Reports of the Academy of Sciences of the USSR

1. Volume 162, No. 6

MATHEMATICS

Yu. I. ERMAKOV

ON THE INVARIANT EQUIPPING OF CERTAIN SURFACES OF SPECIAL TYPE IN PROJECTIVE SPACE

(Presented by Academician A. D. Aleksandrov, 10 XII 1964)

Consider, in projective \(n\)-space \(P_n\), an \(m\)-surface defined by the equation

\[ \bar{x}=\bar{x}(\eta^a)\qquad (a,b,c,d,e=1,\ldots,m). \tag{1} \]

The points of the projective space \(P_n\) may be interpreted (see \((^1)\)) as one-dimensional subspaces \(B_1\) of a certain vector \((n+1)\)-dimensional space \(B_{n+1}\), called the associated vector space. In view of this, each point \(\bar{x}\) of the surface (1) is simultaneously a contravariant vector of the associated space \(B_{n+1}\) and a basis vector of the corresponding one-dimensional vector subspace \(B_1\). A renormalization of the vector \(\bar{x}\) corresponds to a transformation of the basis in the one-dimensional vector space \(B_1\), which is called the radial vector space.

The

\[ \binom{m+v}{v} \]

vectors \(\bar{x}, \bar{x}_a,\ldots,\bar{x}_{a_1\ldots a_v}\)
\(\left(\bar{x}_{a_1\ldots a_u}=\partial^u\bar{x}/\partial\eta^{a_1}\cdots\partial\eta^{a_u}\right)\) of the associated space \(B_{n+1}\) define a vector subspace \(B_{k_v}\) of dimension \(k_v\), which depends neither on the choice of admissible coordinates on the surface nor on the choice of basis in the radial \(B_1\). This subspace is called the osculating space of order \(v\). One says that an equipping of the surface (1) is given if, with each of its points, there is associated a decomposition of the space \(B_{n+1}\) into the direct sum \(B_{n+1}=B_{m+1}+B_{n-m}\), where \(B_{m+1}\) is the osculating space of first order, and \(B_{n-m}\) is some subspace complementary to it, called the equipping space.

An \(m\)-surface in projective \(n\)-space will be called a surface of special type if all its osculating spaces, beginning with some order \(p\) satisfying the inequalities

\[ 2\le p,\qquad \binom{m+p}{p}\le n+1, \]

do not have the maximum possible number of dimensions. The currently known schemes of invariant equipping, i.e. equipping determined by the surface (1) itself, are applicable to surfaces with the maximum possible dimensions of osculating spaces; these schemes cannot be extended in their entirety to surfaces of the special type indicated.

In the present article an invariant equipping is constructed for a surface of dimension \(m\) in projective \(n\)-space under the condition that the numbers \(m\) and \(n\) are related by

\[ n=\binom{m+q-1}{q-1}-1 \]

(a) \(q=3,\ m=3,4;\) b) \(q>3,\ m\ge2\)) and that the dimension of the osculating space of order \(q-1\) is decreased by one. To simplify the exposition, we shall restrict ourselves to the case \(q=4\).

Assume that the surface (1) is a surface of special type and that

\[ n+1=\binom{m+3}{3}. \]

Let the \(n+1\) vectors \(\bar{x}, \bar{x}_a, \bar{x}_{ab}, \bar{x}_{abc}\)

of the attached space \(B_{n+1}\) are linearly dependent, but among them there are exactly \(n\) linearly independent ones. The indicated vectors determine, in an invariant way, an \(n\)-dimensional subspace of the attached \(B_{n+1}\); the determining pseudocovector \(\bar{\xi}(\xi_a)\) \((a=1,\ldots,n+1)\) of this subspace is uniquely found from the system

\[ \bar{\xi}\bar{x}=0,\qquad \bar{\xi}\bar{x}_a=0,\qquad \bar{\xi}\bar{x}_{ab}=0,\qquad \bar{\xi}\bar{x}_{abc}=0, \tag{2} \]

which is a homogeneous system of rank \(n\) with respect to the components \(\xi_a\) of the pseudocovector \(\bar{\xi}\).

