TRANSVECTION AND DIFFERENTIAL INVARIANTS OF PARAMETRIZED CURVES

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1 TRANSVECTION AND DIFFERENTIAL INVARIANTS OF PARAMETRIZED CURVES G. MARÍ BEFFA AND JAN A. SANDERS Abstract. In this paper we describe an sl 2 representation in the space of differential invariants of parametrized curves in homogeneous spaces. The representation is described by three operators, one of them being the total derivative D. We use this representation to find a basis for the space of differential invariants of curves in a complement of the image of D, and so generated by transvection. These are natural representatives of first cohomology classes in the invariant bicomplex. We describe algorithms to find these basis and study most well-known geometries. 1. Introduction In classical invariant theory, that is in the study of the invariants of the action of SL 2 (R) on binary forms, the basic computational tool is that of transvection. Given any two covariants (or finite dimensional irreducible sl 2 -representations in modern language) one can construct a number of (possibly) new covariants by computing the transvectants. The simplest example is given by two linear forms, a o X + a 1 Y and b 0 X + b 1 Y. Their first transvectant is the determinant a 0 a 1 b 0 b 1 of the coefficients. Another example is the discriminant a 0 a 2 a 2 1 of a quadratic form a 0 X 2 +2a 1 XY +a 2 Y 2, which is the second transvectant of the quadratic form with itself. The process of transvection generates the kernel of a certain operator F. This operator is obtained as follows. Differentiation can be thought of as a map of u, u 1, u 2, onto itself using the rule D : u k u k+1, combined with linearity. (This map can be extended to (tensor) products by prescribing the usual product rule, which makes D a derivation.) If one restricts to a finite number of derivatives, say n, then this leads to a nilpotent matrix which can be imbedded into an sl 2 by the Jacobson-Morozow lemma (or by hand). If one now lets n go to infinity, one 1991 Mathematics Subject Classification. Primary: 37K; Secondary: 53A55. Key words and phrases. Differential invariants of curves, classical invariant theory, representations, transvectants, Clebsch-Gordan. 1

2 2 G. MARÍ BEFFA AND JAN A. SANDERS recovers in the limit a Heisenberg algebra, with operators D = E = F = k=0 k=1 k=1 u k+1 u k, u k. u k ku k 1, u k This has been described in [15, 12, 14]. Motivated by modular function theory, John McKay asked the second author of the present paper in December 1999 whether the Schwarzian derivative was in ker F. The answer was: Let g i = u i+1 /(i + 1)!. Then the Schwarzian derivative S(u) = u3 u maps to (g u 2 0 g 2 g1)/g 2 0, 2 which 1 lies in ker F. In the u i -coordinates the operator algebra {F, E, D} gives a representation of sl 2 defined as F = k(k 1)u k 1 u k k=1 u 2 2 (1.1) E = 2ku k u k k=0 D = k=0 u k+1. u k As we will see later, the kernel of F is generated by recurrent transvection of u 1, first with itself, then with the produced transvectants, as is done in classical invariant theory of SL 2 (R). The fact that the Schwarzian derivative is in the kernel of F hints to a possible connection between transvection and the differential invariants of parametrized curves. Indeed, the Schwarzian derivative is the generator of projective differential invariants of u : R RP 1. That is, any other differential invariant can be written as a function of the Schwarzian and its derivatives. The Schwarzian lies in the kernel of the operator F and can be generated by transvection of u 1. After a natural generalization to several dependent variables, a very simple calculation shows that ( ) 2 u 2 u 2 u 1 u 1 u1 u 2 u 1 u 1 is in the kernel of F. The expression is the square of the Euclidean curvature of u : R R n, expressed in terms of an arbitrary parametrization. It can be generated by transvection of u 1 (in fact, of its components u α i ) with itself and its transvectants. The Euclidean curvature is also one of the generating differential invariants for Euclidean curves. A slightly longer calculation shows that torsion of u, when written in terms of an arbitrary parametrization, also lies in the kernel of the operator F and it is also a function of transvectants of u 1 with themselves. We found other examples of this situation for curves in the Möbius sphere, the local model for flat conformal manifolds. These examples strongly suggest a connection between transvection and differential invariants of parametrized curves.

3 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS 3 This paper is based on a very basic and simple property. The operators F, D and E all commute with the prolongation of vector fields, insofar the action of the group does not affect the parameter. They define a representation of sl 2 in the space of differential invariants of curves. In this paper we show how such a representation can be used to show that transvection can take the role of differentiation in the process of finding differential invariants of parametrized curves. By doing so we are capable of finding basis of differential invariants for most well-known geometries which are always in the kernel of F. If we study polynomials on u i or other simple cases, the kernel of F is the complement of the image of D, so in that sense we are producing generators that do not include the derivative of lower order invariants. In many cases one can follow a process that ensures the result is also invariant under reparametrizations. As it is shown in the examples, one can also consider simple fractional expressions in u i, rather than polynomials. It is not clear the method works in general for fractional expressions or how far one can go weakening this assumption. In section 2 we introduce basic notions and describe the simple theorems that allow us to use transvection to generate differential invariants in general. We show how, under certain assumptions, one can always find a generating set of differential invariants produced by transvection. In section 3 we study the case of affine geometries. We describe an algorithm to find a system of independent relative invariants in the kernel of F using iterative transvection of u 1 with an appropriately chosen initial differential invariant. Group invariant contractions of these relative invariants produce a basis of differential invariants. We also show how transvection produces results which are invariant under reparametrization. Section 4 is perhaps a more surprising one. There we show how to apply this same algorithm to some non affine geometries. In the case of differential invariants of projective curves Wylczinski proved in [17] that one could lift a curve in RP n to a curve in R n+1 in the standard way, multiply the lift by a factor and define a different vector µ R n+1. He then proceeded to recurrently differentiate this vector and to produce a basis of differential invariants for projective curves by taking determinants of the derivatives (this was not his original idea, but that is what the process boils down to). It just happens that the vector µ is a relative invariant of the prolonged projective action with constant weight and Wylczinski process mirrors the transvection process we describe in section 3. In section 5 we show that transvection can substitute differentiation in Wylczinski s original method. Furthermore, we show that in many other geometric manifolds G/H the method works identically for both differentiation and transvection. That is, one can lift the curve to a vector in a higher dimensional R m where the group acts linearly. We can modify the vector to produce a relative differential invariant and we can then apply recurrent transvection of it with a properly chosen differential invariant. By doing so we can produce enough relative invariants and combine them to generate a basis for the space of differential invariants of curves in G/H all of them in the kernel of the operator F. We show explicitly the projective, conformal and Grasmann Lagrangian cases. As a conclusion, it is not at all surprising that Schwarzian and Euclidean curvature lie in the kernel of the operator F and can be written as transvectants of the derivative u 1. In fact, being differential invariants of lowest order within their

