UDC 513

MATHEMATICS

E. T. IVLEV

ON A MULTIDIMENSIONAL SURFACE IN PROJECTIVE SPACE

(Presented by Academician A. D. Aleksandrov on 14 IX 1967)

One of the fundamental problems of the differential geometry of a multidimensional surface is the problem of invariant equipment. A sufficiently complete survey of works on this problem was given by G. F. Laptev in \((^1)\) (see also \((^{2-10})\)). As G. F. Laptev notes, following the problem of the invariant equipment of a multidimensional surface there arises the problem of constructing a canonical frame. In the present paper a canonical frame is constructed for an \(m\)-parametric manifold \(S_m(l_\alpha)\), consisting of \(m\)-surfaces \(S_m\), through each point \(S\) of which pass \(m\) lines \(\Gamma_\alpha\) belonging to it, with tangents \(l_\alpha\) in general position. It is assumed here that \(2<m<n-2\). The canonical frame of the manifold \(S_m(l_\alpha)\) is a semicanonical frame in the sense of \((^{14,16})\) of an arbitrary \(m\)-surface \(S_m\) in \(P_n\).

We shall write the derivation formulas of a certain frame \(\{A_i\}\) of the manifold \(S_\alpha(l_m)\) in \(P_n\) in the form \(dA_i=\omega_i^k A_k\) \((i,k=0,1,\ldots,n)\), where \(\omega_i^k\) are Pfaffian differential forms satisfying the structure equations \(D\omega_i^k=[\omega_i^j\omega_j^k]\) \((j=0,1,\ldots,n)\) and the relation \(\omega_i^i=0\). Put \(k_a=m_a-m_{a-1}=C_{m+a}^{a+1}\). Introduce the system of indices \(\alpha,\beta,\gamma,\alpha_0,\beta_0,\gamma_0,\alpha^q,\beta^q,\gamma^q\) \((q\ge 1)=1,2,\ldots,m;\ \alpha_a,\beta_a,\gamma_a,\alpha_a^q,\beta_a^q,\gamma_a^q=m_{a-1}+1,\ldots,m_a\) \((a,b,c=1,\ldots,p;\ m=m_0);\ \alpha_p,\beta_p,\gamma_p=m_{p-1}+1,\ldots,n;\ \alpha_{a\ldots b}=m_{a-1}+1,\ldots,m_a,\ldots,m_{b-1}+1,\ldots,m_b\) \((a\le b);\ \alpha_{a\ldots a}=\alpha_a;\ \alpha,\beta,\gamma=m+1,\ldots,n\). Let \(A_0=S,\ l_\alpha=(A_0A_\alpha)\). Then the systems of differential equations \(\omega_0^{\hat\alpha}=0\) and \(\omega_\alpha^\beta=\Lambda_{\alpha\gamma}^{\beta}\omega^\gamma\) \((\alpha\ne\beta)\) determine the \(m\)-surface \(S_m\) and the system of lines \(\Gamma_\alpha\) on it, respectively. Successive prolongations of the system \(\omega_0^{\hat\alpha}=0\) lead to a sequence of fundamental objects \(\{\Lambda_{\alpha^1\alpha^2\ldots\alpha^r}^{\hat\alpha}\}\) \((^{17})\), whose components are symmetric with respect to the lower indices and satisfy the system of differential equations \((^{1,3})\) in \((^4)\). The fixing of the frame, carried out analytically, gives

\[ \Lambda_{\alpha^1\alpha^2\ldots\alpha^{b+1}}^{\,b+1\ldots p}=0,\qquad \bar b_{\beta_a}^{\alpha_a}=\delta_{\beta_a}^{\alpha_a},\qquad \Lambda_{0\beta}^{\hat\alpha}=0,\qquad \Lambda_{0\beta}^{\alpha}=\delta_\beta^\alpha, \]