Suppose now that the surface under consideration is partially framed with respect to the osculating space of third order \(B_{k_3}\), i.e., for each point of the surface (1) a decomposition of the corresponding vector space \(B_{k_3}\) into the direct sum \(B_{k_3}=B_{k_2}+\bar H_r\) is specified, where \(B_{k_2}\) denotes the osculating space of second order, and \(\bar H_r\) denotes some vector subspace complementary to it of dimension

\[ r=\binom{m+2}{3}-1, \]

which is called a partially framing space with respect to \(B_{k_3}\). Let \(\bar n_i\) \((i,j=1,\ldots,r)\) be some fixed basis of the partially framing \(\bar H_r\). Denoting by \(\bar l_p\) \((p=1,\ldots,k_3-r)\) some basis of the osculating space of second order \(B_{k_2}\), we can represent each vector \(\bar{x}_{abc}\) in the form

\[ \bar{x}_{abc}=h^i_{abc}\bar n_i+T^p_{abc}\bar l_p. \tag{3} \]

The coefficients \(h^i_{abc}\) occurring in the expansion (3) are the components of a connecting affinor of three centro-affine spaces \(E_m\), \(E_r\), and \(E_1\), respectively isomorphic to the tangent vector space \(B_m^{(1)}\), the partially framing \(\bar H_r\), and the radial \(B_1\). The connecting affinor \(h^i_{abc}\) is a contravariant vector both in \(E_r\) and in \(E_1\), and a covariant tensor in \(E_m\). Under a change of the partially framing space \(\bar H_r\), the object \(h^i_{abc}\) does not change and, consequently, is determined invariantly by the surface itself.

Suppose that the connecting affinor \(h^i_{abc}\) has a nonzero relative invariant \(I\). The indicated invariant can be constructed, for example, as follows. Let \(\Delta\) be a determinant of order \(r+1\), composed in some ordered way of the essential components of the affinor \(h^i_{abc}\) and of some tensor \(X_{abc}\). Expanding the determinant \(\Delta\) by the elements of the row that contains only the components \(X_{abc}\), we obtain the invariant equality \(\mathfrak A^{abc}X_{abc}=\Delta\), where \(\mathfrak A^{abc}\) is a connecting tensor density having weights \(3(r+1)/m\), \(-1\), and \(-r\) with respect to the spaces \(E_m\), \(E_r\), and \(E_1\), respectively. Consider the algebraic comitant \(I\) of the tensor density \(\mathfrak A^{abc}\), equal to the discriminant of this density, i.e. \(I=\operatorname{Dis}(\mathfrak A^{abc})\). Since the discriminant \(I\) is a homogeneous function of the components of the connecting tensor density \(\mathfrak A^{abc}\) of degree \(\lambda=mw/3\), where \(w=3\cdot 2^{m-1}\), the weights of the connecting scalar density \(I\) of the spaces \(E_m\), \(E_r\), and \(E_1\) are respectively \(wr\), \(-\lambda\), and \(-\lambda r\). The existence of a nontrivial invariant \(I\) makes it possible (see \((^2)\)) to introduce into consideration the affinor

\[ h^{abc}_{\ \ i}=\frac{1}{I}\frac{\partial I}{\partial h^i_{abc}}, \tag{4} \]

satisfying the relations

\[ \text{a) }\ h^{abc}_{\ \ j}h^i_{abc}=\lambda\delta^i_j,\qquad \text{b) }\ h^{abc}_{\ \ i}h^i_{dbc}=\frac{wr}{3}\delta^a_d. \tag{5} \]

