MATHEMATICS

S. I. Savel’ev

SURFACES WITH PLANE GENERATORS ALONG WHICH THE TANGENT PLANE IS CONSTANT

(Presented by Academician P. S. Aleksandrov, 5 III 1957)

1. Developable surfaces of three-dimensional space can be generalized to \(N\)-dimensional space in two ways. The properties of being developable onto a plane, or of being the envelope of a one-parameter family of planes, also hold in \(N\)-dimensional space for \(n\)-dimensional surfaces with \((n-1)\)-dimensional plane generators along which the tangent plane is constant.

By virtue of the property of developable surfaces of three-dimensional space of consisting of rectilinear generators along which the tangent plane is constant, they are generalized to a broader class of \(n\)-dimensional surfaces with \(p\)-dimensional \((n-1 \ge p \ge 1)\) plane generators, along which the tangent plane is constant, in \(N\)-dimensional space. Surfaces of \(n\) dimensions with \(p\)-dimensional plane generators along which the tangent plane is constant are of great interest also in connection with the problem of bending multidimensional surfaces \((^{1,2})\).

Conical \(n\)-dimensional surfaces formed by \(p\)-dimensional planes joining a fixed \((p-1)\)-dimensional plane vertex with the points of an \((n-p)\)-dimensional directrix surface of general type, and conical \(n\)-dimensional surfaces formed by \(q\)-dimensional planes joining a fixed \((q-1)\)-dimensional plane vertex with the points of an \((n-q)\)-dimensional surface with \((p-q)\)-dimensional plane generators \((q=p-1,\ldots,1)\), along which the tangent \((n-q)\)-dimensional plane is constant, constitute the most numerous class of \(n\)-dimensional surfaces with \(p\)-dimensional plane generators along which the tangent plane is constant, in \(N\)-dimensional space. But only those distinct from conical surfaces with plane generators along which the tangent plane is constant are of great interest. However, of these surfaces only the developable \((p=n-1)\) \(n\)-dimensional surfaces \((^3)\) and bendable hypersurfaces of rank 2 in Euclidean \(N\)-dimensional space \(E_N\) \((^1)\) have been well studied.

2. The property of a surface of having plane generators along which the tangent plane is constant is projective in character, and the study of such surfaces is naturally carried out in projective space \(P_N\). To each point of the surface we attach a frame consisting of \(N+1\) points \(M_0, M_1,\ldots,M_N\), not lying in one hyperplane. The point \(M_0\) is the current point of the surface.

The points \(M_1, M_2,\ldots,M_n\) of the frame are placed in the tangent plane to the surface at the point \(M_0\), with \(p\) of them, \(M_{n-p+1},\ldots,M_n\), in the plane generator passing through the point \(M_0\). The points \(M_{n+1},\ldots,M_N\) are placed outside the tangent plane at the point \(M_0\). The infinitesimal displacement of the frame is written in the form

\[ dM_J=\omega_J^K M_K \qquad (J,K=0,1,\ldots,N). \]

The surface is determined by the Pfaff system of equations

\[ \omega_i^{\hat{\alpha}}=\omega_i^{\hat{\alpha}}=0 \quad (\beta,\gamma=1,2,\ldots,n-p;\ i=n-p+1,\ldots,n; \]

\[ \omega_i^\beta=a_{i\gamma}^{\beta}\omega^\gamma,\quad \omega_\beta^{\hat{\alpha}}=a_{\beta\gamma}^{\hat{\alpha}}\omega^\gamma;\quad \hat{\alpha}=n+1,\ldots,N), \]

where the coefficients are subject to the relations

\[ a_{\beta\gamma}^{\hat{\alpha}}=a_{\gamma\beta}^{\hat{\alpha}},\quad a_{i\beta}^{\nu}a_{\nu\gamma}^{\hat{\alpha}}-a_{i\gamma}^{\nu}a_{\beta\nu}^{\hat{\alpha}}=0 \quad (\beta,\gamma,\nu=1,2,\ldots,n-p). \]