\[ \Lambda_{12}^{\tau}=\Lambda_{m-1,m}^{1}=\Lambda_{m-1,m}^{2}=1,\qquad \Lambda_{0\alpha}^{0}=0,\qquad \Lambda_{(\alpha\gamma)}^{\beta}=0,\qquad \Lambda_{i\alpha}^{i}=0, \]

\[ \Lambda_{\alpha_1\ldots p-2\beta}^{\alpha}=0,\qquad \Lambda_{\alpha-1\beta}^{\alpha+1-p}=0,\qquad \Lambda_{\alpha_1\ldots p-2\beta}^{0}=0, \]

\[ \Lambda_{\alpha_{p-2}[\alpha}^{\alpha_{p-1}}\Lambda_{1\beta_{p-1}|\beta]}^{0}=0,\qquad \Lambda_{\alpha_{p-2}[\alpha}^{\alpha_{p-1}}\Lambda_{1\beta_{p-1}|\beta]}^{\gamma}=0,\qquad \Lambda_{\beta_{f+1}\ldots g^{\alpha}}^{\alpha_f}=0, \]

\[ \Lambda_{\alpha^1\ldots\alpha^{f+1}}^{\beta_f}\Lambda_{\beta_f\alpha}^{\alpha_f} = \frac{1}{f!}\Lambda_{\sigma(\alpha^1\ldots\alpha^f}^{\alpha_f} \Lambda_{\alpha^{f+1})\alpha}^{\sigma},\qquad \Lambda_{\alpha_{p-2}[\alpha}^{\alpha_{p-1}}\Lambda_{1\beta_{p-1}|\beta]}^{\alpha_1\ldots i}=0, \]

\[ \Lambda_{\alpha_{p-1}^{1}}^{\beta_f}=0\quad (n\le m_{p-1}+k_{p-1});\qquad \Lambda_{\alpha_{p-1}^{1}}^{\beta_f}=\Lambda_{\alpha_{p-1}^{\nu+1}}^{\beta_f}=0 \]

\[ (q(p-1,s)<n\le q(p-1,s+1)). \]

Here and in what follows the following notation and numbering of indices are adopted:

1) \(\bar b_{\beta_a}^{\alpha_a}=\Lambda_{\alpha^1\ldots\alpha^{a+1}}^{\alpha_a}\), where the system of quantities \(\Lambda_{\alpha^1\ldots\alpha^{a+1}}^{\alpha_a}\), symmetric in any pair of lower indices, for each fixed upper index consists of \(k_a\) independent components \(\Lambda_{\alpha^1\ldots\alpha^{a+1}}^{\alpha_a}\), in which to each value of the collection of indices there is assigned a definite value of the index \(\beta_a\), i.e. \(\beta_a\leftrightarrow(\alpha^1\ldots\alpha^{a+1})\), and the indices \(\alpha^1,\ldots,\alpha^{a+1}\) are arranged successively so that each subsequent index is not less than the preceding one;

2) \(f=1,2,\ldots,p-1;\ g=f+1,\ldots,p;\)

3) \(\alpha_{p-1}^0=m_{p-2}+1,\ldots,m_{p-2}+n-m_{p-1};\ \beta_p^0\leftrightarrow(\alpha_{p-1}^0 1),\ \alpha_{p-1}^0\leftrightarrow(\alpha^1\ldots\alpha^p),\ \beta_p^0=m_{p-1}+1,\ldots,n\) (in the case \(m_{p-1}<n\le m_{p-1}+k_{p-1}\));