4 4 G. MARÍ BEFFA AND JAN A. SANDERS corresponding geometrical background (projective and Euclidean respectively) they were very likely to be so. This research was funded by a grant from the Netherlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). The first author would like to thank the Vrije Universiteit in Amsterdam for their support during a very pleasant visit. 2. Basic definitions and results Let M be a manifold of dimension n and let G be a group acting on M. Let J (k) (R, M) be the kth order jet bundle or set of equivalence classes of curves under the equivalence relation of kth order contact. If we introduce coordinates u = (u α ) on M, x is the parameter and we denote by u α r, we can introduce coordinates in J (k) (R, M) given by (x, u (k) ) = (x, (u α ), (u α 1 ), (u α 2 ),..., (u α k )) = (x, u, u 1,..., u k ). Very often one works in the infinite jet J ( ) (R, M) as a way to avoid specifying at each step the highest order involved. See [11] for more details. = dr u α dx r Definition 1. If a transformation group G acts on M, the action preserves the order of contact between curves and so there is an induced action in J (k) (R, M) called the prolonged action or kth prolongation. If the parameter x is preserved, it is locally given by g (x, u (k) ) = g (x, u, u 1,..., u k ) = (x, g u, (g u) 1,..., (g u) k ). where by (g u) k we denote the formula on u r obtained after differentiating k times. A (kth order) differential invariant is a local scalar function I : J (k) (R, M) R invariant under the prolonged action of G, that is, such that I(g (x, u (k) )) = I(x, u (k) ) for all g G. A kth order relative differential invariant with Jacobian weight is a vector function V : J (k) (R, M) R n such that ( ) V g (x, u (k) ) = J g (u)v (x, u (k) ) where J g is the Jacobian matrix of the map φ g : M M given by φ g (u) = g u (see [11] for more details on the Jacobian multiplier representation). Classical frames (that is, an invariant curve in the frame bundle) and relative differential invariants with Jacobian weight are the same concept, the first one emphasizes geometric properties while the second emphasizes the invariance under the prolonged action (see [11]). Given a group acting on a manifold, let v = ξ n x + φ α u α, α=1 be a infinitesimal generator of the action, where ξ = ξ(x, u) and φ α = φ α (x, u). The infinitesimal generators of the prolonged action are well known. They are given by the prolongation formula v (n) = ξ x + ( D k (φ α ξu α 1 ) + ξu α (2.1) k+1) u α, k k=0 where D is the total derivative with respect to the parameter x. Infinitesimally, a differential invariant is a local scalar function in the kernel of the infinitesimal generators of the prolonged action.

5 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS 5 Next we will define operators on the infinite jet space forming an sl 2 representation. Definition 2. Consider the following operators defined on J ( ) (R, M) n F λ = k(λ + k 1)u α k 1, λ C α=1 k=1 u α k (2.2) E λ = n (λ + 2k)u α k α=1 k=0 u α k D = n u α k+1 u α α=1 k=0 k In the geometric context, λ = 0 will be the natural value when the operators are acting on the coordinates, by going to tensor products other integer values of λ will become relevant. It is very simple to check that these operators have the following commutation relations (2.3) [F λ, D] = E λ [E λ, F λ ] = 2F λ [E λ, D] = 2D. The infinite dimensional representation theory of sl 2 is described in [4], to which we refer at the appropriate places. The value λ = 0 is particularly interesting. Later on, in Theorem 3, we will see that if ξ 1 = 0 one has (2.4) [F 0, v (n) ] = [D, v (n) ] = [E 0, v (n) ] = 0, so that these three operators will form an sl 2 representation in the space of differential invariants of parametrized curves. Definition 3. We define the (abstract) space V λ = v 0, v 1,, (, represents the linear span) as an irreducible sl 2 -module on which sl 2 acts as follows (2.5) F λ v k = k(λ + k 1)v k 1, E λ v k = (λ + 2k)v k, Dv k = v k+1. and where v 0 is a given element in the kernel of F λ with E λ v 0 = λv 0. The sl 2 - module V λ is called a Lowest Weight Module. Example 1. The space of polynomials (or tensors) in u α k is an sl 2-module. We now give two constructions, the second based on the first, of Lowest ( Weight ) Modules, u starting with the coordinate functions u i j. Let n = 2 and v 1 0 = 1, with u i ker F 0 for i = 1, 2. Then ( ) ( ) F v 0 = F u 1 1 F0 u u 2 = F 0 u 2 = 0, 1 where we denote by F the induced operator on the space of vectors in R 2 when applying F to each entry. Furthermore ( ) ( ) ( ) u Ẽv 0 = Ẽ 1 1 E0 u u 2 = 1 1 u 1 1 E 0 u 2 = u 2 = 2v u 2 1

