy assumption ψ ∗ (d ξ 1 ∧ d ξ 2 ∧ d ξ 3 ) 6 = 0, this yields (9) ˜ U = ( A + ˆ ap t + D U )( B + ˆ mp t + C U ) − 1 . F or this article, consider only PDEs on u : R 3 → R of the form (10) F ( u 11 , u 12 , u 13 , u 22 , u 23 , u 33 ) = 0 . 6 A. D. SMITH Instead of considering the orbit of this PDE under all con tact transformations, consider only those con tact transformations that preserv e the class of Hessian-only equations, those of the form in Equation (10). That is, consider only those con tact transformations ψ suc h that ψ ∗ (d U ij ) ≡ 0 mo dulo { d U kl } . Then it must b e that ˆ a = ˆ m = 0 in Equation (7). Th us, the con tact trans- formations that preserv e the Hessian-only form of Equation (10) are elements of the conformal symplectic group, C S p (3) = { g ∈ GL (6 , R ) : σ ( g v , g w ) = λσ ( v , w ) ∀ v , w (any λ ∈ R ∗ ) } =  g =  B C A D  : 0 = A t B − B t A = D t C − C t D , λI 3 = D t B − C t A  . (11) In the most general case, the completed domain of all Hessian-only F ’s is the Lagrangian Grass- mannian, (12) Λ = J 2 / J 1 = { U ∈ Gr 3 ( R 6 ) : σ | U = 0 } , with lo cal co ordinates given, for example, b y its non- ξ -v ertical op en subset (13) Λ o = { U ∈ Λ : d ξ 1 ∧ d ξ 2 ∧ d ξ 3 | U 6 = 0 } . The conformal symplectic group forms a bundle, Π : C S p (3) → Λ. In fact, when considering lo cal transformations of PDEs, only the transformations in C S p (3) o = Π − 1 (Λ o ) are p ermissible, as the condition d ξ 1 ∧ d ξ 2 ∧ d ξ 3 6 = 0 m ust be preserv ed. As ab o v e, one could eliminate the conformal factor and consider only S p (3) o actions by pro jectivizing, but there is little utility in doing so. T o b e explicit regarding the bundle, supp ose U ∈ Λ o . The open manifold Λ o has coordinates giv en by the comp onen ts of symmetric matrices, U ij = U j i , and d p i ≡ U ij d ξ j mo dulo J 2 . Then (14) ψ ∗  d ξ d p  =  B C A D   d ξ d p  . The assumption that d ξ and ψ ∗ (d ξ ) b oth hav e maximum rank when pulled back to N implies that ( B + C U ) is nonsingular. Moreo v er, Equation (14) sho ws that C S p (3) o acts on Λ o b y (15) g : U 7→ g ( U ) = ( A + D U )( B + C U ) − 1 . The ﬁb er subgroup o ver U ∈ Λ o is the stabilizer of U , so (16) Π − 1 ( U ) = { g : U = g ( U ) = ( A + D U )( B + C U ) − 1 } ∼ = { g : A = 0 } . Because I 6 is in the ﬁb er o ver 0 ∈ Λ o , the pro jection Π : C S p (3) o → Λ o can b e computed from Equation (15) as (17) Π : g 7→ g (0) = AB − 1 . The most important fact ab out Equation (10) is that M = F − 1 (0) admits a natural GL (2)- structure wherever the PDE is h yp erb olic. Recall that a second-order PDE is called h yp erb olic if its leading symbol is a non-de generate matrix with split signature, and this property is in v ariant under con tact transformations. A related notion of h yp erb olicity for h yp erplanes sitting in pro jective space is needed to describ e the induced GL (2)-structure. The complex V eronese v ariet y is the 2-dimensional pro jective v ariet y (18) { [ Z 1 Z 1 : Z 1 Z 2 : Z 1 Z 3 : Z 2 Z 3 : Z 3 Z 3 ] , Z ∈ CP 2 } ⊂ CP 5 . The intersection of a generic h yp erplane with the V eronese v ariet y is a 1-dimensional rational normal curve, and this curve is uniquely given b y such an intersec tion. The real case needed for the present PDE theory requires a bit more detail to describ e accurately . Consider RP 5 with co ordinates [ W 1 : · · · : W 6 ], RP 3 with co ordinates [ Z 1 : Z 2 : Z 3 ], and the V eronese v ariet y deﬁned GL(2) PDES 7 o v er R as ab ov e. A generic h yp erplane in RP 5 is deﬁned uniquely (up to scale and sign) by a single equation (19) a 11 W 1 + a 12 W 2 + a 13 W 3 + a 22 W 4 + a 23 W 5 + a 33 W 6 = 0 . The intersection of this h yp erplane with the V eronese v ariet y yields an equation on R 3 : (20) a 11 ( Z 1 ) 2 + a 12 Z 1 Z 2 + a 13 Z 1 Z 3 + a 22 ( Z 2 ) 2 + a 23 Z 2 Z 3 + a 33 ( Z 3 ) 2 = 0 . Dep ending on the (real) co eﬃcients a ij , Equation (20) may or ma y not hav e real solutions Z . If Equation (20) has real solutions, then the de-pro jectivized solution in R 3 is a real quadric surface. This quadric may or may not b e degenerate. The existence of real nondegenerate solutions is an op en condition on the h yp erplane in the top ology of Gr 5 ( R 6 ). If this condition is satisﬁed, the h yp erplane deﬁned b y { a ij } is called hyperb olic. The precise algebraic condition for h yp erb olicit y is that the real symmetric matrix ( a ij ) is nonsingular and has split signature [36]. Using the iden tiﬁcations R 6 = Gr 5 ( R 6 ) = Sym 2 ( R 3 ), the cone o ver the V eronese v ariety (V eronese cone) is identiﬁed with the space of symmetric matrices ( a ij ) such that rank( a ij ) ≤ 1. A h yp erb olic h yp erplane in R 6 in tersects the V eronese cone in a rational normal cone, and ev ery rational normal cone in R 6 can b e written (uniquely , up to scale) this wa y [20]. Lemma 1.2. L et F : Λ o → R b e a smo oth function, and supp ose U ∈ Λ o such that F ( U ) = 0 , d F U 6 = 0 , and ker(d F U ) is hyp erb olic as a hyp erplane in T U Λ o = Sym 2 ( R 3 ) . Then ther e is an op en 5-dimensional submanifold M ⊂ Λ o deﬁne d by F | M = 0 in a neighb orho o d of U , and M admits a distribution C of r ational normal c ones. That is, M admits a GL (2) -structur e. Pr o of. Aside from the condition on hyperb olicit y of the tangent space, the lemma is simply a statemen t of the implicit function theorem. Here is an explanation of hyperb olicity and its relation to GL (2)-structures. Fix a coframe s on Λ o suc h that s U : T U Λ o → Sym 2 ( R n ). (An ob vious choice is s = d U , using the symmetric matrices as coordinates for Λ o .) So, there is a distribution of 3-dimensional V eronese cones deﬁned by { P ∈ T Λ o : rank s ( P ) ≤ 1 } . This distribution of cones is C S p (3)-inv arian t, for in the ﬁrst part of [9, Theorem XX], Cartan prov es that the ab ov e bundle C S p (3) is exactly the frame bundle ov er Λ whose action on T U Λ ' Sym 2 ( R n ) is to act irreducibly on the co eﬃcients of the equation of a cone, for example Equation (20). In the con text of the lemma, Equation (19) describ es the intersection of ker(d F U ) with the V eronese cone, so the symmetric matrix ( a ij ) is, up to scale, the leading symbol of F at U . Therefore, if U is a regular point for the regular v alue 0 of F : Λ o → R and if the PDE F = 0 is h yp erb olic at U , then F − 1 (0) admits a ﬁeld of rational normal cones near U given b y these in tersections.  