UDC 513

MATHEMATICS

E. T. IVLEV

GENERALIZED EQUIPARAMETRIC MANIFOLDS IN A MULTIDIMENSIONAL PROJECTIVE SPACE

(Presented by Academician A. D. Aleksandrov on 10 VII 1967)

As is known \((^1)\), the problem of invariantly equipping an \(m\)-surface \(S_m\) in an \(n\)-dimensional projective space \(P_n\) is the construction of a field of \((n-m)\)-planes invariantly associated with \(S_m\). Therefore it is of interest to study the \(m\)-parametric manifold \(E(0,n-m,m)\), whose element is an \((n-m)\)-plane \(L_{n-m}\) (the basic \((n-m)\)-plane) with a fixed point \(L\) in it. In a deeper study of the geometry of \(S_m\) in \(P_n\), an essential role is played by the manifold \(E(L,\hat L_m,\hat L_{m+1})\), whose element consists of \(n-m\) linearly independent \((m+1)\)-planes \(L_{\hat m+1}\) \((\hat\alpha=m+1,\ldots,n)\), passing through the \(m\)-plane \(L_m\) in which the point \(L\) is given, where \(L_m\) is the tangent \(m\)-plane of the \(m\)-surface \(S_m\) described by the point \(L\). These manifolds are the subject considered in the present work. Analogous equiparametric manifolds in three-dimensional space were studied in \((^2)\).

1. The derivation formulas of a certain frame, consisting of analytic points \(A_0,A_1,\ldots,A_n\), in the \(n\)-dimensional projective space \(P_n\) will be written 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 \(\omega_i^i=0\). An analytic fixation of a semicanonical frame in the sense of \((^3,^4)\) of the manifold \(E(0,n-m,m)\) in \(P_n\) gives
   \(\omega_i^k=\Lambda_{i\alpha}^k\omega_0^\alpha\) \((\alpha=1,2,\ldots,m)\), where \(\Lambda_{i\alpha}^k\) satisfy the relations

\[ \Lambda_{0\beta}^{\alpha}=\delta_\beta^\alpha,\qquad \Lambda_{0\beta}^{\hat\alpha}=0,\qquad \Lambda_{\alpha\beta}^{\hat\beta}=\Lambda_{\beta\alpha}^{\hat\beta},\qquad A_{\underbrace{0\ldots0}_{m-1}\hat\alpha}=0,\qquad A_{\underbrace{\hat\alpha\ldots\hat\alpha\hat\beta}_{m-1}}=0 \quad(\hat\alpha\ne\hat\beta), \]

\[ A_{\underbrace{\hat\alpha\ldots\hat\alpha}_{m}}=m,\qquad \Lambda_{n\beta}^{0}=0,\qquad \Lambda_{m+1,\alpha}^{0}=1 \]

\[ (\alpha,\beta,\gamma=1,\ldots,m;\quad \hat\alpha,\hat\beta=m+1,\ldots,n), \]

\[ \Lambda_{n\beta}^{\gamma}\Lambda_{\gamma\alpha}^{0} -\Lambda_{n\alpha}^{\gamma}\Lambda_{\gamma\beta}^{0} = \Lambda_{\hat\beta\beta}^{0}\Lambda_{n\alpha}^{\hat\beta} -\Lambda_{\hat\beta\alpha}^{0}\Lambda_{n\beta}^{\hat\beta} \qquad(\beta\ne\alpha,\ \hat\beta\ne n). \]

Here it is assumed that \(m>2\) and \(n<m(m+1)\), and the quantities

\[ m!\,A_{\tilde\alpha_1\tilde\alpha_2\ldots\tilde\alpha_m} = \Lambda_{(\tilde\alpha_1|1|}^{[1}\Lambda_{\tilde\alpha_2|2|}^{2}\cdots \Lambda_{\tilde\alpha_m|m|)}^{m]} \qquad (\tilde\alpha_1,\ldots,\tilde\alpha_m=0,m+1,\ldots,n) \]

