UDC 513

MATHEMATICS

S. M. CHAPESHNIKOV

GEOMETRY OF AN \(m\)-SURFACE IN AFFINE SPACE

(Presented by Academician I. G. Petrovskii on 22 V 1965)

The geometry of an \(m\)-surface in affine space for \(m=n-1\) and \(m=n-2\) was studied by V. V. Wagner \((^{1,9})\). A. E. Liber \((^{2})\) considered the geometry of an \(m\)-surface under the condition that \(m\) and \(n\) satisfy the inequality \(2n \le m(m+3)\). Later, A. E. Liber \((^{3-5})\) and P. I. Shveikin \((^{6,7})\) studied the geometry of an \(m\)-surface for arbitrary \(m\), but under the condition that the tangent spaces of higher order have the greatest possible dimension. We impose no restrictions on \(m\) and \(n\) and do not require the greatest possible dimension of the tangent spaces of higher order, but assume only that all points of the surface are ordinary.

Let

\[ x^\alpha = l^\alpha(\eta^a) \qquad (\alpha,\beta,\gamma,\ldots=1,\ldots,n;\ a,b,c,\ldots=1,\ldots,m) \tag{1} \]

be the equations of an \(m\)-surface. We shall call an \(m\)-surface \(p\)-planar if it lies in some \(p\)-dimensional plane. Otherwise the surface will be called nonplanar. Since the study of a planar surface can be reduced to the study of a nonplanar surface in some affine space of smaller dimension, it is natural to restrict ourselves to considering the case of a nonplanar surface.

Let the tangent spaces of the \(m\)-surface up to order \(\nu\) inclusive \((\nu \ge 1)\) have the greatest possible dimension, but less than \(n\), and let, beginning with order \((\nu+1)\), they have dimension lower than the greatest possible. Then there exists an \(s\) such that the dimension of the tangent space spanned by the vectors

\[ \partial_{a_1}\ldots \partial_{a_i} l^\alpha \qquad (i=1,\ldots,\nu+s), \tag{2} \]

is equal to \(n\), while the dimension of the tangent space of order lower than \((\nu+s)\) is less than \(n\). A surface of this type will be called a surface of differential class \((\nu+s)\).

If \(N_\alpha^{p_s}\) \((p_s=1,\ldots,m_s)\) is a solution of the system of equations

\[ N_\alpha \partial_{a_1}\ldots \partial_{a_i} l^\alpha = 0 \qquad (i=1,\ldots,\nu+s-1), \tag{3} \]

then the associated affinor

\[ H_{a_1\ldots a_{\nu+s}}^{p_s} = N_\alpha^{p_s}\partial_{a_1}\ldots \partial_{a_{\nu+s}} l^\alpha \tag{4} \]

is different from zero. Suppose that there exists a relative invariant \(\overset{(s)}{W}\) of the associated affinor (4), not equal to zero. As is known \((^{3})\), the weights \(k_1\) and \(k_2\) of the relative invariant \(\overset{(s)}{W}\) are in the ratio \(m_s(\nu+s):(-m)\). Raising \(\overset{(s)}{W}\) to a suitable power, we obtain a relative invariant \(\overset{(s)}{I}\) with weights \(k_1=m_s(\nu+s)\) and \(k_2=-m\).

The associated affinors (4) and

\[ H_{p_s}^{a_1\ldots a_{\nu+s}} = \frac{1}{I^{(s)}}\, \frac{\partial I}{\partial H_{a_1\ldots a_{\nu+s}}^{p_s}} \tag{5} \]

satisfy the conditions

\[ H_{a_1\ldots a_{\nu+s-1}a}^{p_s} H_{p_s}^{a_1\ldots a_{\nu+s-1}b} = m_s\delta_a^b, \tag{6} \]

\[ H_{a_1\ldots a_{\nu+s}}^{p_s} H_{q_s}^{a_1\ldots a_{\nu+s}} = m\delta_{q_s}^{p_s}, \tag{7} \]

and therefore the vectors

\[ n_{p_s}^{\alpha} = \frac{1}{m} H_{p_s}^{a_1\ldots a_{\nu+s}} \partial_{a_1}\ldots \partial_{a_{\nu+s}}l^{\alpha} \tag{8} \]

will be basis vectors of a certain vector space \(B_{m_s}\).