The fact that ev ery h yp erb olic PDE of the form in Equation (10) admits a GL (2)-structure w ould only b e an algebraic curiosity , except that the in tegrability of the PDE is in timately related to the 2,3-integrabilit y of the GL (2)-structure. Theorem 1.3 (Theorem 3 in [14]) . Fix a hyp erb olic PDE of the form in Equation (10) and its c orr esp onding GL (2) -structur e π : B → M . Then B is 2-inte gr able. Mor e over, the PDE is inte gr able via 3-p ar ameter hydr o dynamic r e ductions if and only if B is also 3-inte gr able. Theorem 1.3 as presen ted in [14] apparen tly requires that, for each 3-secan t N 3 ⊂ M 5 , the c har- acteristic net deﬁned b y the in tersection T N ∩ C is a co ordinate net. According to Corollary 4.2, this requirement is redundan t. As discussed in [14], for PDEs in three indep enden t v ariables, the existence of 3-parameter h ydro dynamic reductions parametrized by three functions of one v ariable implies the existence of k -parameter hydrodynamic reductions parametrized by k functions of one v ariable for all k ≥ 3. 8 A. D. SMITH Corollary 1.4. F ( u ij ) = 0 is inte gr able via k -p ar ameter hydr o dynamic r e ductions for al l k ≥ 2 if and only if the induc e d GL (2) -structur e over F − 1 (0) is 2,3-inte gr able. Deﬁnition 1.3 (Hessian hydrodynamic PDEs) . A PDE on u : R 3 → R is Hessian hydr o dy- namic if and only if it is hyp erb olic, is of the form F ( u 11 , . . . , u 33 ) = 0 , and is inte gr able by me ans of 3-p ar ameter hydr o dynamic r e ductions. The remainder of this article is a study of Hessian h ydro dynamic PDEs via their associated 2,3-in tegrable GL (2)-structures. 2. A Preferred Connection In this section, Cartan’s metho d of equiv alence is applied to GL (2)-structures. Cartan’s metho d of equiv alence is a standard to ol in the ﬁeld of exterior diﬀerential systems; it is an algorithm for ﬁnding all of the diﬀeren tial in v ariants of a geometric structure, and the ﬁrst step is to ﬁx a preferred connection among all the possible pseudo-connections of a geometric structure [4, 19, 22]. (The distinction b etw een a connection and a pseudo-connection is simply whether it was obtained via suc h an algorithm.) The result of the algorithm is a preferred global coframe for the GL (2)-structure that splits into the semi-basic “tautological” form, ω , and the v ertical gl (2)-v alued “connection” form, θ . F or a GL (2)-structure π : B → M , let ω denote the tautological 1-form deﬁned by ω b = b ◦ π ∗ : T b B → V 4 . As alw ays, ω is a globally-deﬁned 1-form on B with linearly indep enden t comp onen ts, and it is semi-basic, meaning ω | ker π ∗ = 0. A pseudo-connection on B is a 1-form θ taking v alues in the non-trivial represen tation of gl (2) in gl ( V 4 ) such that d ω = − θ ∧ ω + T ( ω ∧ ω ) for some torsion T ( b ) : V 4 ∧ V 4 → V 4 . The goal is to ﬁx a particular θ that minimizes the torsion T . T o carry out this pro cess, one needs to understand the S L (2) represen tation theory on V n . The decomposition of the tensor pro duct V m ⊗ V n in to irreducible comp onen ts is (21) V m ⊗ V n = V | m − n | ⊕ V | m − n | +2 ⊕ · · · ⊕ V m + n − 2 ⊕ V m + n . The pro jections onto the v arious comp onents are giv en b y the Clebsch–Gordan [5, 21] pairings h· , ·i p : V m ⊗ V n → V m + n − 2 p , which are pro vided by the form ula (22) h u, v i p = 1 p ! p X k =0 ( − 1) k  p k  ∂ p u ∂ x p − k ∂ y k · ∂ p v ∂ x k ∂ y p − k . This pairing has some imp ortant prop erties. Notice that h u, v i p = ( − 1) p h v , u i p and that the pairing is non trivial for 0 ≤ p ≤ min { m, n } . Hence, the tensor decomp osition can b e further reﬁned in terms of the symmetric and alternating tensors: V n ◦ V n = V 2 n ⊕ V 2 n − 4 ⊕ · · · ⊕ V 0 or 2 , V n ∧ V n = V 2 n − 2 ⊕ V 2 n − 6 ⊕ · · · ⊕ V 2 or 0 . (23) Notice to o that h· , ·i n : V n ⊗ V n → V 0 = R is a non-degenerate symmetric or sk ew bilinear form. Hence, for ﬁxed u ∈ V n the map h u, ·i n : V n → V 0 = R 1 pro vides a natural identiﬁcation, V n = V ∗ n , and one need nev er distinguish b et ween dual spaces when considering representations. F or an y deriv ation ov er R [ x, y ], a Leibniz rule o v er the pairing holds. Because S L (2) is inﬁnites- imally generated b y X , Y , and H , this means that the pairings are S L (2)-equiv arian t. That is, α ( h u, v i p ) = h α ( u ) , v i p + h u, α ( v ) i p for any α ∈ sl (2), which implies that a · h u, v i p = h a · u, a · v i p for any a ∈ S L (2). The pairing is not GL (2)-equiv ariant, but the scaling action is easily computed easily where required. The geometric ob jects encoun tered here are pro jectively deﬁned, so this v ariance in scaling is of little concern. GL(2) PDES 9 Most importantly , the pairing can be generalized to binary-p olynomial-v alued diﬀeren tial forms o v er a manifold. If u ∈ Γ( ∧ r T ∗ B ⊗ V m ) and v ∈ Γ( ∧ s T ∗ B ⊗ V n ), then extend the deﬁnition b y using the wedge pro duct: (24) h u, v i p = 1 p ! p X k =0 ( − 1) k  p k  ∂ p u ∂ x p − k ∂ y k ∧ ∂ p v ∂ x k ∂ y p − k . In this generalization, the symmetry of the pairing is further altered by the degree of the forms: h u, v i p = ( − 1) rs + p h v , u i p . If λ is an R -v alued 1-form acting by the scaling action λI m on V m , then λ ∧ u ma y b e written as the trivial pairing h λ, u i 0 = ( − 1) r h u, λ i 0 ∈ Γ( ∧ r +1 T ∗ B ⊗ V m ). As S L (2) representations, gl (2) = sl (2) ⊕ R = V 2 ⊕ V 0 . Th us, a gl (2)-v alued pseudo-connection θ decomp oses as ( ϕ, λ ) with ϕ ∈ Γ( T ∗ B ⊗ V 2 ) and λ ∈ Γ( T ∗ B ⊗ V 0 ). The torsion of a generic θ is T : B → V 4 ⊗ ( V ∗ 4 ∧ V ∗ 4 ) = V 4 ⊗ ( V 2 ⊕ V 6 ) = ( V 2 ⊕ V 4 ⊕ V 6 ) ⊕ ( V 2 ⊕ V 4 ⊕ V 6 ⊕ V 8 ⊕ V 10 ) , so T = ( T 2 2 + T 2 4 + T 2 6 ) + ( T 6 2 + T 6 4 + T 6 6 + T 6 8 + T 6 10 ) . (25) In this notation, eac h T r n is a distinct irreducible comp onent of T . The lo wer index n indicates the weigh t of the representation in which T r n tak es v alues, and the upp er index r indicates the summand from which it w as obtained. Th us, Cartan’s ﬁrst structure equation for a GL (2)-structure ma y b e written in either vector form or p olynomial form: d ω = − θ ∧ ω + T ( ω ∧ ω ) = − h ϕ, ω i 1 − h λ, ω i 0 +  T 2 2 , h ω , ω i 3  0 +  T 2 4 , h ω , ω i 3  1 +  T 2 6 , h ω , ω i 3  2 +  T 6 2 , h ω , ω i 1  2 +  T 6 4 , h ω , ω i 1  3 +  T 6 6 , h ω , ω i 1  4 +  T 6 8 , h ω , ω i 1  5 +  T 6 10 , h ω , ω i 1  6 . (26) Explicitly , the connection term is (27) θ ∧ ω =       − 8 ϕ 0 + λ 8 ϕ − 2 0 0 0 − 2 ϕ 2 − 4 ϕ 0 + λ 6 ϕ − 2 0 0 0 − 4 ϕ 2 λ 4 ϕ − 2 0 0 0 − 6 ϕ 2 4 ϕ 0 + λ 2 ϕ − 2 0 0 0 − 8 ϕ 2 8 ϕ 0 + λ       ∧       ω − 4 ω − 2 ω 0 ω 2 ω 4       , so the matrix represen tation of θ is (2 ϕ − 2 X − 2 ϕ 0 H + 2 ϕ 2 Y + λI 5 ), whic h tak es v alues in gl (2) ⊂ gl ( V 4 ). Theorem 2.1. A generic GL (2) -structur e π : B → M admits a two-dimensional family of c onne c- tions such that the essential torsion T de c omp oses irr e ducibly as (28) T = T 2 + T 6 + T 8 + T 10 ∈ V 2 ⊕ V 6 ⊕ V 8 ⊕ V 10 ⊂ V 4 ⊗ ( V ∗ 4 ∧ V ∗ 4 ) . The two-dimensional family is p ar ametrize d by the p ossible e quivariant inclusions of V 2 ⊕ V 6 ⊕ V 8 ⊕ V 10 into V 4 ⊗ ( V ∗ 4 ∧ V ∗ 4 ) , and onc e such an inclusion is chosen, the c onne ction is unique. Pr o