6 6 G. MARÍ BEFFA AND JAN A. SANDERS So we can see that v 0 generates V 2, if we define ( ) u 1 v i = i+1 u 2. i+1 Let us denote this sl 2 -module by U 2. Remark that the operator F can be written as F 2 abstractly, that is, on V 2. Next we can consider U 2 U 2. We see that v 0 v 0 is the lowest weight vector of a Lowest Weight Module V 4, since ˆF (v 0 v 0 ) = F v 0 v 0 + v 0 F v 0 = 0 and Ê(v 0 v 0 ) = Ẽv 0 v 0 + v 0 Ẽv 0 = 4v 0 v 0. However, this V 4 is not the whole of U 2 U 2, as we shall see later. One should realize that the sl 2 -module V 4 on the one hand abstractly is spanned by v 0, v 1,, but concretely is given by tensor products like v 0 v 0 and v 1 v 0 + v 0 v 1, with v i U 2. The following is a list of properties that hold for any representation of the Lie algebra sl 2. We will use these properties in the next section. Lemma 1. The following relations hold true: [E λ, D n ] = 2nD n, [E λ, Fλ n] = 2nF λ n; [F λ, D n ] = nd n 1 (E λ + n 1); [Fλ n n 1, D] = nfλ (E λ n + 1). We denote F 0 and E 0 by F and E, respectively. If Ep = λp, we denote λ by ω p and we call it the weight or eigenvalue of p. Definition 4. Let V be an sl 2 -module. For p V we define its depth d(p) as the lowest m N such that F m p 0, but F m+1 p = 0. Definition 5. Let V be an sl 2 -module such that every element in V can be written as v ι, where the v ι are E-eigenvectors with eigenvalue ω v ι 0 and the sum is finite. Then we say that V is a nonnegative sl 2 -module. Notice that the tensor product of two nonnegative sl 2 -modules is again a nonnegative sl 2 -module, but a nonnegative sl 2 -module can be a tensor product of modules that are not necessarily nonnegative. Theorem 1. Let V be a nonnegative sl 2 -module. Then p V can be written uniquely as p = v 0 + D d(p) j=1 Dj 1 v j, with v j ker F, that is, p v 0 im D. If, moreover, ω v j > 2, then v j / im D. Proof. First we remark that d(p) <, otherwise the eigenvalues would become negative. We first prove the last statement. If v j im D kerf, then v j would be part of a V λ with λ < 0. Indeed, for v j V λ to be in ker F, one needs to have v j = D j v 0 and F v j = jd j 1 (E + j 1)v 0 = j(λ + j 1)D j 1 v 0 so that j = 0 or j = 1 λ. If j = 0, it cannot be in im D, so j = 1 λ 1 if v j im D. This implies that λ < 0. This is in contradiction with the fact that V is an sl 2 -module spanned by eigenvectors with nonnegative eigenvalues. Since p can be written as a finite sum of E-eigenvectors, we write p = k i=1 pi. Let q i = F d(pi) p i for i fixed. Its E-eigenvalues are ω q i = ω p i 2d(p i ). By the assumptions on V one has ω q i 0. We consider first the case ω q i > 0, then we assume ω q i = 0. We let v i = (ω q i 1)!F m p i m!(ω q i +m 1)!, where m = d(pi ). We now show that

7 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS 7 Lemma 2. One has and hence d(p i v i m) < d(p i ). F m (p i v i m) = 0. Once we have shown this, we can replace p i by p i v i m and start again. Proof. We compute F m (p i v i m) = F m (1 D m (ω q i 1)!F m m!(ω q i + m 1)! )pi = F m p i F m 1 md m 1 (ω q i 1)!F m (E + m 1) m!(ω q i + m 1)! pi = F m p i F m 1 md m 1 (ω q i 1)!F m (ω q i + m 1) m!(ω q i + m 1)! pi = F m p i F m 1 D m 1 (ω q i 1)!F m (m 1)!(ω q i + m 2)! pi = F m 1 (1 D m 1 (ω q i 1)!F m 1 (m 1)!(ω q i + m 2)! = = F (1 D F )F m 1 p i ω q i )F pi = 0 and this proves the lemma. We now consider the case ω q i = 0. It follows that DF m p i = 0, that is to say, F m p i is a constant. The simplest example is 1 u 2 2 u 1, which goes to 1 under F. Clearly there is no way back since we obtain zero when we apply D. In this case, if the constant F m p i happens to be the last element before zero, one can conclude that the element before that is (a multiple of) τ1 τ, with τ ker F, which is D log(τ) im D, and proceed as before by defining v i = log(τ) 2m!(m 1)! F m p i. The sl 2 -module is of the type U(0, 0) as defined in [4, Chapter II, section 1.2]. Here log(τ) is defined to be 1 τ1 D τ, we can extend the differential module to include D 1 in the standard way. See [?]. The following Lemma justifies the correction factor. Lemma 3. F m D m log(τ) = 2m!(m 1)!. Proof. We show this by induction on m. For m = 1 the statement is easy to verify. Then, using Lemma 1 we see that F m+1 D m+1 log(τ) = (m + 1)F m (E m)d m log(τ) and this proves the statement. = m(m + 1)F m D m log(τ) = 2m!(m + 1)!, We only need to prove uniqueness to conclude the proof of Theorem 1. But uniqueness is quite simple: if the depth is zero, it is obvious. one can then apply

8 8 G. MARÍ BEFFA AND JAN A. SANDERS induction concentrating on the highest depth term. The difference would lie in the kernel of D, which is zero. Corollary 1. If r = F m p i R, then F m (p i r 2m!(m 1)! Dm log(τ)) = 0, and so one has d(p i v i m) < d(p i ) Comment 1. For the benefit of those readers who worry about the sudden appearance of logarithmic in their computations, we add that if one reads the theorem like: Then p V can be written uniquely as p = v 0 +w, with v 0 ker F, and w im D, then there are no logarithmic terms in either v 0 or w. We now give two examples, one where Theorem 1 applies, and one where it does not. After that we formulate Lemma 4 as a corollary to Theorem 1, covering both examples. Example 2. Let p = u4 u 1. Then F p = 12 u 3 u 1, F 2 p = 72 u 2 u 1, F 3 p = 144. So d(p) = 3 and ω F 3 p = 0. Then v 3 = 6 log(u 1 ) and Let q = p D 3 v 3 = 5 u4 u u2u3 u 2 1 Dv 3 = 6 u 2 u 1, D 2 v 3 = 6 u 3 u 1 6 u2 2 u 2 1 D 3 v 3 = 6 u 4 18 u 2u 3 u 1 u u3 2 1 u 3 1 F q = 24( u 3 3 u 2 1 so d(q) = 1 and ω F q = 4, v 1 = 6( u3 12 u3 2. Then u 3 1 u 2 2 u 2 1 u 1 3 u u 2 1 ), F 2 q = 0. Then v 0 = q Dv 1 = u4 u 1 6 u2u3 + 6 u3 u 2 2 ker F. So 1 u 3 1 p = u 4 6 u 2u 3 u 1 u u3 2 1 u 3 6D( u u 2 1 ) and Dv 1 = 6 u4 u u2u3 u 2 1 u 2 2 u 2 1 ) + 6D 3 log(u 1 ). Example 3. Take p = 1 u 1. Then Dp = u2, D 2 p = u3 u 2 1 u u2u3 6 u3 u u 4 1 of Theorem 1 would give a division by zero. u4 u u3 2. u u2 2, and D 3 p = u 3 1 ker F. Thus if one starts with u5, the algorithm in the proof u 2 1 Lemma 4. Consider p = P (u 1,, u k )/u m 1, where P is a polynomial. Clearly, d(p) <. If the u-degree of the polynomial P is m, Theorem 1 applies as in Example 2, if < m it does not as in Example 3.

