MATHEMATICS

A. S. UPALOV

ON THE THEORY OF CURVES IN PROJECTIVE SPACE OF \(n\) DIMENSIONS

(Presented by Academician I. G. Petrovskii on 4 IV 1962)

In the present paper a natural frame and differential invariants of the least differential order are constructed for a curve in projective space \(P_n\) of \(n\) dimensions, and an application is given of the results obtained to linear differential equations.

1. We shall interpret a point in \(P_n\) as a central ray in the central-affine \((n+1)\)-dimensional space \(E_{n+1}\). To a curve \(C\) in \(P_n\) there will correspond in \(E_{n+1}\) a two-dimensional central conical surface \(S\) with directrix \(K\), whose equation is \(\mathbf r=\mathbf r(t)\), and the admissible transformation of the directrix
   \[ \tilde{\mathbf r}=\varphi^{-1}\mathbf r, \]
   where the function \(\varphi=\varphi(t)\), differentiable a sufficient number of times and nonzero, is at the same time an admissible transformation of the homogeneous coordinates of the point in \(P_n\).

Since every generator of the surface \(S\) is a central-affine one-dimensional space \(E_1\), with the curve \(C\) there is associated a fibered space \(E_1(K)\). We shall assume that in this space a certain field of local coordinate systems has been chosen and that, under the transformation of the directrix \(\tilde{\mathbf r}=\varphi^{-1}\mathbf r\), the field of local coordinate systems is transformed as follows: \(x^*=\varphi^{-1}x\). Since, moreover, the curve \(K\) is parametrized and, consequently, so is \(X_1\), there is naturally associated with it the holonomized tangent fibered space, which, for convenience of reference, we shall denote by \(T_1(X_1)\). Therefore with the curve \(C\) in \(P_n\) there is associated the doubled fibered space\(^*\) \(E_1\times T_1(X_1)\) \((^3)\).

If the curve \(C\) does not lie in a plane of dimension less than \(n\), then among the vectors
\[ \mathbf r^k=\frac{1}{k!}\frac{d^k\mathbf r}{dt^k},\quad k=0,1,\ldots, \]
the first \(n+1\) are linearly independent, and there is a unique relation
\[ \mathbf r^{\,n+1}+\sigma_1\mathbf r^{\,n}+\cdots+\sigma_n\mathbf r^{\,1}+\sigma_{n+1}\mathbf r^{\,0}=0. \tag{1} \]

We shall assume that, under transformations of coordinates in \(E_1\) and in \(T_1\), the coefficients \(\sigma\) are transformed in such a way that equation (1) remains invariant. Then the connecting object \(\sigma_k\) \((k=1,2,\ldots,n+1)\) determines the curve \(C\) up to automorphisms of the space \(P_n\).

Let in \(E_1\times T_1(X_1)\) a geometric differential object \(\Omega_a\) \((a,b=1,2,\ldots,N)\) be given with transformation laws
\[ \widetilde{\Omega}_a=\Phi_a(\Omega_b,\varphi_k); \]
\[ {}^*\Omega_a=F_a(\Omega_b,f_k) \]
under transformations of coordinates in \(E_1\) and in \(T_1\), respectively, where
\[ \varphi_k=\frac{1}{k!}\frac{d^k\varphi}{dt^k},\quad f_k=\frac{1}{k!}\frac{d^k f}{dt^k},\quad k=0,1,\ldots . \]
Introduce the operators
\[ D_m\Omega_a=\left(-\frac{\partial\Phi_a(\Omega_b,\varphi_k)}{\partial\varphi_m}\right)_{\varphi_k=\delta_k^0}, \quad L_m\Omega_a=\left(-\frac{\partial F_a(\Omega_b,f_k)}{\partial f_m}\right)_{f_k=\delta_k^1}^{(2)}, \]
for which we obtain the recurrence relations
\[ D_m\frac{d^k\Omega_a}{dt^k} = \frac{d}{dt}D_m\frac{d^{k-1}\Omega_a}{dt^{k-1}} + mD_{m-1}\frac{d^{k-1}\Omega_a}{dt^{k-1}}, \]
\[ L_m\frac{d^k\Omega_a}{dt^k} = \delta_m^1\frac{d^k\Omega_a}{dt^k} + \frac{d}{dt}L_m\frac{d^{k-1}\Omega_a}{dt^{k-1}} + mL_{m-1}\frac{d^{k-1}\Omega_a}{dt^{k-1}}, \tag{2} \]
where \(\delta_k^s\) are the Kronecker symbols.