of. Changes of pseudo-connection are of the form ˆ ϕ = ϕ + P ( ω ) and ˆ λ = λ + Q ( ω ) where P ∈ V 2 ⊗ V ∗ 4 = V 2 ⊕ V 4 ⊕ V 6 and Q ∈ V 0 ⊗ V ∗ 4 = V 4 . Preferred connections are obtained b y analyzing the skewing map δ that describ es ho w c hanges of pseudo-connection (equiv alently , 10 A. D. SMITH c hanges of horizontal section of the frame bundle) aﬀect the torsion [4]: (29) 0 → gl (2) (1) ( V 2 ⊕ V 0 ) ⊗ V ∗ 4 V 4 ⊗ ( ∧ 2 V ∗ 4 ) H 0 , 2 ( gl (2)) → 0 . B - - δ - H H H H H H H Y P,Q 6 T        * [ T ] F or current purp oses, gl (2) (1) and H 0 , 2 ( gl (2)) are deﬁned b y the exactness of the sequence [19]. They depend on the represen tation of the group GL (2), in this case the irreducible representation V 4 . T o ﬁnd the space of essen tial torsion, H 0 , 2 ( gl (2)), one must compute δ P and δ Q . Fix P ∈ V 2 ⊕ V 4 ⊕ V 6 , where V 2 3 P ( ω ) = h P 2 , ω i 2 + h P 4 , ω i 3 + h P 6 , ω i 4 . Let δ P ∈ V 4 ⊗ ∧ 2 ( V 4 ) hav e comp onen ts δ P = δ P 2 2 + δ P 2 4 + δ P 2 6 + δ P 6 2 + δ P 6 4 + δ P 6 6 + δ P 6 8 + δ P 6 10 , similar to the decomp osition of T . Since these are irreducible components and the action of δ m ust b e S L (2)-equiv ariant, Sc hur’s lemma implies δ must preserve the weigh ts of the representations. In particular there must exist constan ts a 2 , a 4 , a 6 , b 2 , b 4 , and b 6 suc h that δ P 2 2 = a 2 P 2 , δ P 6 2 = b 2 P 2 , and so on. Th us, 0 = h P ( ω ) , ω i 1 − δ P ( ω , ω ) = hh P 2 , ω i 2 , ω i 1 + hh P 4 , ω i 3 , ω i 1 + hh P 6 , ω i 4 , ω i 1 − h a 2 P 2 , h ω , ω i 3 i 0 − h a 4 P 4 , h ω , ω i 3 i 1 − h a 6 P 6 , h ω , ω i 3 i 2 − h b 2 P 2 , h ω , ω i 1 i 2 − h b 4 P 4 , h ω , ω i 1 i 3 − h b 6 P 6 , h ω , ω i 1 i 4 . (30) Carrying out this computation shows that (31) a 2 = 3 10 , b 2 = 1 5 , a 4 = 1 2 , b 4 = 0 , a 6 = − 1 5 , b 6 = − 1 20 . Similarly , ﬁx Q ∈ V 4 , where V 0 3 Q ( ω ) = h Q 4 , ω i 4 . Let δ Q ∈ V 4 ⊗ ∧ 2 ( V 4 ) ha v e comp onen ts δ Q = δ Q 2 2 + δ Q 2 4 + δ Q 2 6 + δ Q 6 2 + δ Q 6 4 + δ Q 6 6 + δ Q 6 8 + δ Q 6 10 , but again the image must hav e the same w eigh t as the domain. In particular there must exist constan ts c 4 , and d 4 suc h that 0 = h Q ( ω ) , ω i 0 − δ Q ( ω , ω ) = hh Q 4 , ω i 4 , ω i 0 − h c 4 Q 4 , h ω , ω i 3 i 1 − h d 4 Q 4 , h ω , ω i 1 i 3 . (32) Carrying out this computation shows that (33) c 4 = − 1 40 , d 4 = − 1 160 . Fix a generic pseudo-connection ( ϕ, λ ) with torsion T . Consider another pseudo-connection ˆ ϕ = ϕ + P ( ω ), ˆ λ = λ + Q ( λ ) with torsion ˆ T . Then ˆ T ( ω , ω ) = d ω + h ˆ ϕ, ω i 1 + D ˆ λ, ω E 0 = d ω + h ϕ, ω i 1 + h P ( ω ) , ω i 1 + h Q ( ω ) , ω i 0 = ( T + δ P + δ Q )( ω , ω ) . (34) GL(2) PDES 11 Using Equation (31) and Equation (33), the absorption of torsion is dictated by the solv abilit y of the following equations in terms of P and Q for ﬁxed T and ˆ T : ˆ T 2 2 = T 2 2 + 3 10 P 2 , ˆ T 2 4 = T 2 4 + 1 2 P 4 − 1 40 Q 4 , ˆ T 2 6 = T 2 6 − 1 5 P 6 , ˆ T 6 2 = T 6 2 + 1 5 P 2 , ˆ T 6 4 = T 6 4 − 1 160 Q 4 , ˆ T 6 6 = T 6 6 − 1 20 P 6 , ˆ T 6 8 = T 6 8 , ˆ T 6 10 = T 6 10 . (35) Generally , one ma y choose P 2 to force exactly one linear com bination of ˆ T 6 2 and ˆ T 2 2 to v anish. Similarly , one ma y c ho ose P 6 to force exactly one linear combination of ˆ T 6 6 and ˆ T 2 6 to v anish. Unique Q 4 and P 4 eliminate ˆ T 6 4 and ˆ T 2 4 . All other comp onents of ˆ T are ﬁxed. Thus, gl (2) (1) = sl (2) (1) = 0 and H 0 , 2 ( sl (2)) = V 2 ⊕ V 4 ⊕ V 6 ⊕ V 8 ⊕ V 10 , while H 0 , 2 ( gl (2)) = V 2 ⊕ V 6 ⊕ V 8 ⊕ V 10 .  Corollary 2.2 (Preferred GL(2) Connection) . A GL (2) -structur e B → M admits a unique c on- ne ction ϕ, λ such that B has ﬁrst structur e e quation d ω = − h ϕ, ω i 1 − h λ, ω i 0 + h T 2 , h ω , ω i 1 i 2 + h T 6 , h ω , ω i 1 i 4 + h T 8 , h ω , ω i 1 i 5 + h T 10 , h ω , ω i 1 i 6 . (36) Pr o of. Of the possible connections, c ho ose the one that absorbs T 2 2 and T 2 6 . The remaining essential torsion is T = T 6 2 + T 6 6 + T 6 8 + T 6 10 .  Henceforth, all references to θ , ϕ , λ , and T assume the connection in Corollary 2.2. The speciﬁ- cation of this connection o ver the others is arbitrary , but it do es not aﬀect an y subsequen t theorems in this article, since the V 2 and V 6 comp onen ts of essential torsion turn out to b e unimp ortan t in the study of in tegrable PDEs. 3. 2-Integrability Theorem 3.1. If a GL (2) -structur e π : B → M is 2-inte gr able, then T = T 8 (that is, T 2 = T 6 = T 10 = 0 ), and the bi-se c ant surfac es in M ar e p ar ametrize d by two functions of one variable. Conversely, if B is a smo oth GL (2) -structur e with T = T 8 , then B is 2-inte gr able. Pr o of. T o prov e the theorem, one must ﬁnd the conditions on B that allo w an arbitrary bi-secant plane E ∈ Gr 2 ( T p M ) to b e extended to a bi-secan t surface N ⊂ M . In a neigh b orho o d M 0 of p , Fix u : M 0 → B ( M 0 ), a section of π : B → M ; that is, ﬁx u , a GL (2) coframe on M 0 . Let b = u ( p ). Since E is bi-secan t, b ( E ) ⊂ V 4 is spanned by ( g 1 x − h 1 y ) 4 and ( g 2 x − h 2 y ) 4 with g 1 h 2 6 = g 2 h 1 . Through a GL (2) frame adaptation redeﬁning u , one may assume that b ( E ) = span { x 4 , y 4 } ⊂ V 4 . Let ˜ E = u ∗ ( E ) ∈ Gr 2 ( T b B ). Then ω − 4 ∧ ω 4 | ˜ E 6 = 0. 12 A. D. SMITH Consider the linear Pfaﬃan exterior diﬀerential system I diﬀeren tially generated b y the 1-forms { ω − 2 , ω 0 , ω 2 } with indep endence condition the ω − 4 ∧ ω 4 6 = 0. It suﬃces to prov e the existence of a surface ˜ N ⊂ B that is in tegral to I , b ecause u ( π ∗ ( T ˜ N )) = ω u ( T ˜ N ) = span { x 4 , y 4 } implies that the surface N = π ( ˜ N ) ⊂ M 0 is bi-secan t. In fact, by adapting the mo ving coframe u appropriately , ev ery bi-secant surface through E m ust arise this w ay . The generating 2-forms of I are (37) d   ω − 2 ω 0 ω 2   ≡   − 2 ϕ 2 0 0 0 0 2 ϕ − 2   ∧  ω − 4 ω 4  +   τ − 2 τ 0 τ 2   ω − 4 ∧ ω 4 mo dulo ω − 2 , ω 0 , ω 2 , where τ − 2 = 48 T 2 , − 2 + 8640 T 6 , − 2 + 322560 T 8 , − 2 − 4838400 T 10 , − 2 τ 0 = − 96 T 2 , 0 + 23040 T 6 , 0 − 4838400 T 10 , 0 τ 2 = 48 T 2 , 2 + 8640 T 6 , 2 − 322560 T 8 , 2 − 4838400 T 10 , 2 (38) Because of the indep endence condition, in tegral elements exist only when the torsion can be absorb ed. The torsion component τ 0 can never b e absorbed, so integral manifolds exist only when τ 0 = 0. The condition of 2-integrabilit y means that ev ery 2-secant plane is tangen t to a 2-secant surface, but the GL (2) action is transitiv e on 2-secan t planes in T p M ; therefore, 2-in tegrability implies that τ 0 = 0 for every elemen t in the GL (2) orbit of T . Under the GL (2) action, the co ordinates of the irreducible represen tations of T will c hange, so eac h irreducible representation that app ears in τ 0 m ust v anish iden tically . Hence, 2-in tegrabilit y of M by in tegral manifolds implies (39) T 10 = T 6 = T 2 = 0 . The remaining