4) \(\alpha_{p-1}^\nu=q(p-2,\nu)+1,\ldots,q(p-2,\nu+1),\ \alpha_{p-1}^s=q(p-2,s)+1,\ldots\)
\[ \ldots,m_{p-2}+n-q(p-1,s),\ \beta_p^\nu\leftrightarrow(\alpha_{p-1}^\nu,\nu+1),\ \alpha_{p-1}^\nu\leftrightarrow(\alpha^1\ldots\alpha^p),\ \beta_p^\nu=m_{p-1}+k_{p-1}+1,\ldots,n, \]
\[ \beta_p^*\leftrightarrow(\alpha_{p-1}^*1),\ \alpha_{p-1}^*\leftrightarrow(\alpha^1\ldots\alpha^p),\ \alpha_{p-1}^*=m_{p-2}+1,\ldots,m_{p-1}; \]
\[ \beta_p^*=m_{p-1}+1,\ldots,m_{p-1}+k_{p-1}, \]
where
\[ q(r,t)=m_r+k_r+\ldots+k_r^{t-1},\quad k_r^s=C_{m+r-s}^{r+1},\quad k_r^0=k_r \]
\((s=1,\ldots,m-1,\ s\ \text{fixed};\ t=1,2,\ldots;\ r=p-2,\ p-1);\)

5) \(r=3,\ldots,m\).

The fixation of the canonical frame of the manifold \(S_m(l_a)\), carried out by formulas (1), has the following geometric interpretation. The linear subspace \(L_m=(A_0A_1\ldots A_m)\) is the tangent \(m\)-plane of the surface \(S_m\) at the point \(A_0\), while \(dL_{m_a}=(L_{m_{a-1}}A_{m_{a-1}+1}\ldots A_{m_a})\) is the osculating \(m_a\)-plane of order \(a+1\) of the surface \(S_m\) at the point \(A_0\) in the sense of (11). Each \((m_a+1)\)-plane
\[ L_{m_{a-1}+1}^{\alpha_a}=(L_{m_{a-1}}A_{\alpha_a}) \]
from \(L_{m_a}\) contains, along the corresponding coordinate line \(\Gamma_a\) of the surface \(S_m\), the first differential neighborhood of the corresponding \((m_{a-1}+1)\)-plane
\[ L_{m_{a-2}+1}^{\alpha_{a-1}}=(L_{m_{a-3}}A_{\alpha_{a-1}}) \]
from \(L_{m_{a-2}}\) (for \(a=1\) the role of \(L_{m_{-1}+1}^{\alpha_1}\) is played by the lines \(A_0A_{\alpha_1}\)). The linear subspace
\[ L_{m-1}^*=(A_1A_2\ldots A_m) \]
is a normal of the first kind in the sense of (12), and the \(k_1\)-plane
\[ L_{k_1}^*=(A_0A_{m+1}\ldots A_{m_1}) \]
is chosen so that the space with connection, in the sense of (12, 13), induced by the \(k_1\)-plane \(L_{k_1}^*\) along \(S_m\), whose geometric element (17) is the point \(A_0\) and the \(m\)-plane \(L_m\), is an affine homogeneous space. At the same time the manifold
\[ X_m^{m-1} \]
is an \(m\)-parameter manifold of normals of the second kind—harmonically conjugate to the surface \(S_m\) in the sense of (12). Each \((m_a-1)\)-plane
\[ L_{m_a-1}^*=(L_{m_{a-1}-1}^*A_{m_{a-1}+1}\ldots A_{m_a})\quad(a=1,\ldots,p-1) \]
is an osculating linear subspace of order \(a\) of the manifold \(X_m^{m-1}\) in the element \(L_{m-1}^*\), i.e. it contains all differential neighborhoods of the \((m-1)\)-plane \(L_{m-1}^*\) up to order \(a\) inclusive. Consequently,
\[ l_{m_b}^{\alpha_{b+1}}=(L_{m_b-1}^*A_{\alpha_{b+1}})=L_{m_{b+1}}^{b+1}\cap L_{m_b-1}^* \quad (b=1,\ldots,p-2;\ L_{m_{b+1}}^{\alpha_{b+1}}=(L_{m_b-1}A_{\alpha_{b+1}})). \]
In the case \(m_{p-1}<n\le m_{p-1}+k_{p-1}\), each linear subspace
\[ L_{m_{p-1}}^{*\beta_p^0}=(L_{m-1}^*A_{\beta_p^0}), \]
belonging to \(L_{m_{p-1}+1}^{\beta_p^0}\) and not passing through the point \(A_0\), contains the first differential neighborhood of the \(m_{p-2}\)-plane \(L_{m_{p-2}}^{p-1}\) under the displacement \(\omega^2=\ldots=\omega^m=0\). Analogously characterized are