are absolutely symmetric in all indices. The manifold \(E(0,n-m,m)\), referred to an arbitrary system of one-dimensional submanifolds, is determined with an arbitrariness of \(m^2+n(m-1)-3n\) functions of \(m\) arguments. The constructed semicanonical frame has the following geometric characteristic. The point \(A_0\) coincides with the point \(L\), and \(L_{n-m}=(A_0A_{m+1}\ldots A_n)\) is the basic \((n-m)\)-plane. The \((n-m-1)\)-plane \(L_{n-m-1}=(A_{m+1}\ldots A_n)\) is the \((m-1)\)-st polar, in the sense of \((^5,^6)\), of the point \(A_0\) with respect to the focal algebraic hypersurface \(\Phi_{n-m-1}^{m}\) of order \(m\) of the basic \((n-m)\)-plane. The algebraic surface \(\Psi_{n-m-2}^{m}\) of order \(m\) and dimension \(n-m-2\), belonging simultaneously to \(\Phi_{n-m-1}^{m}\) and

\(L_{n-m-1}\) is called a polar algebraic surface. The points \(A_{\hat\alpha}\) \((\hat\alpha=m+1,\ldots,n)\) in \(L_{n-m-1}\) are chosen so that \(A^{\hat\alpha}=(A_{m+1}\ldots A_{\hat\alpha-1}A_{\hat\alpha+1}\ldots A_n)\) is the \((m-1)\)-st polar of each of these points \(A_{\hat\alpha}\) with respect to the polar algebraic surface. The \((m-1)\)-plane \(L_{m-1}=(A_1A_2\ldots A_m)\) belongs simultaneously to the tangent to \(S_m\) at the point \(A_0\), to the \(m\)-plane \(L_m=(A_0-A_m)\), and to the hyperplane \(Q_{n-1}\) passing through \(L_{n-m-1}\) and through the tangent \(m\)-plane to the \(m\)-surface described by the point \(A_n\). The lines \(A_0A_\alpha\) are tangent to the lines described by the point \(A_0\) under displacements along the one-dimensional coordinate submanifolds. The points \(A_\alpha\) are the points of intersection of the lines \(A_0A_\alpha\) with the \((m-1)\)-plane \(L_{m-1}\).

Let us note two special classes of systems of submanifolds of the manifold \(E(0,n-m,m)\).

A. A system \(S\) of submanifolds, whose assignment is characterized by the relations \(\Lambda_{\hat\alpha\beta}^{\alpha}=0,\ \Lambda_{\hat\alpha\alpha}^{\alpha}\ne\Lambda_{\hat\alpha\beta}^{\beta}\) \((\alpha\ne\beta\), no summation over \(\alpha\) and \(\beta\), \(\hat\alpha\) fixed), consists of one-dimensional submanifolds corresponding to the focal displacements of different foci of the algebraic surface \(\Phi^m_{n-m-1}\) belonging to the line \(A_0A_{\hat\alpha}\). The canonical frame obtained by referring the manifold \(E(0,n-m,m)\) to the system \(S\) is called the principal frame.

B. A system \(\Gamma\) of submanifolds, whose assignment is characterized by the relations \(\Lambda_{\alpha\beta}^{\hat\alpha}=\Lambda_{\alpha\beta}^{\hat\beta}=0,\ \Lambda_{\alpha\alpha}^{\hat\alpha}\Lambda_{\beta\beta}^{\hat\beta}\ne\Lambda_{\alpha\alpha}^{\hat\beta}\Lambda_{\beta\beta}^{\hat\alpha}\) \((\alpha\ne\beta,\ \hat\alpha\ne\hat\beta,\ \alpha\) and \(\hat\beta\) fixed), consists of one-dimensional submanifolds corresponding to the focal displacements of different hyperplanes \(L^{\hat\alpha}_{n-1},\ \hat\alpha,\hat\beta=(L_m A_{m+1}\ldots A_{\hat\alpha-1}A_{\hat\alpha+1}\ldots A_{\hat\beta-1}A_{\hat\beta+1}\ldots A_n,\ \Lambda_{\alpha\alpha}^{\hat\alpha}A_{\hat\alpha}+\Lambda_{\alpha\alpha}^{\hat\beta}A_{\hat\beta})\) (no summation over \(\hat\alpha\) and \(\hat\beta\)), containing under these displacements the first differential neighborhood of the \(m\)-plane \(L_m\) and passing through the \((m+1)\)-planes \(L^{\hat\gamma}_{m+1}=(L_mA_{\hat\gamma})\) \((\hat\gamma=m+1,\ldots,n;\ \hat\gamma\ne\hat\alpha,\hat\beta)\). The canonical frame obtained by referring the manifold \(E(0,n-m,m)\) to the system \(\Gamma\) is called a \(\Gamma\)-frame.