torsion comp onents, τ − 2 and τ 2 , are absorb ed by setting π 1 = 2 ϕ 2 − 322560 T 8 , − 2 ω 4 and π 2 = − 2 ϕ − 2 − 322560 T 8 , 2 ω − 4 , so (40) d   ω − 2 ω 0 ω 2   ≡   π 1 0 0 0 0 π 2   ∧  ω − 4 ω 4  , mo d ω − 2 , ω 0 , ω 2 . This prov es the torsion condition in the theorem. Con v ersely , to establish the existence and parametrization of bi-secan t surfaces, one can apply Cartan’s test for in volutivit y to the tableau in Equation (40) [3, 22]. F or a generic ﬂag of T p N obtained from generic linear com binations of ω − 4 and ω 4 , the tableau has Cartan characters s 1 = 2 and s 2 = 0. Let ˜ ν : ˜ N → B denote the em b edding of the integral surface. The space of integral elemen ts for the EDS ( I , ω − 4 ∧ ω 4 ) is 2-dimensional, as parametrized by the co eﬃcients p 1 , 4 and p 3 , − 4 that app ear in the pulled-bac k forms ˜ ν ∗ ( π 1 ) = p 1 , 4 ˜ ν ∗ ( ω 4 ) and ˜ ν ∗ ( π 2 ) = p 3 , − 4 ˜ ν ∗ ( ω − 4 ). Therefore, if π : B → M is 2-integrable, then bi-secan t surfaces in M dep end on tw o functions of one v ariable. With the Cartan c haracters computed, Cartan’s test for in volutivit y applies, so integral surfaces for the linear Pfaﬃan system exist in the real-analytic category . More can b e said b y employing modern theorems regarding hyperb olic exterior diﬀeren tial sys- tems [34]. The c haracteristic v ariety of the tableau consists of tw o real p oints, and b ecause a generic tableau is in volutiv e with s 1 = 2, Y ang’s generalization of the Cartan–K¨ ahler theorem to smo oth hyperb olic systems implies that in tegral surfaces exist and are parametrized b y tw o func- tions of one v ariable in the smo oth category [34, Theorem 1.19]. A sp ecial case of this observ ation is revisited in Section 6.1.  GL(2) PDES 13 By restricting the torsion, Theorem 3.1 provides the necessary and suﬃcient ﬁrst-order condi- tions for 2-in tegrabilit y . These ﬁrst-order conditions imply syzygies on the second-order inv arian ts via the Bianchi iden tit y . Corollary 3.2. Supp ose the GL (2) -structur e π : B → M is 2-inte gr able with torsion T . L et S = ∇ ( T ) denote the c ovariant derivative of T , and let Q = T ◦ T denote the symmetric pr o duct of T . Then B has structur e e quations of the form d ω = − h ϕ, ω i 1 − h λ, ω i 0 + h T , h ω , ω i 1 i 5 d ϕ = − 1 2 h ϕ, ϕ i 1 +  R 2 0 , h ω , ω i 3  0 + h 48 Q 4 + 42 S 4 , h ω , ω i 3 i 2 + h 45 S 6 , h ω , ω i 1 i 5 + h 33 S 8 , h ω , ω i 1 i 6 + h− 8 Q 4 − 12 S 4 , h ω , ω i 1 i 4 d λ = 960 h S 6 , h ω , ω i 1 i 6 (41) for a sc alar curvatur e function R 2 0 : B → V 0 . Pr o of. W rite ∇ = d + θ for the cov arian t deriv ativ e on B deﬁned b y the connection 1-form ( ϕ, λ ). Second-order consequences of 2-in tegrability arise from the Bianchi iden tity , ∇ ( θ ) ∧ ω = ∇ ( T ( ω ∧ ω )). Curv ature app ears in ∇ ( θ ), which splits into R ( ω ∧ ω ) = d ϕ + 1 2 h ϕ, ϕ i 1 and r ( ω ∧ ω ) = d λ . The cov arian t deriv ativ e of the torsion t w o-form, ∇ ( T ( ω ∧ ω )), expands as ∇ ( T ( ω ∧ ω )) = ∇ ( T )( ω ∧ ω ) + 2 Q ( T , T )( ω ∧ ω ∧ ω ), so the Bianc hi identit y for a GL (2)-structure is (42) R ( ω ∧ ω ) ∧ ω + r ( ω ∧ ω ) ∧ ω = ∇ ( T )( ω ∧ ω ) + 2 Q ( T , T )( ω ∧ ω ∧ ω ) . Eac h of R , r , ∇ ( T ) and Q is a function on B to the appropriate S L (2) mo dule: R : B → sl (2) ⊗ ( V ∗ 4 ∧ V ∗ 4 ) , r : B → R ⊗ ( V ∗ 4 ∧ V ∗ 4 ) , ∇ ( T ) : B → H 0 , 2 ( gl (2)) ⊗ V ∗ 4 , and Q : B → Sym 2 ( H 0 , 2 ( gl (2)) ∩ ( V 4 ⊗ ∧ 3 V ∗ 4 ) . (43) Since B is 2-integrable, T tak es v alues only in V 8 ⊂ H 0 , 2 ( gl (2)). Using the Clebsch–Gordan decomp osition, the irreducible comp onen ts of these functions are (omitting the domain B for brevit y): R = ( R 2 0 + R 2 2 + R 2 4 ) + ( R 6 4 + R 6 6 + R 6 8 ) ∈ V 2 ⊗ ( V 2 ⊕ V 6 ) , r = r 2 + r 6 ∈ V 0 ⊗ ( V 2 ⊕ V 6 ) , ∇ T = S 4 + S 6 + S 8 + S 10 + S 12 ∈ V 8 ⊗ V 4 , and Q = Q 4 + Q 8 ∈ Sym 2 ( V 8 ) ∩ ( V 4 ⊗ ( V 2 ⊕ V 6 )) . (44) Th us, Equation (42) and Sc hur’s lemma together imply linear relations among the irreducible comp onen ts listed in Equation (44). T o ﬁnd these relations, one can expand Equation (42) using the Clebsch–Gordan pairing; for example, one of the terms is (45) h∇ T , h ω , ω i 1 i 5 = hh S 4 , ω i 0 + h S 6 , ω i 1 + h S 8 , ω i 2 + h S 10 , ω i 3 + h S 12 , ω i 4 , h ω , ω i 1 i 5 . The result is S 10 = 0, R 6 8 = 33 S 8 , R 6 6 = 45 S 6 , r 6 = 960 S 6 , R 6 4 = − 12 S 4 − 8 Q 4 , R 2 4 = 42 S 4 + 48 Q 4 , r 2 = 0, and R 2 2 = 0. In particular, R 2 0 is the only irreducible comp onen t of r and R that is not an algebraic function of T and ∇ ( T ).  14 A. D. SMITH 4. 3-Integrability Theorem 4.1. If an analytic GL (2) -structur e π : B → M is 3-inte gr able, then tri-se c ant 3-folds in M ar e lo c al ly p ar ametrize d by thr e e functions of one variable. F or 3-in tegrable GL (2)-structures that arise from PDEs of hydrodynamic t yp e as in Theorem 1.3 (and are therefore also 2-in tegrable), this parametrization by three functions of one v ariable con- ﬁrms the computation presen ted in [14]. Pr o of. This theorem is pro v en b y applying Cartan–K¨ ahler theory to a diﬀerential ideal whose in tegral manifolds are tri-secant 3-folds N ⊂ M through an arbitrary tri-secan t elemen t E ∈ Gr 3 ( T p M ). As in the pro of of Theorem 3.1, consider a lo cal GL (2) coframe u , and let b = u ( p ). Since E is tri-secan t, b ( E ) is spanned b y ( g 1 x − h 1 y ) 4 , ( g 2 x − h 2 y ) 4 , and ( g 3 x − h 3 y ) 4 , distinct, but u can b e adapted so that b ( E ) is spanned by x 4 , y 4 , and ( x + y ) 4 . Therefore (46) b ( E ) = { ( A + B ) x 4 + B (4 x 3 y + 6 x 2 y 2 + 4 xy 3 ) + ( B + C ) y 3 : A, B , C ∈ R } ⊂ V 4 . Just as in Theorem 3.1, lift to ˜ E = u ∗ ( E ) ∈ Gr 2 ( T b B ), whic h is in tegral to the linear Pfaﬃan system I generated by κ − 2 = ω − 2 − ω 0 and κ 2 = ω 2 − ω 0 with the indep endence condition ω − 4 ∧ ω 0 ∧ ω 4 6 = 0. Again, the pro jection to M of any in tegral 3-fold of I will be a tri-secan t 3-fold that passes through E , and ev ery tri-secant 3-fold arises this wa y (up to a GL (2) frame adaptation). The tableau and torsion of I are giv en by (47) d  κ − 2 κ 2  ≡  π 1 π 3 0 0 − π 1 − π 2 − π 3 π 2  ∧   ω − 4 ω 0 ω 4   + X a − 6. Since d T is a vertical 1-form on B , the v alue of T only v aries in the ﬁb er of B . The reduced structure GL(2) PDES 21 equations are d ω − 4 = 24 ϕ 0 ∧ ω − 4 − 8 ϕ − 2 ∧ ω − 2 + 2 · 322560 ω 0 ∧ ω 4 , d ω − 2 = 20 ϕ 0 ∧ ω − 2 − 6 ϕ − 2 ∧ ω 0 + 322560 ω 2 ∧ ω 4 , d ω 0 = 16 ϕ 0 ∧ ω 0 − 4 ϕ − 2 ∧ ω 2 , d ω 2 = 12 ϕ 0 ∧ ω 2 − 2 ϕ − 2 ∧ ω 4 , d ω 4 = 8 ϕ 0 ∧ ω 4 , d ϕ 0 = 0 , d ϕ − 2 = 4 ϕ 0 ∧ ϕ − 2 . (57) These equations can b e easily in tegrated, so B x 8 has coordinates ξ − 4 , ξ − 2 , ξ 0 , ξ 2 , ξ 4 , a , and b such that ϕ 0 = a − 1 d a ϕ − 2 = a 4 d b, ω 4 = a 8 d ξ 4 , ω 2 = a 12  d ξ 2 − 2 b d ξ 4  , ω 0 = a 16  d ξ 0 − 4 b d ξ 2 + 4 b 2 d ξ 4  , ω − 2 = a 