9 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS 9 Comment 2. If we require that the u-degree of the polynomial P is > m, it follows that after applying F enough times on p, we end up with a polynomial. Once we have a polynomial, applying F will not change that fact, and so the final eigenvalue ω F d(p) p will be strictly positive. If the u-degree of the polynomial P is = m, then we are in the situation of lemma 3. For the algorithm to work one needs a condition that guarantees ω F d(p) p > 0, a consequence of the condition we give here. Definition 6. Let V 1 and V 2 be sl 2 -modules. We define the n-transvectant τ (n) : V 1 V 2 V 1 V 2 as follows. Let f V 1 and g V 2 be eigenvectors of E. We denote f i = D i f, g i = D i g. Then (2.6) τ (m) f g = (f, g) (m) = ( ) ( ) ( 1) i ωf + m 1 ωg + m 1 f i g j. j i The definition is extended to the span of the eigenvectors by bilinearity. (See example 4.) Lemma 5. If f and g are both in ker F, then τ (m) f g ker F, where F (f g) = F f g + f F g, as usual. Furthermore, for general f and g, w (f,g) (m) = w f + w g + 2m. Proof. F τ (m) f g = F ( ) ( ) ( 1) i wf + m 1 wg + m 1 f i g j j i ( ) ( ) = ( 1) i wf + m 1 (D i F + id i 1 wg + m 1 (E + i 1))f g j j i ( ) ( ) + ( 1) i wf + m 1 wg + m 1 f i (D j F + jd j 1 (E + j 1))g j i ( ) ( ) = ( 1) i wf + m 1 wg + m 1 i(w f + i 1)f i 1 g j j i ( ) ( ) + ( 1) i wf + m 1 wg + m 1 f i j(w g + j 1)g j 1 j i ( ) ( ) = ( 1) i wf + m 1 wg + m 1 i(j + 1)f i 1 g j j + 1 i ( ) ( ) + ( 1) i wf + m 1 wg + m 1 f i j(i + 1)g j 1 j i + 1 = ( ) ( ) ( 1) i wf + m 1 wg + m 1 (i + 1)(j + 1)f i g j j + 1 i ( ) ( ) + ( 1) i wf + m 1 wg + m 1 f i (j + 1)(i + 1)g j j + 1 i + 1 = 0. 1

10 10 G. MARÍ BEFFA AND JAN A. SANDERS The eigenvalue computation is similar but easier. Eτ (m) f g = E ( ) ( ) ( 1) i wf + m 1 wg + m 1 f i g j j i ( ) ( ) = ( 1) i wf + m 1 (D i E + 2iD i wg + m 1 )f g j j i ( ) ( ) + ( 1) i wf + m 1 wg + m 1 f i (D j E + 2jD j )g j i ( ) ( ) = ( 1) i wf + m 1 wg + m 1 (w f + 2i)f i g j j i ( ) ( ) + ( 1) i wf + m 1 wg + m 1 f i (w g + 2j)g j j i = (w f + w g + 2m) ( ) ( ) ( 1) i wf + m 1 wg + m 1 f i g j. j i Notice that for the last computation f and g need not be in ker F, see Theorem 2. Comment 3. If w g + m = 1, then τ (m) f g = ( w f ) +m 1 m f0 g m ker F. Theorem 2. Let V λ and V µ be lowest weight sl 2 -modules and assumne λ+µ / Z. Then V λ V µ = V λ+µ+2i, i=0 where the projections are τ i : V λ V µ V λ+µ+2i. If f is the lowest weight vector of V λ, and g of V µ, then (f, g) (i) is the lowest weight vector of V λ+µ+2i. Proof. See [4]. Corollary 2. Let p ker F be an E-eigenvector in V λ V µ with E-eigenvalue ν and with λ + µ N. Then p is a 1 2 (ν λ µ)-transvectant of elements in ker F. Proof. It follows from Theorem 2 that p is the generator of one of the V λ+µ+2i. Therefore ν = λ + µ + 2i. The result follows. Corollary 3. Let p(u 1,, u k ) be a polynomial of degree m in ker F. Then p can be written as a the sum of repeated transvectants of u 1. Proof. Consider u 1 as the generator of the space V 2. Then the polynomial is an element in the symmetrization of V2 m. Applying Theorem 2 repeatedly, we can express p as the sum of repeated transvectants of u 1. Proposition 1. Consider p = P (u 1,, u k )/(u 1 ) m ker F, where P is a polynomial. Clearly, d(p) <. If the u-degree l of the polynomial P is > m, p can be written as a polynomial in transvectants of u 1 and 1 u 1 (the generator of V 2 ).