geometrically the \(m_{p-1}\)-planes \(L_{m_{p-1}}^{*\beta_p^\nu}=(L_{m_{p-1}-1}^{*}A_{\beta_p^\nu})\) and \(L_{m_{p-1}}^{*\beta_p^{*}}=(L_{m_{p-1}-1}^{*}A_{\beta_p^{*}})\) in the case \(q(p-1,s)<n\leq q(p-1,s+1)\). Thus the hyperplane \(L_{n-1}=(A_1A_2\ldots A_n)\) is geometrically determined by the fact that it passes through all \(L_{m_b}^{*\alpha_{b+1}}\) and \(L_{m_{p-1}}^{*\alpha_p}\).

Each linear subspace \(L_{k_1+\cdots+k_a}^{*\alpha_{a+1}}=(L_{k_1+\cdots+k_a-1}^{*}A_{\alpha_{a+1}})\) is such that
\[ L_{k_1+\cdots+k_a}^{*\alpha_{a+1}} = L_{m_{a+1}}^{\alpha_{a+1}}\cap L_{k_1+\cdots+k_{a+1}}^{*}, \]
where \(L_{k_1+\cdots+k_{a+1}}^{*}\) is the tangent linear subspace of order \(a\) of the variety \(X_m^{k_1-1}\)—the \(m\)-parameter variety of \((k_1-1)\)-planes \(L_{k_1-1}^{*}=L_{k_1}\cap L_{m_1-1}^{*}\). In the case \(m_{p-1}<n\leq m_{p-1}+k_{p-1}\), every \((m_{p-1}-m)\)-plane
\[ L_{k_1+\cdots+k_{p-1}}^{*\beta_p^0} = (L_{k_1+\cdots+k_{p-1}-1}^{*}A_{\beta_p^0}) \]
contains the first differential neighborhood of the linear subspace \(L_{k_1+\cdots+k_{p-2}}^{*\alpha_{p-1}^0}\) under a displacement \(\omega^2=\omega^3=\cdots=\omega^m=0\). The \((m_{p-1}-m)\)-planes \(L_{k_1+\cdots+k_{p-1}}^{*\beta_p}\) and \(L_{k_1+\cdots+k_{p-1}}^{*\beta_p^\nu}\) are characterized analogously in the case \(q(p-1,s)<n\leq q(p-1,s+1)\). Thus the normal of the first kind in the sense of \((12)\)—the \((n-m)\)-plane
\[ L_{n-m}=(A_0A_{m+1}\ldots A_n) \]
—is geometrically characterized by the fact that it passes through the point \(A_0\) and all linear subspaces \(L_{k_1+\cdots+k_a}^{*\alpha_{a+1}}\) \((a=1,\ldots,p-2)\) and \(L_{k_1+\cdots+k_{p-1}}^{*\alpha_p}\). Since the normals of the first and second kind are geometrically determined, the surface \(S_m\) is normalized in the sense of A. P. Norden \((12)\). Each linear subspace
\[ L_{k_{b-1}+k_b-1}^{*}=(L_{k_{b-1}-1}^{*}A_{m_b+1}\ldots A_{m_{b+1}}) \]
is chosen so that it induces, along \(X_m^{k_{b-1}-1}\)—the \(m\)-parameter variety of linear subspaces \(L_{k_{b-1}-1}^{*}=(A_{m_{b-2}+1}\ldots A_{m_{b-1}})\) \((b=2,\ldots,p-1)\)—a space with connection that is an affine homogeneous space. Consequently,
\[ L_{k_{b+1}-1}^{*}=(A_{m_b+1}\ldots A_{m_{b+1}})= L_{k_{b+1}+k_{b-1}-1}^{*}\cap L_{k_1+\cdots+k_{b+1}-1}^{*}. \]
Each \((k_{b+1}+\cdots+k_a-1)\)-plane
\[ L_{k_{b+1}+\cdots+k_a-1}^{*} = (L_{k_{b+1}+\cdots+k_{a-1}-1}^{*}A_{m_{a-1}+1}+\cdots+A_{m_a}) \]
\((b=1,\ldots,p-1;\ a=b+1,\ldots,p)\) is a tangent linear subspace of order \(a-b\) of the variety \(X_m^{k_{b+1}-1}\)—the \(m\)-parameter variety of linear subspaces \(L_{k_{b+1}-1}^{*}\). Therefore
\[ L_{k_{b+1}\ldots k_a}^{\alpha_a} = (L_{k_{b+1}+\cdots+k_{a-1}-1}^{*}A_{\alpha_a}) = L_{k_{b+1}+\cdots+k_{a-1}}^{\alpha_a}\cap L_{m_{a-1}+1}^{\alpha_a}. \]