20  d ξ − 2 − 6 b d ξ 0 + 12 b 2 d ξ 2 − 8 b 3 d ξ 4 − 322560 ξ 4 d ξ 2  , ω − 4 = a 24  d ξ − 4 − 8 b d ξ − 2 + 24 b 2 d ξ 0 − 32 b 3 d ξ 2 + 16 b 4 d ξ 4 + 8(322560) ξ 4 b d ξ 2 − 2(322560) ξ 4 d ξ 0  . (58) The ﬁeld of rational normal cones can no w b e written in these lo cal co ordinates, since C p = {h u p ( v ) , u p ( v ) i 2 : v ∈ T p M } for a lo cal section u of B . Eac h of the 3-dimensional ro ot t yp es is a single GL (2)-orbit. Again, T ( B ) = [ v ] for any v in the ro ot t yp e, so an arbitrary representativ e v ma y b e c hosen for any ( B , M , p ) 2 , 3 . Each v has a 1-dimensional stabilizer, which is a Lie subgroup of GL (2). The corresp onding Lie algebras are easy to compute [2, 27]. In all cases, d T has b oth v ertical and semi-basic comp onents, so the em b edding of the stabilizer ﬁber group v aries ov er M . The reduced structure B v is a 6-dimensional Lie group. If dim[ v ] = 4, then B v is a ﬁnite co v er of M , but there are only eight such structures, since eac h ro ot t yp e with three ro ots is a single GL (2)-orbit. This con trasts with the case dim[ v ] ≥ 5. If dim[ v ] ≥ 5, then B v is a ﬁnite co ver of M , but there is no reason to believe that T ( B ) ⊂ [ v ] implies T ( B ) = [ v ] when dim[ v ] ≥ 5. Consider [ v ] = { 2 , 2 , 2 , 2 } . The quotient space { 2 , 2 , 2 , 2 } /GL (2) has orbifold singularities; for example, the p oint x 2 y 2 ( x + y ) 2 ( x − y ) 2 has an eight-elemen t stabilizer group, but all nearb y p oints hav e trivial stabilizer groups. The existence of these orbifold singu- larities implies that there cannot b e a surjectiv e smo oth map M → [ v ] /GL (2). If T ( B ) is a proper subset of [ v ] for all B represen ting v , then b y Theorem 5.3 a ﬁnite sequence of 2,3-in tegrable GL (2)- structures connects any t wo p oin ts in the leaf, but inﬁnitely many lo cally distinct GL (2)-structures are required to cov er the en tire leaf. This b eha vior is closely related to the top ology of CP n with k mark ed p oints, up to GL (2 , C ) action, which is a diﬃcult and well-kno wn problem in complex algebraic geometry . The real case encoun tered here is b oth harder and less well-kno wn than the classical complex case. The most relev ant result is [10], whic h sho ws that the orbifold singularities presen t in the larger leav es of V 8 are so bad that they preclude the e xistence of Riemannian metrics on the leav es. In the open case, dim[ v ] = 9, so J ( T ) has maxim um rank, and T itself pro vides lo cal co ordinates on B . This discussion is summarized by the shap e of the no des in Figure 2. 22 A. D. SMITH 6. GL(2) PDEs This section contains v arious conclusions regarding Hessian h ydro dynamic PDEs that can b e inferred from Theorem 1.3 and Theorem 5.3. Recall that a PDE F = 0 is said to ha ve k symmetries if the Lie algebra of p oin t symmetries of F − 1 (0) has dimension k . Lemma 6.1. A Hessian hydr o dynamic PDE F ( u ij ) = 0 r epr esenting v ∈ V 8 has k symmetries if and only if dim[ v ] = 9 − k . Pr o of. The contact transformations by C S p (3) induce the lo cal automorphisms on B , but an automorphism ψ : B → B near b is a symmetry if and only if the structure equations are preserv ed b y ψ ∗ . This can happ en if and only if T ( b ) is preserved near ψ ( b ). In particular, ψ ∗ (d T ) = d T if and only if ψ ∗ is the identit y on the range of d T , whic h has dimension dim[ v ].  Corollary 6.2. Ther e is no Hessian hydr o dynamic PDE with exactly eight symmetries. Pr o of. There is no ro ot t yp e of dimension one.  The classiﬁcation provides even more b oun tiful information ab out the Hessian h ydro dynamic PDEs, b ecause Theorem 4.3 allows a con v erse of Theorem 1.3. Theorem 6.3. Every ( B , M , p ) 2 , 3 is r e alize d, lo c al ly ne ar p , by a Hessian hydr o dynamic PDE. Pr o of. The structure ( B , M , p ) 2 , 3 is realized, lo cally near p , b y a Hessian hydrodynamic PDE as in Lemma 1.2 if and only if there exists a neigh b orho o d M 0 ⊂ M of p and an em b edding i : M 0 → Λ o suc h that the distribution of rational normal cones i ∗ ( C ( M 0 )) ov er i ( M 0 ) is the same as the intersection of i ∗ ( T M 0 ) with the distribution of V eronese cones ov er Λ o . Since the ﬁbers B are exactly the symmetries of C and the ﬁb ers of C S p (3) o are exactly the symmetries of the distribution of V eronese cones, it suﬃces to establish a bundle immersion h : B ( M 0 ) → C S p (3) o co v ering an em b edding i : M 0 → Λ o . Let µ denote the Maurer–Cartan form of C S p (3), whic h is of the form (59) µ =  β γ α − β t  , α = α t , γ = γ t , d µ + µ ∧ µ = 0 . (One ma y assume that the conformal scaling has b een incorp orated in to β , as it is b elo w.) By Equations (16) and (17), the comp onen ts β and γ are v ertical for Π, and α is semi-basic. Recall the F undamental Lemma of Lie Groups [22, Theorem 1.6.10]: If there exists η : T B → csp (3) such that d η + η ∧ η = 0, then for an y b ∈ B , there exists a neighborho o d B 0 of b and a map h : B 0 → C S p (3) suc h that h ∗ ( µ ) = η . Moreov er, if h ∗ ( α ) is semi-basic, then the ﬁb ers of B 0 immerse into the ﬁb ers of C S p (3). Therefore, it suﬃces to construct a csp (3)-v alued Maurer–Cartan form η on B suc h that the en tries of the lo wer-left symmetric submatrix of η , namely h ∗ ( α ij ), are semi-basic on B . F or brevit y , the pull-backs h ∗ ( · ) are dropp ed from the notation henceforth. The condition that α is semi-basic is that α ij = A ij a ω a . Note that α : V 4 → Sym 2 ( R 3 ) = Sym 2 ( V 2 ), and recall that for u, v ∈ V 2 the symmetric tensor is given b y (60)   u − 2 u 0 v 2   ◦   v − 2 v 0 v 2   = 1 2   u − 2 v − 2 + v − 2 u − 2 u − 2 v 0 + v − 2 u 0 u − 2 v 2 + v − 2 u 2 u 0 v − 2 + v 0 u − 2 u 0 v 0 + v 0 u 0 u 0 v 2 + v 0 u 2 u 2 v − 2 + v 2 u − 2 u 2 v 0 + v 2 u 0 u 2 v 2 + v 2 u 2   . Therefore, to resp ect the w eights in the GL (2) representation, α m ust hav e the following form for constan ts A − 4 , A − 2 , A 0 , A 0 0 , A 2 , and A 4 : (61) α =   A − 4 ω − 4 A − 2 ω − 2 A 0 ω 0 A − 2 ω − 2 A 0 0 ω 0 A 2 ω 2 A 0 ω 0 A 0 ω 2 A 4 ω 4   . GL(2) PDES 23 W riting β = β ϕ ( ϕ ) + β λ ( λ ) + β T ( ω ), it is apparen t that β ϕ m ust b e the representation sl (2) → M 3 × 3 ( R ) such that the natural action of M 3 × 3 ( R ) on Sym 2 ( V 2 ) is induced by the natural action of sl (2) on ω ∈ V 4 . One can now easily v erify the follo wing formulas: α =   ω − 4 ω − 2 ω 0 ω − 2 ω 0 ω 2 ω 0 ω 2 ω 4   , β =   4 ϕ 0 2 ϕ 2 0 − 4 ϕ − 2 0 4 ϕ 2 0 − 2 ϕ − 2 − 4 ϕ 0   − 1 2 λI 3 + β T ( ω ) . (62) Here, the − 1 2 λI 3 comp onen t of β is simply the scaling action of GL (2) as represented by the scaling action in C S p (3). If P GL (2) and S p (3) w ere used instead, it would not app ear. All that remains is to ﬁnd β T and γ such that d η + η ∧ η = 0. This is arithmetic, and solutions exist. The simplest η (the one where the undetermined co eﬃcien ts in β T are set to 0) is provided in [26].  