11 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS 11 Proof. We consider p as an element of the symmetrization of V 2 m V2 l = V 2l 2m+2i. Since l > m, one can always apply Theorem 2 such that the sum of the eigenvalues is strictly positive. Corollary 4. The boundary case l = m can also be treated: One has V 2 m V2 l = V 2 V 2i+2 = (V 2 V 2 ) V 2i. i=0 This implies that if the eigenvalue of p is strictly positive, it has to be a repeated transvectant. The only difficult case is when the eigenvalue is zero, which will correspond to the case p R. Comment 4. When w f + w g = 0, one has (f, g) (0) = f g and (f, g) (1) = w f (f 0 g 1 + f 1 g 0 ) = w f D(f, g) (0). This contradicts the pretty picture V wf V wf = V 2i, since the term generating V 2 is already generated in V 0. Therefore one cannot have the direct sum decomposition generalizing the Clebsch-Gordan formula to the infinite dimensional case as given by Theorem 2. This problem arises in general whenever one has V λ V µ with λ+µ an integer 0. Compare [4, Chapter 2, Exercise 6]. Let w f = 1 and w g = 1. The Casimir operator C = E 2 + 2(DF + F D) has on i=0 the basis f 0 g 1, f 1 g 0 the matrix [ This has Jordan normal form [ In fact, C(f 0 g 1 f 1 g 0 ) = 8(f 0 g 1 + f 1 g 0 ) and C(f 0 g 1 + f 1 g 0 ) = 0. This means that the representation of sl 2 on V 1 V 1 is not quasisimple. 3. A generating set of differential invariants created by transvection In this section we will first describe how, given a finite dimensional manifold M and a group action G, the operators above take always differential invariants of curves to differential invariants of curves, as far as the action of the group does not affect the parameter. This result is a corollary of the following Theorem and of standard differential invariance theory. Recall that the space of differential invariants for curves on a manifold M of dimension n is generated by a set of n functionally independent differential invariants of minimal order. We call such a set a minimal basis. It is simple to see that [D, v (n) ] = ξ 1 D. So we assume ξ 1 to be zero in the sequel. i=0 ]. ]. i=1

12 12 G. MARÍ BEFFA AND JAN A. SANDERS Theorem 3. Assume v is a vector field on M given by (3.1) v = ξ n x + α=1 φ α u α then, if ξ 1 = 0, the commutator of F λ and v (n) is given by n [F λ, v (n) ] = λ kd k 1 (u φ u φ) u α. k α=1 k=1 In particular, F = F 0 and v (n) commute. This singles out F as the geometric choice to complement D. Proof. The proof of this theorem is a simple calculation. Indeed, straightforward calculations give us [F λ, v (n) ] = n α=1 k=0 F λ(d k (φ ξu α 1 + ξu α k+1 ) u α k n α=1 k=0 (λ + k)(k + 1)(Dk (φ α ξu α 1 ) + ξu k+1 ). We use the fact that to rewrite this as [F λ, ] = (s + 1)(λ + s) u s [F λ, v (n) ] = n ( = α=1 k=0 n α=1 k=0 u s+1 kd k 1 (λu α φ + (k 1)φ) uα D k φ(k + 1)(λ + k) u α k+1 From here one easily gets the expression in the Theorem. These two formulas result in the following Corollary.. ) u α k u α k+1 Corollary 5. Assume that the action of G leaves the parameter x invariant. Then, the operators F, D and E take differential invariants to differential invariants. Notice that the fact that D and v (n) commute if x is left invariant by the action follows from invariant theory since D is the invariant differentiation. From now on we will assume the parameter x is invariant. First of all, the following simple lemma. Lemma 6. Given u : I G/H generic. Assume a minimal basis of differential invariants can be found as rational functions of u α k s. Then, there exists a basis of differential invariants of u formed by invariants which are eigenvectors of the operator E. Proof. Notice that, if the action of G is rational, a minimal basis of differential invariants can be found as rational functions of u k, using the algebraic formulation of the moving frame method found in [5]. Notice also that such a basis can always be decomposed in terms that are eigenvectors of E.

13 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS 13 Assume k is any differential invariant for u, and assume k = ν +µ where ν and µ are eigenvectors of E, that is, E(ν) = αν and E(µ) = βµ, α β. Since E and v (r) commute for any r, we conclude that v (r) (ν) and v (r) (µ) will also be eigenvectors with the same eigenvalue as ν and µ. Hence, if v (r) (ν + µ) = 0. This implies that v (r) (ν) = v (r) (µ) and so E(v (r) (ν)) = αv (r) (ν) = E(v (r) (µ)) = βv (r) (µ) = βv (r) (ν). If α β we have v (r) (ν) = v (r) (µ) = 0. Both ν and µ are differential invariants. This is also clearly the case if instead of two terms we have any number of them. Let {k 1,..., k n } be a minimal basis of differential invariants. Splitting these into eigenvectors, we can then produce a system of generators which are eigenvectors {ν 1,..., ν m }. Since the dimension of the space of differential invariants is n, and, being minimal, the number of generators at each degree r is determined by the rank of the r prolonged action, following a standard procedure we can extract a subset of n elements from {ν 1,..., ν m } formed by independent and generating invariants. Notice that after splitting k 1,..., k n we will always have enough differential invariants at each order r to cover the needed number of generators, it suffices to choose the terms with the highest order, for example. The lemma follows. Using Theorem 1, Corollary 2 and Lemma 6 we get the following result. Theorem 4. Let M = G/H be a homogeneous space. Given a curve u : I M, assume that we can find a generating set of differential invariants forming a nonnegative sl 2 -module, and assume that each member of this generating set can also be found to be elements of some V λ V µ with λ + µ N. Then, there exists a generating set for differential invariants of curves that lies in the kernel of F. This set is generated by transvection of elements in the kernel of F. Proof. The proof of this theorem is the direct application of Theorem 1 and its consequences. If we start with a bases {k i } of differential invariants, we can choose a bases which are also E-eigenvectors following Lemma 6. Therefore, without loss of generality we can assume they are eigenvectors with nonnegative eigenvalues. Therefore, applying Theorem 1, we can find a set {vj i }, all of them in the kernel of F, and such that k i = d(k i ) k=0 D j v i j. Since k i are all differential invariants, the algorithm shown in the proof of Theorem 1 to produce vj i ensures that they are also differential invariants. Indeed, they are linear combinations of elements obtained after applying F and D repeatedly to k i and these operators take differential invariants to differential invariants. Since {k i } generate all differential invariants, {vj i} also do. Observe that since the vi j are computed using the elements of sl 2, they are part of the nonnegative sl 2 -module, and therefore have nonnegative eigenvalues. This implies that if they lie in a tensorproduct V λ V µ, one must have λ + µ 0. Finally, also due to the process followed to generate vj i, they are guaranteed to belong to tensor products of V α factors. Applying Corollary 2 we obtain that vj i are generated by transvection of elements in the kernel of F.