\[ \overline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]

\(^*\) We use the term “doubled fibered space” instead of “doubled composite manifold” (see \((^2)\), p. 98).

An object \(W\) is then and only then a density of weight \((k,p)\) (i.e., a relative invariant of weight \(k\) in \(E_1\) and weight \(p\) in \(T_1\)) when (see \((2)\))

\[ D_m W = k\delta_m^0 W;\qquad L_m W = p\delta_m^1 W. \tag{3} \]

1. Using the known differentiation formulas or, successively applying relations (2), we obtain
   \(D_m {}^{k}\mathbf r = {}^{k-m}\mathbf r\),
   \(L_m{}^{k}\mathbf r=(k-m+1){}^{k-m+1}\mathbf r\), and everywhere it is to be assumed that an object with a negative index is zero (for example, \({}^{-s}\mathbf r=0\)). Applying condition (3) to the left-hand side of equation (1), we obtain

\[ D_0\sigma_k=0;\quad D_m\sigma_k=-\sigma_{k-m},\quad m>0;\quad k=1,2,\ldots,n+1\quad(\sigma_0=1); \tag{4a} \]

\[ L_1\sigma_k=k\sigma_k;\quad L_m\sigma_k=-(n-k+1)\sigma_{k-m+1},\quad m>1; \quad k=1,2,\ldots,n+1\quad(\sigma_0=1). \tag{4b} \]

The vectors

\[ \mathbf r_k=\sum_{s=0}^{k}\sigma_{k-s}\,{}^{s}\mathbf r \quad\text{and}\quad \widetilde{\nabla}\mathbf r_k=\frac{d}{dt}\mathbf r_k+\sigma_1\mathbf r_k, \quad k=0,1,\ldots,n, \]

form a connecting object with vector components in \(E_1\times T_1(X_1)\), which is a density of weight 1 in \(E_1\), and the first of them are linearly independent. Then there is a unique decomposition

\[ \widetilde{\nabla}\mathbf r_0=\mathbf r_1;\qquad \widetilde{\nabla}\mathbf r_k=(k+1)\mathbf r_{k+1} +\sum_{s=0}^{k} I_{k-s+1}\mathbf r_s, \quad k=1,2,\ldots,n, \tag{5} \]

where one should set \(\mathbf r_{n+1}=0\), and the coefficients \(I_k\), called the projective invariants of the curve, are equal to

\[ I_1=0;\qquad I_{k+1}=\sum_{h=1}^{k}\frac{d}{dt}\sigma_{k-h} \sum^{(h)}(-1)^p\frac{p!}{i_1!i_2!\cdots i_h!} \sigma_1^{i_1}\sigma_2^{i_2}\cdots\sigma_h^{i_h} \]
\[ {}+(k+1)\sum^{(k+1)}(-1)^p\frac{(p-1)!}{i_1!i_2!\cdots i_{k+1}!} \sigma_1^{i_1}\sigma_2^{i_2}\cdots\sigma_{k+1}^{i_{k+1}} . \tag{6} \]

(\(\sum^{(h)}\) everywhere denotes summation over all ordered systems of nonnegative integers \(i_1,i_2,\ldots,i_h\) satisfying the equation
\(i_1+2i_2+\cdots+hi_h=h\), and the number \(p\) under the summation sign is to be taken equal to
\(p=i_1+i_2+\cdots+i_h\).) The projective invariants \(I_k\)
\((2\le k\le n+1)\) coincide with the projective invariants of N. F. Rzhekhina \((^{1})\), have order \(n+2\) (the smallest possible), and, together with \(\sigma_1\), determine the curve up to automorphisms of the space \(P_n\). Denoting

\[ b_k=\frac{1}{n+2}I_k,\quad 2\le k\le n+1;\qquad b_0=1, \]

according to (2) and (4) we have

\[ L_1b_k=kb_k;\qquad L_m b_k=(k-m)b_{k-m+1},\quad m>1;\quad 2\le k\le n+1. \tag{7} \]

It follows that the collection of quantities \(b_0,b_2,b_3,\ldots,b_{n+1}\) is an object in \(T_1\).

