INVARIANTS OF ANALYTIC CURVES

In this article we introduce a complete system of geometric invariants for an analytic curve. No restrictions are imposed on the curve and the invariants can be easily computed.

1 .In differential geometry the usual system of invariants of a curve c in Rn consists of the curvature functions ([2, p. 94]).Their definition involves the assumption that the derivatives c', . . ., c (n-1) are linearly independent .This excludes curves (in R3) like straight lines or (t + t2, t3 , t4).In the following we will show how to overcome these difficulties in the context of analytic curves.
The key lies in the observation that the linear dependence of successive deri- vaties c' . . . . . .(') in a (non-degenerate) interval is equivalent to the fact that c is contained in a linear manifold of dimension at most m (Proposition 1).This statement is false for C°°-curves as an example in section 5 shows; and this explains why we restrict our attention to analytic curves which include the vast majority of natural, regular curves.
The linear dependence can be determined using Gram--determinants, and these determinants (suitably normalized) turn out to furnish a complete set of invariants for every analytic curve (Theorem 1 and Theorem 2).First we want to discuss how these expressions change if c is exposed to a motion or a change of parameter .Here a motion M : Rn -+ Rn is given by M(x) = Rx + m with fixed m E Rn and an orthogonal matrix R with det R = 1; and a change of parameter is an analytic, bijective function 0 I' -+ I with 0' > 0 on I' (it respects orientation) .Defining c = M o c o 0 one immediately verifies Theorem 1 .Gj( c) = Gj( c) o (1 <_ j _< n), ¡.e.these functions are invariant under motions and channgs of parameter.

. A curve c :
In order to show that they in fact constitute a complete system of invariants we first exhibit some of their properties .for j < n where D (1, . ..,j) is the determinant of some (j,j) -submatrix of We want to find the first derivative of G; which does not vanish.Thus we first discuss the derivatives of D (1, . . ., j) for fixed v and j .Differentiation of a determinant leads to a sum of determinants each of which is obtained by differentiating one column .Since determinants with two equal columns vanish identically D(P)(1, . ..,j) is a sum of terms D(i1, . ..,ij) with 1 _< i 1 < . . .< ij and p = Ek-1(ik -k) E N.This suggests that we should look for linearly independent derivatives _¿('I), . . ., c~`> > such that Ek-1ik is minimal.They can be obtained inductively by defining i1 = 1, ik E N as the smallest number such that c( 'k ) (a) is linearly independent of c('1>(a), . .., (a) (2 < k < m) .That they minimize Ek-1ik for each j(1 <_ j < m) can be seen by looking at a different sequence 1 <_ il < < ií .
where the vectors c('k ) are independent .If k is the first index with ik 7É ik then by construction ik < ik and we can replace one of the derivatives c ('k) , . . ., c~'> by c'k without destroying the linear independence [1, p. 102] .Hence Ek-1ik was not minimal.Thus we conclude that p.i = Ek-1(ik -k) is the first number such that some DP,' does not vanish.Therefore the first non-vanishing derivatives of G; are G~Zp' 1 as long as j < n and j < m resp.GP^for j = m = n .
(iii) =~> (i) : Let a E I be chosen such that G, (a) :~0.Then c'(t), . . ., c(~)(t) are linearly independent near a and from them c~m+1>(t) can be obtained as a linear combination .This is nothing but an m-th order linear differential equation for c' .Thus c'(t) lies in an m-dimensional subspace of R" for all t near a, and hence for all t E I since c' is analytic.This shows that the dimension of c is at most m.But it cannot be less since otherwise G,,, = 0.
During the proof we introduced the numbbes pi(a) E No which denote half of the (resp.the full) multiplicity of a as zero of G; for 1 < n (resp .j = n) if G; 56 ~0.We also use the convention po = 0.
Proof.(i),(ii) and (iii) are obvious .And (iv) follows from p.i(a) _ E',= 1 (z -v) as in the proof of Proposition 1 where the i are strictly increasing .
The power series of the functions G;, property (iv) in Proposition 2 and the definitions of Ej show that all ice are analytic on I. Thus the Frenet-Serret formulae - [e 1 5 . .., have a unique solution on I with é;(a) equal to the i-th vector of the canonical basis [2, p. 96] .Furthermore e1, . . ., én form a positive orthonormal system .Thus c(t) = fáé1(T)d7is an analytic curve with 11 c' 1 1= 1. Moreover 7 0 ) = K1 . . .Kj _ 1 -_ e j -+-rj where rj is a linear combination of (The empty product is taken to be 1.) .Then we compute every Gj(c) whose value is unchanged if we add a multiple of one occuring derivative to another .This leads to Gj (c) = Ihirc~~~-°) which is easily seen to be G;(1 _< j < n-1) .Moreover Gn(c) = II~-i K~-°is analytic, has the same absolute value as Gn and the two coincide at a .Hence Gn(c) = Gn (notice that Gn :~0 at a unless Gn -0) .Now consider another curve c1 with the same properties.By Proposition 1 both curves have dimension m and applying appropriate motions to them we may assume that they lie in Rm x {O}n -n`.If m = 1 the curves are subsets of straight lines where the nature of the subset (bounded, ray or full line resp. open, closed etc.) is completely determined by I. Therefore c 1 can be obtained from c by a motion and we assume from now on m > 2. For the moment we restrict our attention to a neighborhood U of a where Gj :~0 (for all j < m).
Assume first that m = n.Then the Kj are the usual curvature functions [2,   p.93] and the theory of the Frenet-Serret formulae shows that c1 (t) is obtained from c(t) by a fixed motion for t E U.But since both curves are analytic, this must hold consequently for all t E I. If m < n we try the same approach in the 0 -K1 0 . .This proof shows that, in principle, we continued the functions tc ; analytically to all of I.This was also done in [4] but only for curves in R 3 by different methods .The disadvantages of these continued functions K ; are that their computation is involved and that their signs depend on the orientation of the Frenet-vectors j1 , . . ., e , .This was incorporated in our choice of the functions e j (t) .
5 .Our invariants G; are not only easy to compute, but do not depend on arbitrary normalizations and do still furnish a complete system of invariants for analytic curves .Furthermore they occur in natural connection with the geometric concept of the dimension of the curve c.E.g. plane curves are cha- racterized by Gi -0 (dm >_ 3) and straight lines in R 3 possess the invariants G1 -1, G 2 -0, G3 = 0.
That our considerations cannot be transferred to C°°-curves is demonstrated by the curves (t, e -1 / t2 , 0) t 7É 0 71 t > 0 71 They are not linked by a rigid motion, since 7 1 has dimension 2, but -¿2 has dimension 3. Yet the corresponding functions G,,, coincide for all m E N and t E R, and the same is true for a and r whose limits at t = 0 exist.I would like to thank K. Nomizu whose stimulating article [3] roused my interest in these questions and W.B. Jurkat whose comments were of great help when I prepared this article .
by disregarding the remaning n -m coordinates (which are 0) .The only problem is that the orientation of the vectors -¿('1 . . . . . .1-1 (in R-) may lead to the conclusion that our tc ,-1 has the wrong sign .Then we first apply the rotation R to c which is obtained from the identity-matrix by replacing the elements in the (1,1)-resp .(m + 1, m + 1)-position by -1's.If necessary we do the same to c1.Then we can use (in R-) the same reasoning as above.