Theorem 6.3 allows explicit construction of several of the most-symmetric Hessian h ydro dynamic PDEs. Each of the ro ot t yp es with three or fewer ro ots is itself closed under GL (2); hence, the torsion of any GL (2)-structure cov ers the en tire ro ot t yp e. This allows a ﬁrst step at constructing all Hessian h ydro dynamic PDEs. Ho w ever, Theorem 6.3 do es not say that there are only 55 Hessian h ydro dynamic PDEs up to C S p (3) actions. There is no reason to b elieve that the torsion map T : B → V 8 is surjective for the larger root t yp es. When T fails to b e surjective, there cannot b e a single PDE that represen ts all possible torsions in its ro ot type. Nonetheless, the leav es of dimension at most 4 can b e used to fully describ e the most symmetric Hessian hydrodynamic PDEs, as in the following theorems. Theorem 6.4. The r o ot typ e { 0 } is r epr esente d by the wave e quation, (63) u 22 = u 13 and this r epr esentation is unique up to C S p (3) . Ther efor e, the wave e quation is the unique Hessian hydr o dynamic PDE with nine symmetries. Theorem 6.5. The r o ot typ e { 8 } is r epr esente d by the ﬁrst ﬂow of the dKP hier ar chy, (64) u 22 = u 13 − 1 2 ( u 33 ) 2 and this r epr esentation is unique up to C S p (3) . Ther efor e, the ﬁrst ﬂow of the dKP hier ar chy is the unique hydr o dynamic PDE with seven symmetries. Theorem 6.6. The r o ot typ e { 7 , 1 } is r epr esente d by the PDE (65) u 22 = u 13 − 1 48 u 33 + 1 2 u 33 u 23 and this r epr esentation is unique up to C S p (3) . Theorem 6.7. The r o ot typ e { 6 , 2 } is r epr esente d by the PDE (66) u 22 = u 13 + 7 u 23 5 u 33 − 14 and this r epr esentation is unique up to C S p (3) . 24 A. D. SMITH Theorem 6.8. The r o ot typ e { 6 , 1 , 1 } is r epr esente d by the PDE (67) u 22 = u 13 + 7 u 23 ( u 23 − u 33 ) (5 u 33 − 14) + 49( − ( u 33 ) 2 + 14 u 33 − 28) 12(5 u 33 − 14) − 49 6  − (5 u 33 − 14) 14  2 / 5 and this r epr esentation is unique up to C S p (3) . Theorems 6.4 through 6.8 are pro ven via the same tec hnique. F or illustration, consider the simplest non-trivial case, Theorem 6.5. Pr o of of The or em 6.5. Consider a 2,3-in tegrable GL (2)-structure π : B → M with T ( b ) = x 8 1 322560 , so [ T ( B )] = { 8 } . The constan t is chosen for the aesthetic app eal of the resulting PDE. The goal is to describ e Theorem 6.3’s em b edded h yp ersurface i ( M 0 ) = { Π( h ( q )) ∈ Λ o : q near b } as the lo cus of a single equation F ( U ) = 0. As C S p (3) o is a matrix group, the op en set { g = h ( q ) : q near b } can b e describ ed as { exp I ( η b ( v )) : v ∈ T b B } . Moreo ver, only v ∈ ker( ϕ, λ ) need b e considered, since the ﬁb ers of π : B → M immerse in to the ﬁb ers of Π : C S p (3) o → Λ o . Fix an arbitrary v ∈ k er( ϕ, λ ) ⊂ T b B and write v in comp o- nen ts using the tautological 1-form, v ω = ( v − 4 , v − 2 , v 0 , v 2 , v 4 ) ∈ R 5 . Of course, v ω do es not actually provide lo cal co ordinates on B or M ; ho wev er, the matrix η b ( v ) still represents a generic p oin t in h ∗ ( T b ( B )), as seen here: (68) η b ( v ) =        0 0 0 0 0 0 0 0 0 0 0 0 − v 4 0 0 0 0 0 v − 4 v − 2 v 0 0 0 v 4 v − 2 v 0 v 2 0 0 0 v 0 v 2 v 4 0 0 0        . Therefore, (69) exp I ( η b ( v )) =         1 0 0 0 0 0 0 1 0 0 0 0 − v 4 0 1 0 0 0 v − 4 − 1 6 v 3 4 v − 2 + 1 2 v 4 v 2 v 0 + 1 2 v 2 4 1 0 v 4 v − 2 − 1 2 v 4 v 2 v 0 v 2 0 1 0 v 0 − 1 2 v 2 4 v 2 v 4 0 0 1         . So, using Equation (17), a generic p oin t U ∈ i ( M 0 ) lo oks like (70) U = Π(exp I ( η p ( v ))) =   v − 4 − 1 6 v 3 4 + ( v 0 + 1 2 v 2 4 ) v 4 v − 2 + 1 2 v 4 v 2 v 0 + 1 2 v 2 4 v − 2 + 1 2 v 4 v 2 v 0 v 2 v 0 + 1 2 v 2 4 v 2 v 4   . There is a single relation b et w een the en tries of such U : (71) U 22 = U 13 − 1 2 ( U 33 ) 2 . When U is interpreted as the Hessian of u : R 3 → R , Equation (71) is the ﬁrst ﬂow of the dKP hierarc h y , a w ell-kno wn example of a Hessian h ydro dynamic equation.  The only change for the other ro ot t yp es is that η is more complicated; hence, its exp onential is (immensely) more diﬃcult to compute, and the relation F ( U ) = 0 is more diﬃcult to recognize. Note also that U = 0 is alwa ys in the locus of the equation obtained b y this pro cedure. Therefore, PDEs suc h as the Boy er–Finley equation, u xx + u y y = e u tt (whic h has six symmetries and must represen t one of the 3-dimensional ro ot t yp es [14]), will not directly app ear as represen tativ es GL(2) PDES 25 via this procedure. None-the-less, ev ery C S p (3) equiv alence class of Hessian h ydro dynamic PDEs m ust arise this w ay . 6.1. Hyp erb olic Planar PDEs. This section presents some preliminary but in triguing observ a- tions regarding the h yp erb olic linear Pfaﬃan system I describing bi-secant surfaces in Theorem 3.1 and its relation to hyperb olic second-order planar PDEs, (72) f ( ξ 1 , ξ 2 , z , z 1 , z 2 , z 11 , z 12 , z 22 ) = 0 . Equation (72) deﬁnes a 7-dimensional manifold Σ f = f − 1 (0) ⊂ J 2 ( R 2 , R ) whose structure equations are obtained by pulling back the con tact system [18, 30]. Such Σ f admit a p oin t-wise classiﬁcation in to Monge–Amp` ere, Goursat, or generic-type equations. Theorem 6.9. Consider ( B , M , p ) 2 , 3 with T ( b ) = v for some b ∈ B p . Over a neighb orho o d M 0 of p , ther e is a bund le W → M 0 with 7-dimensional ﬁb er and a submersion f : B ( M 0 ) → W such that the ide al I fr om The or em 3.1 describing the existenc e of bi-se c ant surfac es thr ough p is the pul l-b ack of a hyp erb olic line ar Pfaﬃan ide al ¯ I on W . This W admits a c ofr ame ( β 1 , . . . , β 7 ) such that d β 1 ≡ β 2 ∧ β 4 + β 3 ∧ β 6 , mo d β 1 d β 2 ≡ U 1 β 3 ∧ β 7 + β 4 ∧ β 5 , mo d β 1 , β 2 d β 3 ≡ U 2 β 2 ∧ β 5 + β 6 ∧ β 7 , mo d β 1 , β 3 . (73) for U 1 = − 645120 T − 8 ( µ 3 ) 3 µ 2 µ 1 and U 2 = − 645120 T 8 ( µ 2 ) 3 µ 3 µ 1 for some non-zer o functions µ 1 , µ 2 , and µ 3 on W . Mor e over, ther e is some se c ond-or der hyp erb olic planar PDE f and a diﬀe omorphism ϕ : Σ f → W such that these structur e e quations on W pul l b ack via ϕ ∗ to the c ontact-induc e d structur e e quations on Σ f . Pr o of. Fix ( B , M , p ) 2 , 3 . As in the general case of Theorem 3.1, the linear Pfaﬃan system describing the existence of bi-secan t surfaces through p is diﬀeren tially generated b y ω − 2 , ω 0 , and ω 2 , and it has tableau given b y (74) d   ω − 2 ω 0 ω 2   =   π 1 0 0 0 0 π 2   ∧  ω − 2 ω 4  for π 1 = 2 ϕ 2 − 322560 T − 2 ω 4 and π 2 = − 2 ϕ − 2 − 322560 T 2 ω − 4 . The Lie algebra A ( I ) of Cauc h y c haracteristics of I is spanned b y the duals of λ and ϕ 0 . Therefore, the retracting space C ( I ) = A ( I ) ⊥ is a rank-sev en F robenius system on B , so B admits a foliation b y 2-dimensional Cauch y c haracteristic surfaces [3, Section