Each \((k_{b+1}+\cdots+k_{p-1})\)-plane
\[ L_{k_{b+1}+\cdots+k_{p-1}}^{*\beta_p^0} = (L_{k_{b+1}+\cdots+k_{p-1}-1}^{*}A_{\beta_p^0}) \]
in the case \(m_{p-1}<n\leq m_{p-1}+k_{p-1}\) contains the first differential neighborhood of the \((k_{b+1}+\cdots+k_{p-2})\)-plane
\[ L_{k_{b+1}+\cdots+k_{p-2}}^{*\alpha_{p-1}^0} = (L_{k_{b+1}+\cdots+k_{p-2}-1}^{*}A_{\alpha_{p-1}^0}) \]
under a displacement \(\omega^2=\cdots=\omega^m=0\). The \((k_{b+1}+\cdots+k_{p-1})\)-planes \(L_{k_{b+1}+\cdots+k_{p-1}}^{*\beta_p}\) and \(L_{k_{b+1}+\cdots+k_{p-1}}^{*\beta_p^\nu}\) are also determined geometrically in the case \(q(p-1,s)<n\leq q(p-1,s+1)\). Each \(k_{p-1}\)-plane
\[ L_{k_{p-1}}^{*\beta_p^0} = (L_{k_{p-1}-1}^{*}A_{\beta_p^0}) \]
in the case \(m_{p-1}<n\leq m_{p-1}+k_{p-1}\) contains the tangent to