1. The objects

\[ G_{k-3}=\frac{d}{dt}b_{k-1}-(k+1)b_k-\sum_{i+j=k}' b_i b_j, \quad k=3,4,\ldots,n+1, \tag{8} \]

(\(\sum'\) denotes summation over all positive integers \(i,j\) greater than 1 and such that \(i+j=k\)) satisfy the condition: if \(G_s=0\)
\((s=0,1,\ldots,h-1)\), then \(G_k\) \((k=0,1,\ldots,h-1,h)\) are densities of weight \((0,k+3)\), respectively.

The maximal number \(q\) such that \(G_0=G_1=\cdots=G_{q-1}=0\) is called the class of the curve \(C\) and is an important arithmetic invariant of it, which can take values from \(0\) to \(n-1\) (for \(q=n-1\) one should take \(G_0\ne 0\)).

a) If \(q<n-2\), then \(\dot\gamma=2G_{q+1}/G_q\) is an object of affine connection in \(T_1(X_1)\); b) if \(q=n-2\), then \(\gamma\) is determined by the condition \(\nabla G_{n-2}=0\). In both cases \(\Gamma=\sigma_1+\dfrac n2\gamma\) is an object of affine connection in \(E_1(X_1)\). Finally, c) if \(q=n-1\), then one can introduce such fields of local coordinate systems in \(E_1\times T_1(X_1)\) that the points of the curve \(C\) will have coordinates \((1,t,t^2,\ldots,t^n)\).

Assuming further that \(q<n-1\), and considering the object \(b_1=-2\gamma\), satisfying condition (7) for \(k=1\), we obtain that the vectors
\[ \mathbf R_k=\mathbf r_k+G_1^k\mathbf r_{k-1}+\cdots+G_{k-1}^k\mathbf r_1+G_k^k\mathbf r_0, \]
where
\[ G_h^k=\sum^{(h)}\frac{(k-n-1)(k-n-2)\cdots(k-n-p)}{i_1!i_2!\cdots i_h!}\, b_1^{i_1}b_2^{i_2}\cdots b_h^{i_h}, \tag{9} \]
are densities of weight \((1,k)\), respectively, and therefore form a natural frame of the curve \(C\). It is shown that the order of the natural frame \(n+3\) for \(q<n-2\) and \(n+4\) for \(q=n-2\) is minimal; if \(q=n-1\), the objects \(G_k^h\) cannot be constructed.

1. Having objects of affine connections in \(E_1\times T_1(X_1)\), one can construct the basic differentiation of densities of weight \((k,p)\) \((^3)\). Then from the unique expansion
   \[ D\mathbf R_n+\sum_{k=0}^{n}v_{n-k+1}\mathbf R_n=0 \tag{10} \]
   there are determined densities \(v_k\) of weight \((0,k)\), \(k=1,2,\ldots,n+1\), which have the form
   \[ v_0=0;\qquad v_2=\frac d{dt}b_1-3b_2+b_1^2; \]
   \[ v_{k+3}=\sum_{h=0}^{k}(k-h+1)G_{k-h}\sum^{(h)} \frac{k(k-1)\cdots(k-p+2)}{i_1!i_2!\cdots i_h!}\, b_1^{i_1}b_2^{i_2}\cdots b_h^{i_h}, \tag{11} \]
   where the objects \(G_k\) are found by formula (8), and the densities \(v_k\) \((k>2)\) have minimal order \(n+3\), while \(v_2\) has order \(n+4\) for \(q<n-2\) and order \(n+5\) for \(q=n-2\).