I I.2] [22, Section 6.1]. Let W denote the 7-dimensional (lo cal) leaf space for this foliation, so there is a submersion ˜ π : B → W . Since the ω is semi-basic for the submersion ˜ π , W also admits a submersion onto a neigh b orho o d of p ∈ M . It remains to ﬁnd the structure equations for a coframing on W . W rite α 1 = ω 0 , α 2 = ω − 2 , α 3 = ω 2 , α 4 = π 1 , α 5 = ω − 4 , α 6 = π 2 , α 7 = ω 4 , α 8 = ϕ 0 , and α 9 = λ as a coframe for B . T o simplify the notation, ﬁx the index con ven tion 1 ≤ i, j, k ≤ 7 and 8 ≤ r , s ≤ 9, so the forms α i are semi-basic for the bundle B → W , and the forms α r are vertical for the bundle B → W . W rite d α i = − 1 2 C i j k α j ∧ α k − C i j r α j ∧ α r , where C i j k = − C i kj and C i j a are functions of T as determined b y Theorem 4.3. Note that C i j a = 0 if i 6 = j , and (75) C 1 18 = 0 , C 2 28 = − 4 , C 3 38 = 4 , C 4 48 = 4 , C 5 58 = − 8 , C 6 68 = − 4 , C 7 78 = 8 , (76) C 1 19 = 1 , C 2 29 = 1 , C 3 39 = 1 , C 4 48 = 0 , C 5 58 = 1 , C 6 68 = 0 , C 7 78 = 1 . 26 A. D. SMITH Corollary 2.3 of [3] implies that there exist functions µ i on B suc h that 1 µ i α i is basic. Deﬁne a new co-framing ( ˜ α i ) for B b y setting ˜ α i = 1 µ i α i (no sum) and ˜ α r = α r . Th us, T ∗ B is (lo cally) split into basic 1-forms and v ertical 1-forms with resp ect to the bundle ˜ π : B → W . Of course, ˜ α i = ˜ π ∗ ( β i ) for some independent 1-forms β i , and ( β 1 , . . . , β 7 ) is the desired co-framing for W . Since ˜ α i is basic, the structure equations of ( ˜ α i ) and ( β i ) are identical. There is a lot of freedom in the designation of µ i . W rite d µ i = µ i,j ˜ α j + µ i,r ˜ α r . The condition that ˜ α i is basic is equiv alen t to µ i,r = µ i C i ir for all 1 ≤ i ≤ 7 and 8 ≤ r ≤ 9, but µ i,j is otherwise free, so any non-zero function f i on W ma y b e lifted to B in lo cal co ordinates ( x 1 , . . . , x 9 ) by setting µ i = f i exp( C i i 8 x 8 + C i i 9 x 9 ). A t this p oin t, the coframe ( β i ) satisﬁes d β 1 ≡ 0 , d β 2 ≡ µ 4 µ 5 µ 2 β 4 ∧ β 5 , mo d { β 1 , β 2 , β 3 } d β 3 ≡ µ 6 µ 7 µ 3 β 6 ∧ β 7 , (77) so one ma y eﬀectively eliminate µ 5 and µ 7 b y redeﬁning β 5 as µ 4 µ 5 µ 2 β 5 and β 7 as µ 6 µ 7 µ 3 β 7 . Th us, the structure equations for W satisfy d β 1 ≡ 0 , d β 2 ≡ β 4 ∧ β 5 , mo d { β 1 , β 2 , β 3 } d β 3 ≡ β 6 ∧ β 7 . (78) One ma y now re-lab el the coframe ( β i ) follo wing the pro cedure given in App endix A of [30] to obtain Equation (73). Although U 1 and U 2 as written in the theorem are not explicitly functions on W , this is easily remedied. As noted ab ov e, µ i ma y b e tak en as the lift of f i on W . Also, the action of α 8 and α 9 is diagonal on T : (79) d              T − 8 T − 6 T − 4 T − 2 T 0 T 2 T 4 T 6 T 8              ≡              T − 8 T − 6 T − 4 T − 2 T 0 T 2 T 4 T 6 T 8              α 8 +              16 T − 8 12 T − 6 8 T − 4 4 T − 2 0 − 4 T 2 − 8 T 4 − 12 T 6 − 16 T 8              α 9 , mo d α 1 , . . . , α 7 . So, while there is no natural map from W to O J ( B ), the lo cal Lie group G generated b y the ﬁb er actions of H and I 9 (corresp onding to α 8 and α 9 ) induces a map W → O J ( B ) /G that one could also call T . Theorem 11.1.1 of [29] implies that an y 7-dimensional manifold with structure equations of this form must be diﬀeomorphic to Σ f for some hyperb olic planar PDE f .  Corollary 6.10. F or any Hessian hydr o dynamic PDE, the level sets of the R iemann invariants ar e the solutions of planar hyp erb olic PDEs. Pr o of. The manifold W arises from the ideal describing bi-secan t surfaces in M , and Corollary 4.2 sho ws that these are the level sets of the Riemann in v ariants of the Hessian hydrodynamic PDE deﬁning B .  GL(2) PDES 27 This corollary is not at all surprising, since the entire point of h ydro dynamic reduction is to reduce a h yp erb olic PDE in three v ariables to a family of h yp erb olic planar PDEs deﬁned by the Riemann inv arian ts. In the case of 2,3-integrabilit y , Corollary 6.10 also provides a more explicit justiﬁcation for the claim in Theorem 3.1 that the PDE deﬁning bi-secant surfaces can b e solv ed in the smo oth category with smo oth initial data. One naturally asks “to which Σ f is W equiv alent?” Corollary 6.11. L et Σ f = f − 1 (0) ⊂ J 2 ( R 2 , R ) for a hyp erb olic planar PDE f . Supp ose ther e is a lo c al diﬀe omorphism ψ : Σ f → W with ˜ π ( b ) ∈ ψ (Σ f ) . (1) If x = 0 and y = 0 ar e b oth r o ots of T ( b ) , then Σ f is of the Monge–A mp` er e typ e at ψ − 1 ( ˜ π ( b )) , (2) If exactly one of x = 0 or y = 0 is a r o ot of T ( b ) , then Σ f is of the Goursat typ e at ψ − 1 ( ˜ π ( b )) , (3) If neither x = 0 nor y = 0 is a r o ot of T ( b ) , then Σ f is of the generic typ e at ψ − 1 ( ˜ π ( b )) . Pr o of. These are the three p oin twise t yp es of planar h yp erb olic PDEs, and for structure equations of the form in Equation (73), they are determined by whether U 1 and U 2 v anish [18, 30]. Since U 1 ∼ T − 8 , U 1 = 0 if and only if y = 0 is a ro ot of T ( b ) ∈ V 8 . Similarly , U 2 = 0 if and only if x = 0 is a ro ot of T ( b ) ∈ V 8 .  Corollary 6.11 puts in teresting restrictions on which GL (2)-structures can yield whic h planar PDEs. In particular, x = 0 and y = 0 can b oth b e roots of v = T ( b ) if and only if [ v ] is a ro ot t yp e ha ving tw o distinct real ro ots, and the strictly complex ro ot types cannot hav e x = 0 or y = 0 as ro ots. Notably , the t yp e of W can c hange. F or example, supp ose T ( b ) = x 7 y , so ψ − 1 ( ˜ π ( b )) is of the Monge–Amp ` ere t yp e but nearb y ψ − 1 ( ˜ π ( b 0 )) is of the generic t yp e, since T ( b 0 ) = ( x + ε 1 y ) 7 ( ε 2 x + y ). This type-changing do es not o ccur for the ﬂat structure, and it is easy to compute a change-of- frame from the structure equations of the ﬂat W to the structure equations of Σ f for the planar w a ve equation z 12 = 0. Corollary 6.12. L et ( B , M , p ) 2 , 3 b e ﬂat. Then W is isomorphic to { z 12 = 0 } ⊂ J 2 ( R 2 , R ) . 