The dimension of the osculating planes at the points of an \(n\)-dimensional surface with \(p\)-dimensional plane generators, along which the tangent plane is constant, is equal to \(n+R\), where \(R\) is the number of independent vectors

\[ a_{\beta\gamma}=\{a_{\beta\gamma}^{n+1},\,a_{\beta\gamma}^{n+2},\ldots,a_{\beta\gamma}^{N}\} \quad (\beta,\gamma=1,2,\ldots,n-p) \]

in the parameter space of \(N-n\) dimensions. The number \(n+R\) may, for different surfaces, take values from \(n+1\) to

\[ n+\frac{1}{2}(n-p)(n-p+1). \]

The dimension \(n+R\) of the osculating planes and the number \(n-p\) of parameters on which the tangent plane depends are the most essential numerical parameters of the surface. Prescribing these parameters a priori, we can single out narrow classes of surfaces whose structure is amenable to investigation. Thus, for \(n+R>n+(n-p)\) we have the following theorem.

Theorem 1. If the osculating planes at the points of an \(n\)-dimensional surface with \(p\)-dimensional \((n-p>1)\) plane generators, along which the tangent plane is constant, have dimension greater than \(n+(n-p)\), then the surface is a conical one with a \((p-1)\)-dimensional plane vertex.

Consequently, nonconical \(n\)-dimensional surfaces with \(p\)-dimensional plane generators, along which the tangent plane is constant, have osculating planes of dimension not exceeding the number \(n+(n-p)\).

1. In the case when the osculating planes at the points of the surface have the least dimension \(n+1\), the surface, for \(n-p>1\), is a hypersurface of the \((n+1)\)-dimensional projective space \(P_{n+1}\). The characteristic property of hypersurfaces with plane generators, along which the tangent hyperplane is constant, is given by the following theorem.

Theorem 2. A hypersurface in \(P_{n+1}\) with \(p\)-dimensional plane generators, along which the tangent hyperplane is constant, is the envelope of a family of hyperplanes corresponding, under a correlative transformation, to the points of an \((n-p)\)-dimensional surface in \(P_{n+1}\).

The class of hypersurfaces with \(p\)-dimensional plane generators, along which the tangent hyperplane is constant, is dual to the class of \((n-p)\)-dimensional surfaces without plane generators with a constant tangent plane along them in a projective space of \(n+1\) dimensions.

1. For \(n\)-dimensional surfaces with \((n-2)\)-dimensional plane generators, along which the tangent plane is constant, the dimension of the osculating planes can have the values \(n+1\), \(n+2\), and \(n+3\). The value \(n+3\) satisfies the conditions of Theorem 1; the value \(n+1\) characterizes the hypersurfaces already considered. For \(n\)-dimensional surfaces with \((n-2)\)-dimensional generators whose osculating planes have dimension \(n+2\), we have Theorem 3.

Theorem 3. All nonconical \(n\)-dimensional surfaces with \((n-2)\)-dimensional plane generators, along which the tangent plane is constant, and which have osculating planes of dimension \(n+2\), are divided into the following three types:

a) one-parameter families of \((n-1)\)-dimensional conical surfaces with \((n-2t_0-2)\)-dimensional plane vertices and \(2t_0\)-dimensional

directing surfaces, which are either developable surfaces or conical \(2t_0\)-dimensional surfaces with \((2t_0-2)\)-dimensional vertices, \(t_0=1,2,\ldots,\left[\frac12(n-1)\right]\);

b) the envelopes of the osculating planes of order \(\frac12 n\) to a two-dimensional surface having a conjugate net, or to a two-dimensional surface having one family of asymptotic lines, in the case of even \(n\);

c) the envelopes of the osculating planes of order \(\frac12(n-1)\) to a three-dimensional manifold of points of a two-parameter congruence of lines, or to a three-dimensional manifold of points of the totality tangent to the asymptotic lines of a two-dimensional surface with one family of asymptotic lines, in the case of odd \(n\).

Thus, for \(n-p=2\) we have a complete classification of \(n\)-dimensional surfaces with \((n-2)\)-dimensional plane generators along which the tangent plane is constant.

In conclusion I express my gratitude to S. P. Finikov, under whose supervision this work was carried out.

Bauman Higher Technical School

Received
5 III 1957

REFERENCES

\({}^{1}\) E. Cartan, Bull. Soc. Math., 45, 73 (1916).
\({}^{2}\) Н. Н. Яненко, Тр. Моск. матем. общ., 3, 90, 149 (1954).
\({}^{3}\) E. Cartan, Bull. Soc. Math., 48, 161 (1920).