line described by the point \(A_{\alpha_{p-1}^{0}}\) under the displacement \(\omega^2=\cdots=\omega^m=0\). Analogously, the \(k_{p-1}\)-planes
\(L_{k_{p-1}}^{*\beta_p^*}=(L_{k_{p-1}}^*A_{\beta_p^*})\) and
\(L_{k_{p-1}}^{*\beta_p^0}=(L_{k_{p-1}}^*A_{\beta_p^0})\) are geometrically defined in the case
\(q(p-1,s)<n\le q(p-1,s+1)\). In the case \(m_{p-1}<n\le k_{p-1}+m_{p-1}\), in the linear subspace there exists such a point \(A_{\beta_p^0}\) for each \(\beta_p^0\) that the line \(A_{\alpha_{p-1}^{0}}A_{\beta_p^0}\) is tangent to the line described by the point \(A_{\alpha_{p-1}^{0}}\) under the displacement \(\omega^2=\cdots=\omega^m=0\). The points \(A_{\beta_p^*}\) and \(A_{\beta_p^0}\) are analytically defined in the case \(q(p-1,s)<n\le q(p-1,s+1)\). Thus, the points \(A_\alpha\) belong simultaneously to the lines \(l_\alpha=(A_0A_\alpha)\) and to the normal of the second kind, while each point \(A_{\alpha_\alpha}\) belongs simultaneously to the \((m_\alpha+1)\)-plane
\(L_{m_{\alpha-1}+1}^{\alpha_\alpha}=(L_{m_{\alpha-1}}A_{\alpha_\alpha})\) and to the \((k_\alpha-1)\)-plane
\(L_{k_{\alpha-1}}^*=(A_{m_{\alpha-1}+1}\ldots A_{m_\alpha})\). Therefore all elements of the canonical frame of the manifold \(S_m(l_\alpha)\) are geometrically characterized. In this frame we single out the following two special classes: 1) the manifold \(S_m(l_\alpha)\), determined by the relations \(\Lambda_{\alpha\beta}^{\beta}=0\) (summation over \(\beta\)), exists with an arbitrariness of \(n+m(m-3)\) functions of \(m\) arguments and is characterized by the fact that the points \(A_\alpha\) on the lines \(A_0A_\alpha\) are harmonic poles \((^{18})\) of the point \(A_0\) with respect to the pseudofoci \(F_\alpha^\beta\) \((\alpha\ne\beta)\) \((^{15})\); 2) the manifold \(S_m(l_\alpha)\), determined by the natural equations \(\Lambda_{\alpha\beta}^{\beta}=0\) \((\alpha\ne\beta,\) with no summation over \(\beta)\), exists with an arbitrariness of \(n-m\) functions of \(m\) arguments and is characterized by the fact that all pseudofoci \(F_\alpha^\beta\) of each line \(A_0A_\alpha\) coincide with the point \(A_\alpha\).

Tomsk State University
named after V. V. Kuibyshev

Received
30 I 1967

CITED LITERATURE

\(^{1}\) G. F. Laptev, Itogi nauki, Geometry, 1963, p. 5.
\(^{2}\) N. M. Ostianu, Materials of the Second Baltic Geom. Conf., Tartu, 1965, p. 136.
\(^{3}\) N. M. Ostianu, RZhMat, 7, 7A 539 (1966).
\(^{4}\) N. M. Ostianu, Tr. geom. seminara, 1, Moscow, 1966, p. 239.
\(^{5}\) L. Ya. Berezina, Reports of the Third Siberian Conf. on Mathematics and Mechanics, Tomsk, 1966, p. 180.
\(^{6}\) L. Ya. Berezina, Materials of the Second Baltic Geom. Conf., Tartu, 1965, p. 9.
\(^{7}\) L. Ya. Berezina, On the theory of surfaces of a multidimensional space, Riga, 1965.
\(^{8}\) S. E. Karapetyan, Reports of the Third Siberian Conf. on Mathematics and Mechanics, Tomsk, 1964, p. 193.
\(^{9}\) P. I. Shveikin, Tr. geom. seminara, 1, Moscow, 1966, p. 331.
\(^{10}\) A. E. Liber, Tr. seminara po vektorn. i tenzorn. analizu, 13, Moscow, 1966, p. 407.
\(^{11}\) W. Klingenberg, Math. Zs., 55, 321 (1952).
\(^{12}\) A. P. Norden, Spaces of affine connection, Moscow–Leningrad, 1950.
\(^{13}\) E. N. Karapetyan, Spaces of affine, projective, and conformal connection, Kazan, 1962.
\(^{14}\) R. N. Shcherbakov, Geom. sborn., 3 (Tr. Tomsk. univ., 168), 1963, p. 5.
\(^{15}\) V. T. Bazylev, Izv. vyssh. uchebn. zaved., Matematika, No. 2 (15), 9 (1966).
\(^{16}\) K. Svoboda, V. Havel, I. Kolář, Comm. Math. Univ. Carolinae, 5, 4, 183 (1964).
\(^{17}\) G. F. Laptev, Tr. Mosk. matem. obshch., 2, 275 (1953).
\(^{18}\) G. Casanova, Rev. Math. Spéc., 65, No. 6, 437 (1955).