The systems \(S\) and \(\Gamma\) make it possible to distinguish the following special classes of manifolds \(E(0,n-m,m)\).

I. A manifold \(E(0,n-m,m)\) having, in the principal or \(\Gamma\)-frame, natural equations \(\Lambda_{\alpha\alpha}^{\beta}=0\) \((\alpha\ne\beta)\), is defined with arbitrariness \((m+1)(n-m)\) functions of \(m\) arguments and is characterized by the fact that the osculating 2-planes \(a_\alpha\) to the coordinate lines of the surface \(S_m\) described by the point \(A_0\) intersect the principal \((n-m)\)-plane \(L_{n-m}\) along straight lines \(l_\alpha=\Lambda_{\alpha\alpha}^{\hat\alpha}(A_0A_{\hat\alpha})\), i.e., in accordance with (7), the coordinate lines of the surface \(S_m\) are geodesic lines of the projective-connection space induced by the \((n-m)\)-plane \(L_{n-m}\) along \(S_m\).

II. A manifold \(E(0,n-m,m)\) having, in the principal frame, natural equations \(\Lambda_{\alpha\alpha}^{\hat\beta}=0\), is defined with arbitrariness \(m^2-2m+n\) functions of \(m\) arguments and is characterized by the fact that the coordinate lines of the surface \(S_m\) are asymptotic lines.

III. A manifold \(E(0,n-m,m)\) having, in the principal frame, natural equations \(\Lambda_{\alpha\alpha}^{\beta}=0,\ \Lambda_{\gamma\gamma}^{\hat\alpha}=0\) \((\alpha\ne\beta,\ \alpha,\beta,\gamma=1,\ldots,m)\), is defined with arbitrariness \(n-m\) functions of \(m\) arguments and is characterized by the fact that the coordinate lines of the surface \(S_m\) are straight lines.

2. Fixing the semicanonical frame of the manifold \(E(L,L_m,L^{\hat\alpha}_{m+1})\), carried out analytically, gives \(\omega_i^k=\lambda^k_{i\alpha}\omega^\alpha\), where \(\Lambda^k_{i\alpha}\) satisfy the relations

\[ \Lambda_{0\beta}^{\alpha}=\delta_\beta^\alpha,\quad \Lambda_{0\beta}^{\alpha}=0,\quad \Lambda_{\alpha\beta}^{\hat\beta}=\Lambda_{\beta\alpha}^{\hat\beta},\quad \Lambda_{\hat\gamma\alpha}^{n}=0,\quad \Lambda_{n\alpha}^{m+1}=0,\quad A^{\hat\alpha\ldots\hat\alpha}=m\ (\hat\gamma\ne n), \]

\[ \Lambda_{m+1,\beta}^{\alpha}\Lambda_{\alpha\gamma}^{m+1} +\Lambda_{\gamma\gamma}^{m+1}\Lambda_{n\beta}^{\hat\gamma} = \Lambda_{m+1,\gamma}^{\alpha}\Lambda_{\alpha\beta}^{m+1} +\Lambda_{\gamma\beta}^{m+1}\Lambda_{n\gamma}^{\hat\gamma} \qquad (\hat\gamma\ne n,\ \gamma\ne\beta), \]

\[ \Lambda_{\alpha\gamma}^{n}\Lambda_{\alpha\beta}^{\alpha} = \Lambda_{\alpha\gamma}^{n}\Lambda_{\alpha\gamma}^{\alpha}, \qquad \Lambda_{\beta\alpha}^{\alpha}=0, \qquad \Lambda_{n\beta}^{0}=0, \qquad \Lambda_{m+1,\alpha}^{0}=1 \qquad (\beta\ne\gamma,\ \hat\alpha\ne n), \]

\[ \Lambda_{n\beta}^{\gamma}\Lambda_{\gamma\tau}^{0} - \Lambda_{n\tau}^{\gamma}\Lambda_{\gamma\beta}^{0} = \Lambda_{\hat\beta\beta}^{0}\Lambda_{n\tau}^{\beta} - \Lambda_{\beta\tau}^{0}\Lambda_{n\beta}^{\hat\beta} \qquad (\hat\beta\ne m+1,\ \beta\ne\tau). \]