For the purpose of normalizing the pseudovector \(\xi_\alpha\) determined by the system (2), we introduce the object

\[ \mathfrak D=\operatorname{Det}(\mathfrak X^\alpha,\mathfrak X_a^\alpha,\mathfrak X_{ab}^\alpha,n_i^\alpha,v^\alpha)=\mathfrak d_\alpha v^\alpha, \tag{6} \]

where \(v^\alpha\) denotes the components of an arbitrary vector linearly independent of the vectors \(\mathfrak X^\alpha,\mathfrak X_a^\alpha,\mathfrak X_{ab}^\alpha,n_i^\alpha\), and \(\mathfrak d_\alpha\) denotes the algebraic complements of the corresponding components of the vector \(v^\alpha\) in the determinant \(\mathfrak D\). Under a change of the partially adjoining space \(H_r\), the object \(\mathfrak D\) and, consequently, the vector density \(\mathfrak d_\alpha\) do not change. The components of the vector density \(\mathfrak d_\alpha\) are scalar densities of weight \(-(n-r)\) in \(E_1\), of weight \(m+2\) in \(E_m\), and of weight \(+1\) in \(E_r\).

Let us now consider the \(W\)-vector density of weight \(-1\), determined up to sign,

\[ \mathfrak t_\alpha=|I|^{1/\lambda}\mathfrak d_\alpha, \tag{7} \]

whose components are scalar \(W\)-densities of weight \(-n\) in \(E_1\) and of weight \(3r/m+m+2\) in \(E_m\). It is obvious that the density thus constructed is a solution of the system (2). The fourth-order differential form formed with the aid of the density (7),

\[ \mathfrak t_\alpha \mathfrak X_{abce}^{\alpha}\,d\eta^a d\eta^b d\eta^c d\eta^e = A_{abce}\,d\eta^a d\eta^b d\eta^c d\eta^e \tag{8} \]

is not identically equal to zero on the surface (1); otherwise the surface under consideration lies in a hyperplane of the space \(P_n\). At the basis of the form (8) lies the tensor density \(A_{abce}\), whose discriminant \(\mathfrak A=\operatorname{Dis}(A_{abce})\) is, in the general case, different from zero. (An example of a surface with nonzero discriminant \(\mathfrak A\) is given at the end of the article.)

Using the results of works (3), we can now construct on the surface (1), from the components of the associated tensor density \(A_{abce}\), the tensor densities \(\mathfrak A^{abcd}\) and \(\mathfrak A_{abce}\) of weights \(+4/m\) and \(-4/m\), respectively, connected by the relation \(\mathfrak A^{abcd}\mathfrak A_{abce}=\delta_e^d\), and then define in the tangent stratified space \(E_m(X_m)\)* attached to the surface under study the object of affine connection \(\Gamma_{bc}^{a}\). Associating with each point of the base space \(X_m\) two local spaces \(E_m\) and \(E_1\), we obtain a doubled stratified space \(E_m\times E_1(X_m)\) \((^2)\), the affine connection in which can be uniquely determined by requiring that the covariant derivative of the discriminant \(\mathfrak A\) be equal to zero:

\[ D_b\mathfrak A=\partial_b\mathfrak A+3^{m-1}\{m(n+1)\gamma_b-[(m+1)^2+3(r+1)]\Gamma_{bc}^{c}\}\mathfrak A=0. \tag{9} \]

Indeed, since the quantities \(\Gamma_{bc}^{a}\) are determined, it follows from relation (9) that the affine-connection coefficients \(\gamma_b\) in the radial stratified space \(E_1(X_m)\) are found uniquely:

\[ \gamma_b=\frac{(m+1)^2+3(r+1)}{m(n+1)}\,\Gamma_{bc}^{c} -\frac{3^{1-m}}{m(n+1)}\,\partial_b\ln\mathfrak A. \tag{10} \]

The basis of the invariantly adjoining space \(B_{n-m}\) is the system of the following \(n-m\) vectors:

\[ \bar l_{ab}=D_{(a}D_{b)}\bar{\mathfrak X},\qquad \bar N_i=\frac1m\,h_i^{abc}D_aD_bD_c\bar{\mathfrak X},\qquad \bar N=\frac1m\,\mathfrak A^{abce}D_aD_bD_cD_e\bar{\mathfrak X}. \tag{11} \]