If \(N_{\alpha}^{p_{s-1}}\) \((p_{s-1}=1,\ldots,m_{s-1})\) is a solution of the system of equations

\[ N_{\alpha}n_{p_s}^{\alpha}=0;\qquad N_{\alpha}\partial_{a_1}\ldots \partial_{a_i}l^{\alpha}=0 \quad (i=1,\ldots,\nu+s-2), \tag{9} \]

then the associated affinor

\[ H_{a_1\ldots a_{\nu+s-1}}^{p_{s-1}} = N_{\alpha}^{p_{s-1}} \partial_{a_1}\ldots \partial_{a_{\nu+s-1}}l^{\alpha} \tag{10} \]

will be different from zero. If there exists a relative invariant \(W^{(s-1)}\) of the affinor (10) that is not equal to zero, then, raising it to a suitable power, we obtain a relative invariant \(I\) with weights \(k_1=m_{s-1}(\nu+s-1)\) and \(k_2=-m\). The vectors

\[ n_{p_{s-1}}^{\alpha} = \frac{1}{m} H_{p_{s-1}}^{a_1\ldots a_{\nu+s-1}} \partial_{a_1}\ldots \partial_{a_{\nu+s-1}}l^{\alpha} \tag{11} \]

will be basis vectors of a certain vector space \(B_{m_{s-1}}\). Similarly, one finds the basis vectors \(n_{p_i}^{\alpha}\) \((i=1,\ldots,s-2)\) of the vector spaces \(B_{m_i}\).

It is not difficult to see that

\[ m_1+m_2+\ldots+m_s = n+1-\binom{m+\nu}{m} \tag{12} \]

and the vectors

\[ \partial_{a_1}\ldots \partial_{a_i}l^{\alpha}; \qquad n_{p_k}^{\alpha} \quad (i=1,\ldots,\nu;\quad k=1,\ldots,s) \tag{13} \]

are linearly independent. Therefore the vector spaces \(B_m^k\) \((k=1,\ldots,s)\) partially equip the \(m\)-surface.

If the relative invariants \(W\) of the \(m\)-surface are not equal to zero, then the method proposed by the author [8] for constructing relative invariants of associated affinors of type (4) makes it possible to find the basis vectors \(n_{p_i}^{\alpha}\) of the partially equipping spaces \(B_{m_i}\). Thus, for example, for the two-dimensional surface in six-dimensional space

\[ \bar r\{u;\ v;\ u^2;\ v^2;\ u^3;\ v^3\}, \tag{14} \]

which has differential class \((1+2)\), as the relative invariants \(\overset{(2)}{W}\) and \(\overset{(1)}{W}\) one may take the hyperdeterminant of the matrix

\[ H^{p_2q_2s_2r_2} = H_{\underset{(2)(3)}{1[1}}^{\{p_2} \left[ H_{\underset{(4)(5)(6)}{1[1[1}}^{\{q_2\}} H_{\underset{(2)(3)}{2\ 2]2]}^{\{s_2\}} \right] H_{\underset{(4)(5)(6)}{2\ 2]\ 2]}^{r_2\}}, \tag{15} \]

where symmetrization is carried out over the combined index \(\{pq\}\), and alternation over each \(i\)-th index of adjacent elements separately \((i=2,3,4,5,6)\), and

\[ \overset{(1)}{W}=\operatorname{Det}\left(H^{p_1q_1}\right), \tag{16} \]

where

\[ H^{p_1 q_1}=H^{(p_1}_{1[1}H^{q_1)}_{|2|2]}. \tag{17} \]

Simple computations show that the relative invariants \(\overset{(1)}{W}\) and \(\overset{(2)}{W}\) are not equal to zero, and therefore may be used for constructing partially normalizing spaces.

Since the vectors (13) form a basis, the vector \(\partial_{a_1}\cdots \partial_{a_{\nu+1}}l^\alpha\) admits a unique expansion in these vectors. Let \(T^{b_1\ldots b_\nu}_{a_1\ldots a_{\nu+1}}\) be the coefficients of the expansion at the vectors \(\partial_{b_1}\cdots \partial_{b_\nu}l^\alpha\). Then the object

\[ T^c_{ba}=T^{a_1\ldots a_{\nu-1}c}_{a_1\ldots a_{\nu-1}ba} \tag{18} \]

is transformed according to the law

\[ T^{c'}_{b'a'}=A^{c'}_c A^b_{b'} A^a_{a'}T^c_{ba} +m^{(\nu-1)}A^{c'}_e\partial_{a'}A^e_{b'} \]
\[ +2(\nu-1)m^{(\nu-2)}A^e_e\delta^{c'}_{(a'}\partial_{b')}A^e_{e'} +P(\nu-2)\delta^{c'}_{x'}A^{e'}_e\partial_{b'}A^e_{e'}, \tag{19} \]

where

\[ P(\nu)=\sum_{k=0}^{\nu-1}(\nu-k)m^{\nu-k-1}. \tag{20} \]