7. Concluding Remarks The main results of this article are summarized in Figure 2. In short, Hessian h ydro dynamic PDEs in three indep endent v ariables are equiv alent to lo cal 2,3-in tegrable GL (2)-structures of degree 4, and b oth ob jects admit a geometric, co ordinate-free classiﬁcation by the singular foliation of R 9 sho wn in the ﬁgure. Lemma 5.1 seems to b e a miraculous coincidence. The bluntness of this relationship b et ween v and J ( v ) prompted me to in vestigate relationships b et w een the ro ots of v and the structure of the leaf O J ( v ), yielding this pro ject’s main result, Theorem 5.3. It app ears that no suc h relationship holds for 2,3-integrable GL (2)-structures of degree n ≥ 5, even though versions of Theorem 4.3 and Lemma 4.7 exist in those cases. In general, intransitiv e group oids and pseudo-groups are v ery p o orly understo o d, and it is generally imp ossible to explicitly write down the integral manifolds of an y giv en singular distribution. If it w ere not for the coincidence in this case, the leaf-equiv alence classes would ha ve little utility in understanding the Hessian hydrodynamic PDEs. Of course, Hessian h ydro dynamic PDEs on u : R 3 → R are not the only integrable PDEs of in terest in mathematics and physics. Tw o generalizations are imp ortant to consider: Q1 Integrabilit y should b e a contact-in v ariant prop ert y of a PDE. The Hessian-only form F ( u 11 , u 12 , u 13 , u 22 , u 23 , u 33 ) = 0 is not preserv ed under the full family of contact trans- formations. How ever, the asso ciated GL (2)-structures apparently relied on this form and 28 A. D. SMITH 0 8 7 1 6 2 5 3 4 4 [ 4 4 ] 6 1 1 5 2 1 4 3 1 4 2 2 3 3 2 6 [ 1 1 ] [ 3 3 ] 2 4 [ 2 2 ] 5 1 1 1 4 2 1 1 3 3 1 1 3 2 2 1 2 2 2 2 5 [ 1 1 ] 1 4 2 [ 1 1 ] 3 3 [ 1 1 ] [ 3 3 ] 1 1 [ 3 3 ] [ 1 1 ] 3 [ 2 2 ] 1 [ 2 2 ] 2 2 [ 2 2 ] [ 2 2 ] 4 1 1 1 1 3 2 1 1 1 2 2 2 1 1 4 [ 1 1 ] 1 1 4 [ 1 1 ] [ 1 1 ] 3 2 [ 1 1 ] 1 2 2 2 [ 1 1 ] [ 2 2 ] 2 1 1 [ 2 2 ] 2 [ 1 1 ] 3 1 1 1 1 1 2 2 1 1 1 1 3 [ 1 1 ] 1 1 1 3 [ 1 1 ] [ 1 1 ] 1 2 2 [ 1 1 ] 1 1 2 2 [ 1 1 ] [ 1 1 ] [ 2 2 ] 1 1 1 1 [ 2 2 ] [ 1 1 ] 1 1 [ 2 2 ] [ 1 1 ] [ 1 1 ] 2 1 1 1 1 1 1 2 [ 1 1 ] 1 1 1 1 2 [ 1 1 ] [ 1 1 ] 1 1 2 [ 1 1 ] [ 1 1 ] [ 1 1 ] 1 1 1 1 1 1 1 1 [ 1 1 ] 1 1 1 1 1 1 [ 1 1 ] [ 1 1 ] 1 1 1 1 [ 1 1 ] [ 1 1 ] [ 1 1 ] 1 1 [ 1 1 ] [ 1 1 ] [ 1 1 ] [ 1 1 ] Figure 2. The leaf-classiﬁcation of all ( B , M , p ) 2 , 3 . The n umber of sides on a no de is the dimension of the bundle after the symmetry reduction from Section 5.1. Rep- resen tativ e PDEs for the shaded no des are included in Section 6. its asso ciated C S p (3) transformations. Ho w can this classiﬁcation b e extended to second- order PDEs in three indep endent v ariables that also include lo wer deriv atives? How do es the GL (2) geometry generalize to these PDEs? The observ ations and computations of some recen t articles may pro v e very useful [7] [15]. Q2 V ery few in tegrable PDEs are kno wn to exist in more than three indep endent v ariables, but equations of the form F ( u 11 , . . . , u N N ) = 0 can sometimes yield distributions of rational nor- mal cones of degree n on h yp ersurfaces M n = F − 1 (0) ⊂ Sym 2 ( R N ) with n = 1 2 N ( N + 1) − 2. Results similar to those in Section 4 are kno wn for 2,3-in tegrability for degrees 5 through 20, but the foliation by group oid orbits of V n +4 is not understo o d, and k -integrabilit y is extremely restrictiv e [27]. What can integrable GL(2) geometry sa y about the existence of in tegrable PDEs in more v ariables? On a more detailed level, it would be in teresting to study the foliation that app ears in the presen t case. Despite the signiﬁcan t computational diﬃculties, it is important both to produce represen tativ e PDEs for each ro ot type and to ﬁnd the ro ot types of the well-kno wn Hessian h ydro dynamic PDEs. F or example, if one can pro duce Hessian hydrodynamic PDEs as w ell as con tact-equiv alent PDEs that inv olve low er-order terms, such computations could op en the do or to Q1, abov e. Additionally , the lack of surjectivit y of T : B → O J ( B ) is irritating. What is the exact relationship b etw een tw o PDEs that are leaf-equiv alent but do not ha ve o verlapping torsion? GL(2) PDES 29 Finally , the relationship b et w een W , whic h describ es bi-secant surfaces in M , and Σ f , which arises from a planar hyperb olic PDE, is worth pursuing. What is is the nature of the corresp on- dence? Can every h yp erb olic planar PDE app ear this w ay? Can this corresp ondence pro vide any new information ab out the Riemann inv arian ts of the Hessian hydrodynamic PDE? Appendix A. The Ma trix J(T) F or reference, here is the matrix J ( T ), listed by column. Note! In the ω columns, the common factor of 9216 = 2 10 3 2 has b een remov ed for clarity . The 9216 ω − 4 column:              280 T − 8 T 4 − 280 T − 6 T 2 − 245 T − 4 T 2 + 70 T − 8 T 6 + 175 T − 6 T 4 70 T − 4 T 4 − 210 T − 2 T 2 + 130 T − 6 T 6 + 10 T − 8 T 8 − 175 T 0 T 2 − 35 T − 2 T 4 + 35 T − 6 T 8 + 175 T − 4 T 6 84 T − 4 T 8 + 196 T − 2 T 6 − 140 T 2 2 − 140 T 0 T 4 − 350 T 2 T 4 + 175 T 0 T 6 + 175 T − 2 T 8 350 T 0 T 8 − 350 T 2 4 700 T 2 T 8 − 700 T 6 T 4 − 1400 T 2 6 + 1400 T 8 T 4              The 9216 ω − 2 column:              − 1400 T − 8 T 2 + 1400 T − 6 T 0 − 385 T − 8 T 4 − 840 T − 6 T 2 + 1225 T − 4 T 0 − 280 T − 4 T 2 + 1050 T − 2 T 0 − 70 T − 8 T 6 − 700 T − 6 T 4 − 910 T − 4 T 4 + 280 T − 2 T 2 + 875 T 2 0 − 240 T − 6 T 6 − 5 T − 8 T 8 1540 T 0 T 2 − 952 T − 2 T 4 − 28 T − 6 T 8 − 560 T − 4 T 6 − 105 T − 4 T 8 − 1120 T − 2 T 6 + 1400 T 2 2 − 175 T 0 T 4 2100 T 2 T 4 − 1750 T 0 T 6 − 350 T − 2 T 8 2450 T 2 4 − 1050 T 0 T 8 − 1400 T 2 T 6 − 2800 T 2 T 8 + 2800 T 6 T 4              The 9216 ω 0 column:              − 2800 T − 6 T − 2 + 2800 T − 8 T 0 − 2450 T − 4 T − 2 + 875 T − 8 T 2 + 1575 T − 6 T 0 210 T − 8 T 4 − 2100 T 2 − 2 + 1540 T − 6 T 2 + 350 T − 4 T 0 1890 T − 4 T 2 − 2625 T − 2 T 0 + 35 T − 8 T 6 + 700 T − 6 T 4 1568 T − 4 T 4 + 336 T − 2 T 2 − 2100 T 2 0 + 192 T − 6 T 6 + 4 T − 8 T 8 − 2625 T 0 T 2 + 1890 T − 2 T 4 + 35 T − 6 T 8 + 700 T − 4 T 6 210 T − 4 T 8 + 1540 T − 2 T 6 − 2100 T 2 2 + 350 T 0 T 4 − 2450 T 2 T 4 + 1575 T 0 T 6 + 875 T − 2 T 8 2800 T 0 T 8 − 2800 T 2 T 6              30 A. D. SMITH The 9216 ω 2 column:              2800 T − 4 T − 6 − 2800 T − 2 T − 8 − 1400 T − 6 T − 2 − 1050 T − 8 T 0 + 2450 T 2 − 4 2100 T − 4 T − 2 − 350 T − 8 T 2 − 1750 T − 6 T 0 − 105 T − 8 T 4 + 1400 T 2 − 2 − 1120 T − 6 T 2 − 175 T − 4 T 0 − 952 T − 4 T 2 + 1540 T − 2 T 0 − 28 T − 8 T 6 − 560 T − 6 T 4 − 910 T − 4 T 4 + 280 T − 2 T 2 + 875 T 2 0 − 240 T − 6 T 6 − 5 T − 8 T 8 1050 T 0 T 2 − 280 T − 2 T 4 − 70 T − 6 T 8 − 700 T − 4 T 6 1225 T 0 T 4 − 385 T − 4 T 8 − 840 T − 2 T 6 1400 T 0 T 6 − 1400 T − 2 T 8              The 9216 ω 4 column:              1400 T − 4 T − 8 − 1400 T 2 − 6 − 700 T − 4 T − 6 + 700 T − 2 T − 8 − 350 T 2 − 4 + 350 T − 8 T 0 − 350 T − 4 T − 2 + 175 T − 8 T 2 + 175 T − 6 T 0 84 T − 8 T 4 − 140 T 2 − 2 + 196 T − 6 T 2 − 140 T − 4 T 0 − 35 T − 4 T 2 − 175 T − 2 T 0 + 35 T − 8 T 6 + 175 T − 6 T 4 70 T − 4 T 4 − 210 T − 2 T 2 + 130 T − 6 T 6 + 10 T − 8 T 8 70 T − 6 T 8 − 245 T − 2 T 4 + 175 T − 4 T 6 − 280 T − 2 T 6 + 280 T − 4 T 8              The λ , ϕ − 2 , ϕ 0 , and ϕ 2 columns:              T − 8 T − 6 T − 4 T − 2 T 0 T 2 T 4 T 6 T 8              ,              − 16 T − 6 − 14 T − 4 − 12 T − 2 − 10 T 0 − 8 T 2 − 6 T 4 − 4 T 6 − 2 T 8 0              ,              16 T − 8 12 T − 6 8 T − 4 4 T − 2 0 − 4 T 2 − 8 T 4 − 12 T 6 − 16 T 8              ,              0 2 T − 8 4 T − 6 6 T − 4 8 T − 2 10 T 0 12 T 2 14 T 4 16 T 6              References 1. 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