14 14 G. MARÍ BEFFA AND JAN A. SANDERS Notice that, in general, one cannot guarantee that a minimal bases of differential invariants can be extracted from {vk i }. Indeed, if p = d(p) k=0 Dk v k, the differential order of v k could be higher than that of p. For example, if p = u 3 2, the process in Theorem 1 results in p = v 0 + Dv 1 + D 3 v 3, where v 0 = 3 5 (u3 2 u 1 u 2 u 3 ) u2 1u 4, v 1 = 6 35 (u2 1u u 1u 2 2), v 3 = 1 42 u3 1. One can find non minimal sets of differential invariants that generate all other invariants, are functionally independent and have more elements that the minimal bases. Still, no linear combination of them will produce a minimal set, only functional combinations. This fact gives the impression that generating a bases using transvection is not possible, it is possible only to generate large sets of generating invariants. But that is far from the real situation in specific cases, and in fact this situation does not happen in any of the known geometric cases, where lower order differential invariants are always in the kernel of F. Example 4. Assume that M = O(2) R 2 /O(2) is the Euclidean plane. A system of generators for the differential invariants of a planar Euclidean curve is given by u 1 2 = u 1 u 1 and the curvature k = p 22 p 2 12 where p ij = ui uj u 1 u 1. It is trivial to see that both u 1 2 and k are in the kernel of F. Assume that M = P SL(2)/H = RP 1. It is known that any differential invariant for curves in RP 1 can be written as a function of the Schwarzian derivative of u, S(u) = u 3 u i+j=2 ( ) 2 u2 and its derivatives with respect to x. It is also trivial to check that S(u) is in the kernel of F. We compute, following the procedure in Corollary 4, τ (2) u 1 1 = ( ) 3 u i+1 D j 1 u 1 j u 1 = 3u 1 (2 u2 2 u 3 1 and this contracts by symmetrization to u 1 u 3 u 2 ) 3u 2 u 2 1 u 2 + u u 1 Alternatively, 3 u2 2 u 2 2 u 3 = 2S(u). 1 u 1 and we can write τ (2) u 1 u 1 = 6u 3 u 1 9u 2 u 2, τ (0) u 1 u 1 = u 1 u 1 S(u) = 1 Cτ (2) u 1 u 1 6 Cτ (0) u 1 u 1 where Cp q = p q is an example of a contraction operator (and thus Cτ (0) u 1 u 1 = u 2 1). It follows that there is no unique way to see differential invariants as transvectants, and that one can try and make the choice optimal with respect to the type of general result one wants to prove.

15 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS 15 Finally, the restriction of having a system of generating DI that form a nonnegative module AND are elements of tensor products might seem very restrictive. In reality it is not. Traditional ways of generating differential invariants for curves in many geometric manifolds include the use of bilinear forms or group invariant operators (like the determinant in the case of SL(n)) and their applications to combination of derivatives of the curve. This method naturally produces system of differential invariants which are elements of tensor products and form nonnegative modules. Even for non-affine geometries for which the use of these group invariant forms is not so apparent, it is still possible to effectively apply the method generating a basis of differential invariants directly by transvection of u 1 and basic invariants. In fact, it will be interesting to find a situation in which this method is not applicable. In our next sections we will try to show how one can effectively use this representation to generate differential invariants by recurrent transvection of u 1 starting with a lowest weight invariant which is guaranteed to be generated by transvection. We will do first the simplest case, that of affine geometries. We will also show how the process can be minimally adapted to ensure the result is also invariant under reparametrization. 4. Differential invariant of curves in (G R n ) /G Let M T be the R-module of elements of the form P i u i where the P i R are polynomials in the u α i, i = 1, 2,..., α = 1,... n, divided by elements in the kernel of F. Notice that the elements of M T are vector functions. Define F T on M T by applying F as in (2.2) to each entry of this vector to produce another vector. Likewise we define E T and D T by applying E and D to each entry. We can also extend these operators to tensor products of M T using the standard product rule. We will denote the resulting operators with the same notation, F T, E T and D T. Using Proposition 1 (and its obvious generalization to several dependent variables where one divides by inner products or other combinations of u α 1 ) and its preceding comments, we know that under certain verifiable conditions the kernel of F T is generated by transvectants of the form (2.6). Assume next that we have a bilinear map (or contraction) and assume that C : M T M T R (4.1) F (C(f g)) = C(F T (f g)). Consider the special case of M = R n = (G R n )/G with action given by g u = Au + b for any g = (A, b) G R n. These Klein geometries include Euclidean, symplectic, affine and equi-affine, for example. For these special type of geometric background we will see how the sl 2 representation produced above will describe a method to find differential invariants by recursive transvection. The process can be done to ensure that they are also invariant under reparametrization. We will use the first transvectant to generate all relative differential invariants of Jacobian weight (a classical moving frame) by recurrent transvection of one initial invariant with u 1. The following are very simple observations. The expression φ f

16 16 G. MARÍ BEFFA AND JAN A. SANDERS represents the change of parameter x in f by a diffeomorphism φ. The proofs are straightforward. Proposition 2. If φ f = (φ p 1 f) φ 1, then Ef = 2pf. Notice that the reciprocal statement is not true: take f = u 2. Proposition 3. Assume f and h are real functions defined on J (k) (M) and assume they hold φ f = (φ p 1 f) φ 1, φ g = (φ r 1g) φ 1 for any change of x-variable φ : I R R. Then, (f, g) (1) holds φ (f, g) (1) = (φ r+p+1 1 (f, g) (1) ) φ 1. Likewise, if E(f) = ω f f, then ω (f,g) (1) = ω f + ω g + 2. The basis for the generation process is the following simple observation. Proposition 4. Let k be a differential invariant, and let R be a relative differential invariant with Jacobian weight. Then (k, R) (1) is also a relative differential invariant with Jacobian weight. Proof. The proof of this proposition is rather simple. If k is a differential invariant and R is a relative invariant of Jacobian weight, then v (n) (k) = 0 and v (n) (R) = J v R, where by v (n) (R) we mean the application of the prolongation vector to each one of the entries of R. Since the action is the composition of a linear action and a translation, J v will be a constant matrix for any v g (given in fact by the element in G generating v). Therefore, since D and the prolongation commute, v (n) ((k, R) (1) ) = J v (k, R) (1). Consider now a contraction map associated to the manifold. If the Lie group G is defined as the group preserving a bi-linear form, G = {g GL(n), such that g T Jg = J} then we can define C to be C(v w) = v T Jw R. The main example for which this does not exist is the equi-affine case G = SL(n). In this case we will need to use a slightly different approach that will be described separately. We start by finding a differential invariant for the action. The lowest order differential invariants, let s call it k, will be in the kernel of F and hence it will be written in terms of transvectants. One can then analyze the lower order transvectants to search for a first invariant. The simplest one will be [ k = C(u 1, u 1 ) (0)] 1 2 = [ u T ] 1 1 Ju 2 1. This is indeed a nonzero differential invariant in most geometric cases and E(k) = 2k. One case for which this vanishes is G = Sp(n). In that case we would choose [ k = C(u 1, u 1 ) (1)] 1 3 = [ u T ] 1 2 Ju 3 1. Both choices are in the kernel of F. In the case at hand there is a basic relative differential invariant of Jacobian weight, namely u 1, which trivially holds E T (u 1 ) = 2u 1. We then define V 2 = ((k) r, u 1 ) (1)