It follows from this that the object

\[ G^c_{ba}=\frac{1}{m^{(\nu-1)}}T^c_{ba} -\frac{2(\nu-1)}{mP(\nu)}T_{(a}\delta^c_{b)} -\frac{P(\nu-2)}{m^{(\nu-1)}P(\nu)}T_b\delta^c_a, \tag{21} \]

\[ T_b=T^e_{eb} \tag{22} \]

is the object of the affine connection in \(X_m\) corresponding to the \(m\)-surface.

Using the connection found, we obtain the vectors

\[ l^\alpha_{a_1\ldots a_i}=\nabla_{a_1}\cdots\nabla_{a_i}l^\alpha \qquad (i=1,\ldots,\nu). \tag{23} \]

The vectors

\[ l^\alpha_{a_1\ldots a_i};\qquad n^\alpha_{p_k} \qquad (i=1,\ldots,\nu;\ k=1,\ldots,s). \tag{24} \]

determine an invariant normalization of the \(m\)-surface. The covariant vectors \(L^{a_1\ldots a_i}_{\alpha};\ N^{p_k}_{\alpha}\ (i=1,\ldots,\nu;\ k=1,\ldots,s)\), where the covectors \(L^{a_1\ldots a_i}_{\alpha}\) are determined by the conditions

\[ L^{a_1\ldots a_i}_{\alpha}l^\alpha=0;\quad L^{a_1\ldots a_i}_{\alpha}n^\alpha_{p_k}=0;\quad L^{a_1\ldots a_i}_{\alpha}l^\alpha_{b_1\ldots b_i} =\delta^{a_1\ldots a_i}_{b_1\ldots b_i};\quad L^{a_1\ldots a_i}_{\alpha}l^\alpha_{b_1\ldots b_j}=0 \tag{25} \]

and form the basis reciprocal to the basis (24).

It is not difficult to show that the derivative formulas of the surface under consideration have the form

\[ \nabla_a l^\alpha_{a_1\ldots a_i} =l^\alpha_{aa_1\ldots a_i}, \qquad (i=1,\ldots,\nu-1), \]

\[ \nabla_a l^\alpha_{a_1\ldots a_\nu} =\sum_{i=1}^{\nu}h^{b_1\ldots b_i}_{aa_1\ldots a_\nu}l^\alpha_{b_1\ldots b_i} +H^{p_1}_{aa_1\ldots a_\nu}n^\alpha_{p_1}, \tag{26} \]

\[ \nabla_a n^\alpha_{p_j} =\sum_{i=1}^{\nu}K^{b_1\ldots b_i}_{ap_j}l^\alpha_{b_1\ldots b_i} +\sum_{k=1}^{j+1}M^{q_k}_{ap_j}n^\alpha_{q_k} \]

(where the summation over \(k\) for \(j=s\) is carried out up to \(k=s\)).

Thus we obtain:

Theorem. An \(m\)-dimensional surface of differential class \((v+s)\) is determined in an affine space, up to an automorphism, by specifying the objects of the system of differential equations (26) that satisfy the integrability conditions.

Kirovograd State
Pedagogical Institute named after A. S. Pushkin

Received
20 V 1965

REFERENCES

\(^{1}\) V. V. Wagner, Tr. seminara po vektorn. i tenzorn. analizu, vol. 7, 65 (1949).
\(^{2}\) A. E. Liber, DAN, 85, No. 1, 37 (1952).
\(^{3}\) A. E. Liber, Nauchn. ezhegodn. Saratovsk. univ. for 1954, 669 (1955).
\(^{4}\) A. E. Liber, Tr. seminara po vektorn. i tenzorn. analizu, vol. 10, 193 (1956).
\(^{5}\) A. E. Liber, Tr. III Vsesoyuzn. matem. s”ezda, 1, 157, 1956.
\(^{6}\) P. I. Shveikin, UMN, 10, 3 (65), 181 (1955).
\(^{7}\) P. I. Shveikin, Tr. III Vsesoyuzn. matem. s”ezda, 1, 175, 1956.
\(^{8}\) S. M. Chapchenko, Reports and communications (abstracts) of the reporting scientific conference of the departments of the Pedagogical Institute for 1964, Kirovograd, 1965, p. 39.
\(^{9}\) V. Wagner, Ann. of Math., 49 (1), 141 (1948).