Let us differentiate each vector of the natural frame basically and expand these derivatives in the vectors of the same frame. We obtain
\[ D\mathbf R_0=\mathbf R_1;\qquad D\mathbf R_k=(k+1)\mathbf R_{k+1}+(n-k+1)\sum_{s=0}^{k-1}v_{k-s+1}\mathbf R_s, \tag{12} \]
\[ k=1,2,\ldots,n-1. \]

These relations together with (10) constitute the Frenet formulas for the curve \(C\), from which it follows that the densities \(v_2,v_3,\ldots,v_{n+1}\), together with the objects of affine connections \(\gamma\) and \(\Gamma\), determine the curve up to automorphisms of the space \(P_n\).

Since \(v_q=G_q\) is different from zero, one can introduce the projective arc length of the curve
\[ s=\int |v_q|^{1/q}\,dt \]
and construct its invariants (curvatures) of minimal order
\[ \varkappa_1=|v_q|^{-(q+1)/q}\cdot Dv_q;\qquad \varkappa_k=v_k\cdot |v_q|^{-k/q}, \]
\[ k=2,3,\ldots,q-1,\quad q+1,\ldots,n+1; \tag{13} \]
then from the Frenet formulas it follows that the specification of the invariants \(\varkappa_k\) as functions of arc length determines the curve up to automorphisms of the space \(P_n\), for one can always choose \(\varphi\) so that \(\widetilde\Gamma=0\).

1. Let a homogeneous linear differential equation be given,

\[ y^{(n+1)}+a_1y^{(n)}+\cdots+a_ny'+a_{n+1}y=0. \tag{14} \]

It preserves its form under a change of variable and a linear change of the function \(y=\varphi \tilde y\). With equation (14) there is associated, one-to-one, a curve \(C\) in projective \(n\)-dimensional space \(P_n\), defined by the object

\[ \sigma_k=\frac{(n-k+1)!}{(n+1)!}\,a_k,\quad 1\leq k\leq n+1, \]

up to automorphisms of the space \(P_n\). Therefore all objects invariantly connected with the curve \(C\) will be objects invariantly connected with the equation; in particular, \(I_k\) are called projective invariants, \(\varkappa_k\) invariants, and the number \(q\) the class of the equation. Then the equation is reduced to the form

\[ y^{(n+1)}+g_{q+3}y^{(n-q-2)}+\cdots+g_ny'+g_{n+1}y=0, \]

where the parameter \(z\) is chosen in such a way that \(b_2(z)=0\), and

\[ \varphi=-\exp\int \sigma_1\,dz. \]

Further, if the coefficients \(a_k\) are continuously differentiable once, then it follows from (5) that equation (14) is reducible by a change of the function \(y=\varphi \tilde y\) to an equation with constant coefficients if and only if its projective invariants are constant; here one should put

\[ \varphi=-\exp\int \sigma_1\,dt. \]

Finally, from Frenet’s formulas it follows that equation (14) is reducible by a change of variable and a linear change of the function to an equation with constant coefficients if and only if its invariants \(\varkappa_k\) are constant; here it is required that the coefficients \(a_k\) be differentiable 3 times when \(q<n-2\) and 4 times when \(q=n-2\), although some of them may be differentiable a smaller number of times. The parameter is changed according to the formula

\[ s=\int |v_q|^{1/q}\,dt, \]

and

\[ \varphi=-\exp\int \Gamma(s)\,ds^*. \]

Saratov State University
named after N. G. Chernyshevsky

Received
31 III 1962

REFERENCES

1. N. F. Rzhekhina, Proceedings of the Seminar on Vector and Tensor Analysis, vol. XI, 153 (1961).
2. A. E. Liber, Scientific Notes of Saratov State University named after N. G. Chernyshevsky, 70, 73 (1961); Scientific Yearbook of Saratov State University named after N. G. Chernyshevsky for 1955, p. 34 (1955).
3. A. E. Liber, DAN, 90, 137 (1953).

* The results of the work were presented at the IV All-Union Mathematical Congress.