Here \(m>2\), and the quantities

\[ m!\,A^{\hat\alpha_1\ldots\hat\alpha_m} = \Lambda_{1[1}^{\hat\alpha_1}\Lambda_{2]2}^{\hat\alpha_2}\cdots \Lambda_{|m|m]}^{\hat\alpha_m} \qquad (\hat\alpha_\beta=m+1,\ldots,n) \]

are absolutely symmetric in all indices. The manifold \(E(0,n-m,m)\), referred to an arbitrary system of one-dimensional submanifolds, is determined with arbitrariness \((n-m)^2+m(m-1)\) functions of \(m\) arguments. In this frame the point \(A_0\) coincides with the point \(L\), \(L_m=(A_0A_1\ldots A_m)\), and \(L_{m+1}^{\hat\alpha}=(L_mA_{\hat\alpha})\) \((\hat\alpha=m+1,\ldots,n)\). The lines \(A_0A_{\hat\alpha}\) \((\hat\alpha\ne n)\) are \(B\)-characteristics of the \((m+1)\)-planes \(L_{m+1}^{\hat\alpha}\) \((\hat\alpha\ne n)\) with respect to \(L_{n-1}^{n}=(L_mA_{m+1}\ldots A_{n-1})\), i.e., the entire first differential neighborhood of each of these lines, belonging to \(L_{m+1}^{\hat\alpha}\) \((\hat\alpha\ne n)\), does not leave \(L_{n-1}^{n}\). The line \(A_0A_n\) is a \(B\)-characteristic of the \((m+1)\)-plane \(L_{m+1}^{n}\) with respect to \(L_{n-1}^{m+1}=(L_mA_{m+2}\ldots A_n)\). Therefore the \((n-m)\)-plane \(L_{n-m}=(A_0A_{m+1}\ldots A_n)\) passes through the lines \(A_0A_{\hat\alpha}\). Consequently, with each element of the manifold \(E(L,L_m,L_{m+1}^{\hat\alpha})\) there is invariantly associated the manifold \(E(0,n-m,m)\), and the further geometric characteristic of the frame is evident. The manifold \(ES_m\) is the manifold \(E(L,L_m,L_{m+1}^{\hat\alpha})\) determined by the natural equations

\[ A^{\hat\alpha\ldots\hat\alpha}=0\quad(\hat\alpha\ne\hat\beta). \]

This manifold is determined with arbitrariness the product of \(n-m\) functions of \(m\) arguments, where

\[ m+2<n<\frac12 m(m+3). \]

The totality of all focal hyperplanes \((^{8,9})\) of the \(m\)-surface \(S_m\) forms a certain hypercone \(T_{n-1}^{m}\) of class \(m\) with vertex \(L_m\), defined by the equation

\[ T\equiv A^{\hat\alpha_1\hat\alpha_2\ldots\hat\alpha_m} x_{\hat\alpha_1}x_{\hat\alpha_2}\cdots x_{\hat\alpha_m}=0. \]

For the manifold \(ES_m\) we have: each \((m+1)\)-plane \(L_{m+1}^{\hat\alpha}=(L_mA_{\hat\alpha})\) is an \((m-1)\)-fold plane in the sense of \((^{5,6})\) of the hyperplane

\[ A^{\hat\alpha}=(L_mA_{m+1}\ldots A_{\hat\alpha-1}A_{\hat\alpha+1}\ldots A_n) \]

with respect to \(T_{n-1}^{m}\). Let us note that in the manifold \(ES_m\) the surface \(S_m\) is arbitrary. Therefore the semicanonical frame of the manifold \(ES_m\) may be regarded as the semicanonical frame of an arbitrary \(m\)-surface \(S_m\) in \(P_n\) \(\bigl(m+2<n<\tfrac12 m(m+3)\bigr)\).

Tomsk State University
named after V. V. Kuibyshev

Received
30 I 1967

CITED LITERATURE

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