17 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS 17 and choose r so that E T (V 2 ) = 2V 2, namely r = 1. We repeat the process and define V 3 to be given by V 3 = ((k) r, V 2 ) (1) where r = 1 is chosen so that E T (V 3 ) = 2V 3. And so on. Clearly this process is generating independent relative invariants, assuming the curve is generic so that u 1,..., u n are all independent. If one wishes the system to be also independent under reparametrizations, then we merely need to divide the result by k and apply Proposition 2. Theorem 5. If G is defined by a bilinear form as above and u is nondegenerate, any differential invariant for curves in (G R n )/G can be written as a function of k, k i = C(V i, V i ) (0), i = 2, 3,..., n and their derivatives if G Sp(n) and of k, k i = C(V i, V i+1 ) (0), i = 2,..., n + 1 and their derivatives if G = Sp(n). Furthermore, in either case k and k i lie in the kernel of the operator F and k and k i k are additionally invariant under reparametrizations. Proof. The nondegeneracy asumption is usually guaranteed by {u 1,..., u n } being independent, but one can also think of generic curves. We can use standard differential invariant theory to conclude that for the groups specified in the statement of the Theorem, one has r independent differential invariants at each one of the orders r = 1, 2,..., n, except for the case G = Sp(n), for which we have r 1 of order r for r = 2,..., n 1. That means we have one first (or second for Sp(n)) order invariant, the differential of this plus an extra invariant of order 2 (or 3 for Sp(n)), the differential of these two plus an extra invariant at the next order and so on. Thus, to prove the Theorem it suffices to show that k, k i, i = 2,..., n are all functionally independent. The other two properties (invariant under reparametrizations and in the kernel of F ) are clearly true since they hold for each V i. Indeed, if G Sp(n), it is simple to see that k and k i, i = 2,..., n are independent. Notice that, up to terms and factors depending of k and its derivatives, k 2 is equivalent to u T 2 Ju 2. Up to terms and factors depending on k, k 2 and their derivatives, k 3 is equivalent to u T 3 Ju 3 and, in general, up to the previous invariants and their derivatives, k i is equivalent to u T i Ju i. Clearly, if the curve is nondegenerate, u T i Ju i are all functionally independent and the proof in this case is concluded. In the case G = Sp(n) a similar argument is valid. Up to k and derivatives of k, k 2 is equivalent to u T 2 Ju 3, up to k, k 2 and their derivatives, k 3 is equivalent to u T 3 Ju 4. In general, up to derivatives of k and k s, s r, k r is equivalent to u T r Ju r+1. These are all clearly independent, in the generic case, and so are the original differential invariants. In the case of G = SL(n, R) classical invariant theory tells us that there is one independent differential invariant of order n plus n 1 (other than the derivative of the n order one) independent ones of order n + 1. These can be obtained as above, with some changes. As before, the key is to make an initial choice of differential invariant and relative invariant, but the contraction, since it needs to be invariant under the group, is now different. Theorem 6. If G = SL(n, R), then we can choose k = det(u n,..., u 1 ) 1 r, with r = ( n 2) and produce V i as above, i = 1,... n+1 with V 1 = u 1. Then, a basis for the space of differential invariants of equi-affine curves is given by k and the differential invariants k i = det(v n+1, V n,..., V i+1, V i 1,..., V 1 ), i = 1, 2,..., n 1.

18 18 G. MARÍ BEFFA AND JAN A. SANDERS Note : notice that we are using here the contraction map n C : ˆV 0 R = S(V0 n ) given by C(v 1,..., v n ) = det(v 1,..., v n ). Proof. The proof is as simple as the previous ones. It is quite easy to see that, up to factors of k, k n 1 is equivalent to det(v n+1, V n, u n 2,..., u 1 ). Therefore, again up to functions of k and its derivatives, k n 1 will be determined by det(u n+1, u n, u n 2,..., u 1 ) and det(u n+1, u n 1, u n 2,..., u 1 ). This last one is the derivative of (k) r and so k n 1 is equivalent to ν n 1 = det(u n+1, u n, u n 2,..., u 1 ). Next, one looks at k n 2 and uses the same reasoning to conclude that, up to factors and terms depending on k, k n 1 and their derivatives, k n 2 is equivalent to ν n 2 = det(u n+1, u n, u n 1, u n 3,..., u 1 ). A simple recursion show that k i will be equivalent to ν i = det(u n+1,..., u i+1, u i 1,..., u 1 ). It is well known that {(k) r, ν 1,..., ν i 1 } form a basis for the differential invariants of equi-affine curves, and hence the proof is concluded. Comment 5. From these simple cases one can appreciate certain things. It is in fact the differentiation that generates relative differential invariants - this was of course known -. Still, the transvection process allows us to generate invariants in the kernel of F, the complement to the image of D. So, despite the fact that F has no geometrical meaning here, it is still very useful to project everything on a complement of (the trivial) im D. In this sense, the basis generated by transvection give natural representatives for first cohomology classes of the invariant variational bicomplex. See [6] for more information. We will next analyze some non-affine geometries for which, surprisingly, the same algorithm applies, including the role of differentiating and transvecting in the generation of relative and absolute differential invariants. 5. Differential invariant of curves in G/H, G GL(n, R) semisimple In this section we will analyze three manifolds, PSL(n, R)/H 1 RP n, O(n + 1, 1)/H 2 M n the Möbius sphere or local model for flat conformal manifolds, and Sp(n)/H 3 L n the Grasmann Lagrangian. Each H i is an appropriate isotropy subgroup of a distinguished point. In all cases we will describe how to write a basis of differential invariants for curves in M as a combination of transvectants. Unlike the previous cases, the action of the groups PSL(n, R), O(n + 1, 1) and Sp(n) on their respective manifolds are not linear as it affects second order frames. Still, we can use an analogous method to the one used in the linear case with proper choices of initial invariant and relative invariant. The main difference is that our relative invariants will not be vectors tangent to the manifold any longer. Several other cases of G/H, G GL(n, R) semisimple (G = O(2n, 2n) or O(p + 1, q + 1) for example) follow one of these three models as it is explained in [10]. Their study would be identical to the ones presented here.