In conclusion we give an example of a surface of special form to which the scheme of invariant adjoining set forth above is applicable. Let a three-dimensional surface in nine-dimensional projective space, with some—

* In the literature \((^1,^2)\) the term “compound manifold” is used instead of “stratified space.”

in the second normalization the vector \(\bar{x}\) has the equation

\[ \begin{gathered} \bar{x}=e_0+\eta^1\bar{e}_1+\eta^2\bar{e}_2+\eta^3\bar{e}_3+\eta^1\eta^2\bar{e}_4+\eta^1\eta^3\bar{e}_5+\eta^2\eta^3\bar{e}_6+ \\ +\bigl((\eta^1)^2-(\eta^3)^2\bigr)\bar{e}_7+\bigl((\eta^2)^2-(\eta^3)^2\bigr)\bar{e}_8 +a_{abc}\eta^a\eta^b\eta^c e_9+\bar{f}_4(\eta^a) \end{gathered} \tag{12} \]

\[ (a,b,c=1,2,3), \]

where \(\bar{e}_0,\bar{e}_1,\ldots,\bar{e}_9\) are vectors constituting a basis of the attached space \(B_{10}\); \(a_{abc}\eta^a\eta^b\eta^c\) is an arbitrary harmonic form (a form that is a harmonic function) of the variables \(\eta^a\); \(\bar{f}_4(\eta^a)\) is a vector-function of the scalar variables \(\eta^a\), representing a sum of harmonic forms of order not lower than the fourth. It is obvious that in this normalization the vector \(\bar{x}\) satisfies the differential equation \(\bar{x}_{11}+\bar{x}_{22}+\bar{x}_{33}=0\).

Let us consider this surface in a neighborhood of the point \(P_0\) with curvilinear coordinates \(\eta^\alpha=0\). At the point \(P_0\) the nine vectors \(\bar{x}, \bar{x}_a, \bar{x}_{11}, \bar{x}_{12}, \bar{x}_{13}, \bar{x}_{22}, \bar{x}_{23}\) are linearly independent; by continuity they will also be linearly independent in some neighborhood of this point. The coefficients of the form (8) at the point under consideration have the form \(A_{abc}=6r_\alpha e_9^\alpha a_{abc}\) \((\alpha=1,\ldots,10)\), where \(\alpha e_9^\alpha\ne0\); therefore the discriminant \(\mathfrak A\) vanishes together with the discriminant of the tensor \(a_{abc}\). The requirement that the form \(a_{abc}\eta^a\eta^b\eta^c\) be harmonic, however, does not entail that the discriminant of the tensor \(a_{abc}\) be equal to zero; this can be verified by assigning the components of this tensor the values
\[ a_{221}=a_{113}=1,\quad a_{331}=a_{223}=-1,\quad a_{111}=a_{112}=a_{222}=a_{332}=a_{333}=a_{123}=0. \]
Then, defining partially the osculating space \(H_5\) at the point \(P_0\) by the vectors \(\bar{x}_{11}, \bar{x}_{12}, \bar{x}_{13}, \bar{x}_{22}, \bar{x}_{23}\), we obtain the following values of the essential components of the tensor density \(\mathfrak A^{ab}\): \(\mathfrak A^{11}=\mathfrak A^{22}=\mathfrak A^{33}=1\), \(\mathfrak A^{12}=\mathfrak A^{13}=\mathfrak A^{23}=0\), and, consequently, \(I=\operatorname{Dis}(\mathfrak A^{ab})\ne0\).

Saratov State University
named after N. G. Chernyshevsky

Received
13 XI 1964

CITED LITERATURE

\(^{1}\) A. E. Liber, DAN, 40, 137 (1953).
\(^{2}\) A. E. Liber, Uch. zap. Saratov State Univ. named after N. G. Chernyshevsky, 70, 73 (1961).
\(^{3}\) Yu. I. Ermakov, DAN, 118, No. 6, 1070 (1958); 128, No. 3, 460 (1959).