19 DIFFERENTIAL INVARIANTS OF CURVES AND TRANSVECTANTS Differential invariants for parametrized projective curves. In this section we will show how one can find a generating system of differential invariants for projective RP n expressed in terms of transvection. Given a curve in RP n, it is known that a complete set of generators of the differential invariants of the curve is given by the so-called Wylczinski invariants ([17]). These are given by the formulas k m = det(µ n+1, µ n,..., µ m 1, µ m+1,..., µ) ( u where µ = W 1) 1 n+1, W = det(un,..., u 1 ) and where m = 0, 1,..., n 1. Proposition 5. F (k m ) = (m n)(m + 1)k m+1 E(k m ) = (2(n + 1 i) + n)k m F (W ) = 0 The proof of the propositions is straightforward using the formulas in Lemma 1. It shows how {k m } and their derivatives generate V λ for λ = 3n + 2 and v 0 = k n 1. The functional space generated by V λ would be the space of differential invariants of the curve. Wylczinski originally found the invariants k m as coefficients of the unique n + 1 order scalar differential equation with zero n order term having each entry of µ as solution. Still, one can look at them as contractions of iterated differentiation of µ. According to previous Theorems we can always find a combination of Wylczinski s invariants generated by transvection. Instead of using the algorithm, we will use a method similar to the one used for equi-affine geometry to generate this basis. Indeed, consider the vector µ above. Proposition 6. The vector µ is a relative invariant for the projective action of PSL(n+1, R). Indeed, if g PSL(n+1, R) and we denote by g (r) u (r) the prolonged projective action of that element on J (r) (R, M), then is not hard to check that g (r) µ = gµ where g (r) µ indicates the application of the prolonged action to each entry of the vector µ and where gµ is the standard multiplication of matrices. In fact, this property is an immediate consequence of µ being a column of a left-invariant moving frame for RP n, see [7]. Once we have a relative invariant with constant weight, we can apply Proposition 4 to generate several independent ones. As before, we will need a differential invariant to start the process. We choose the Wylczinski invariant we know to be in the kernel of F already, namely the one with lowest weight k = k n 1. It is not hard to see that one cannot find, in general, a differential invariant that behaves as a one form, that is, such that φ k = φ 1 k. Indeed, consider the case n = 2. In that case there is only one generating differential invariant for the action of PSL(2) on RP 1, namely the Schwarzian derivative of u : R RP 1. It is well known that φ S(u) = S(u) φ 1 + S(φ) φ 1 so that only fractional transformations will preserve S(u). If one can find an invariant in the kernel of F behaving like a one form respect to changes of parameter, then the process can be repeated as in the linear case so ensure that the resulting generators are also invariant under reparametrization, but that is not the case here.

20 20 G. MARÍ BEFFA AND JAN A. SANDERS Let us choose k = k n 1. Indeed, k n 1 = α( 1 W, W )(2) + ( 1 W, det(u n+1, u n, u n 2,..., u 1 )) (0), where α = 1 2n(n(n + 1) + 1)!)Γ( n(n + 1)). Notice that kn 1 (and indeed all projective invariants in the kernel of F ) can be written as transvection of equi-affine differential invariants, although not simply polynomial transvection. Define V i = (k, V i 1 ) (1) for i = 1,..., n + 1, where V 0 = µ. The vectors V i are all in the kernel of F so we need to use the group preserved contraction to generate invariants. Theorem 7. Consider the expressions ν i = det(v n+1, V n,..., V i+1, V i 1,..., V 0 ) for i = 0, 1,..., n 2. Then {k, ν 0,..., ν n 2 } form a basis for the space of differential invariants of projective curves. Clearly, they all lie in the kernel of F. Proof. The proof of this theorem is very simple. Indeed, V 0 = µ and hence, since V 1 is a combination of µ 1 = (V 0 ) x and V 0 with coefficients depending on k and its derivatives, we can safely substitute V 1 by µ 1 in the definition of ν i for the purposes of this proof, as far as V 0 is also present in the determinant. Likewise, for the purpose of the proof of this theorem, V 2 can be substituted by a multiple of kµ 2 since the other terms appearing in V 2 are combinations of µ and µ 1 with coefficients depending on k and its derivatives. And so on. Therefore, ν n 2 is equivalent to det(v n+1, V n, V n 1, µ n 3,..., µ) in the sense that it is equal to a multiple of it with (nonzero) coefficients depending on k. We can now write out this determinant to find it is a combination (again with coefficients depending on k and its derivatives) of k n+1 = 1, k n = 0, k n 1 = k and k n 2, the first two Wylczinski invariants. Very clearly the coefficient of k n 2 is a nonzero function of k and hence we can conclude that {k, ν k 2 } generates the same subspace of differential invariants as {k n 1, k n 2 }. Furthermore they are also functionally independent since {k n 1, k n 2 } are. If we now look at ν n 3 we see it is equivalent to analyzing the generating and independence properties of k, ν n 2 and det(v n+1, V n, V n 1, V n 2, µ n 4,..., µ). As before, this determinant can be written as a combination of k n+1 = 1, k n = 0, k n 1, k n 2 and k n 3, with coefficients depending on k and its derivatives. The coefficient of k n 3 is given by a nonzero function of k. Hence, one can conclude that {k, ν n 2, ν n 3 } generates the same subspace of differential invariants as {k n 1, k n 2, k n 3 }. They are also functionally independent since the Wylczinski invariants are. A simple iteration of the argument proves the theorem.