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MATHEMATICS
M. A. AKIVIS
ON MULTIDIMENSIONAL SURFACES CARRYING A NET OF CONJUGATE LINES
(Presented by Academician P. S. Aleksandrov, 3 IV 1961)
1. In this paper we study surfaces \(V_n\) of the affine space \(E_N\) all of whose asymptotic quadratic forms can be simultaneously reduced to canonical form—surfaces carrying a net of conjugate lines. The question of the structure of such surfaces was considered by V. T. Bazylev in [1]. We study this question in greater detail and show that the structure of the surface \(V_n\) is completely determined by the structure of the matrix of eigenvalues of its asymptotic quadratic forms. We then consider the question of the possibility of adjoining to the surface \(V_n\) a family of normalizers lying in its osculating planes and inducing on it a connection with respect to which the lines of the conjugate net are geodesics. This question also turns out to be connected with the structure of the matrix of eigenvalues of the asymptotic quadratic forms of the surface \(V_n\).
All constructions of the present paper are projectively invariant. However, for formal reasons it has proved more convenient for us to regard the surface \(V_n\) as embedded in the affine space \(E_N\).
2. Let us attach to the surface \(V_n\) an affine frame whose point \(A\) coincides with the current point of the surface, while the vectors \(\mathbf e_1,\ldots,\mathbf e_n\) lie in its tangent plane and are tangent to the lines of the conjugate net passing through the point \(A\). The infinitesimal displacement of this frame will be determined by the equations
\[ dA=\omega^p \mathbf e_p+\omega^\alpha \mathbf e_\alpha,\qquad d\mathbf e_p=\omega_p^q \mathbf e_q+\omega_p^\alpha \mathbf e_\alpha,\qquad d\mathbf e_\alpha=\omega_\alpha^p \mathbf e_p+\omega_\alpha^\beta \mathbf e_\beta, \]
\[ (p,q=1,\ldots,n;\qquad \alpha,\beta=n+1,\ldots,N), \tag{1} \]
where
\[ \omega^\alpha=0,\qquad \omega_p^\alpha=a_p^\alpha\omega^p,\qquad \omega_p^q=\sum_s l_{ps}^q\omega^s\quad (p\ne q), \]
and in the latter equations, and everywhere below, summation over the indices \(p,q,s\) will be carried out only when this is indicated by the sign \(\sum\). The forms \(\omega^p\) will be linearly independent on the surface \(V_n\), and its asymptotic quadratic forms will take the form
\[ \varphi^\alpha=\sum_p a_p^\alpha(\omega^p)^2; \]
the \(p\)-th conjugate line of the surface \(V_n\) is defined by the equations: \(\omega^q=0\) for \(q\ne p\), \(\omega^p\ne0\).
The vectors \(\mathbf a_p=a_p^\alpha \mathbf e_\alpha\), together with the vectors \(\mathbf e_p\), determine the osculating plane of the surface \(V_n\). Its dimension is \(n+n_1\), where \(n_1\le n\). We assume that the surface \(V_n\) is tangentially nondegenerate. This means that none of the vectors \(\mathbf a_p\) on it vanishes.
Let us place the vectors \(\mathbf e_{n+p_1}\) of the frame in the osculating plane of the surface \(V_n\), and the vectors \(\mathbf e_{\alpha_1}\) outside it \((p_1=1,\ldots,n_1;\ \alpha_1=n+n_1+1,\ldots,N)\). Then \(\mathbf a_p=a_p^{\,n+p_1}\mathbf e_{n+p_1}\), and the matrix of the components of these vectors assumes the form:
\[ A_n^{n_1}= \left\| \begin{array}{cccc} a_1^{n+1} & a_2^{n+1} & \ldots & a_n^{n+1}\\ \ldots & \ldots & \ldots & \ldots\\ \ldots & \ldots & \ldots & \ldots\\ a_1^{n+n_1} & a_2^{n+n_1} & \ldots & a_n^{n+n_1} \end{array} \right\|. \tag{2} \]
The rank of this matrix is equal to \(n_1\).
- We shall say that a surface \(V_n\) decomposes into the direct sum of surfaces \(V_m\) and \(V_{n-m}\) if it decomposes into an \((n-m)\)-parameter family of surfaces \(V_m\) and an \(m\)-parameter family of surfaces \(V_{n-m}\), in such a way that through each point of the surface \(V_n\) there passes one surface of each family, intersecting only at this point.
The following theorems show how the structure of the surface \(V_n\) is related to the structure of its matrix \(A_n^{n_1}\).
Theorem 1A. If the matrix \(A_n^{n_1}\) of the surface \(V_n\) can be brought to the form
\[ \left\| \begin{array}{cc} A_m^{m_1} & 0\\ 0 & A_{n-m}^{\,n_1-m_1} \end{array} \right\| \]
\((m_1\leq m;\ n_1-m_1\leq n-m)\), then the surface \(V_n\) decomposes into the direct sum of surfaces \(V_m\) and \(V_{n-m}\), bearing nets of conjugate lines, and its osculating plane will be the direct sum of the osculating planes \(E_{m+m_1}\) and \(E_{n+n_1-m-m_1}\) of the surfaces \(V_m\) and \(V_{n-m}\).
Theorem 1B. For \(n_1=n\) the matrix \(A_n^{n_1}\) of the surface \(V_n\) can be brought to the form
\[ A_n^n= \left\| \begin{array}{cccc} 1 & 0 & \ldots & 0\\ 0 & 1 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & 1 \end{array} \right\| \]
and the surface \(V_n\) is a manifold of a special projective type, considered by Cartan in paper \((^2)\). Such a surface, generally speaking, does not lie entirely in its osculating plane, and the dimension of its second osculating plane is \(2n+n_2\), where \(n_2\leq n\).
Theorem 1C. If, for \(n_1<n\), on the surface \(V_n\) the rank of each subsystem of the system of vectors \(\mathbf a_1,\ldots,\mathbf a_n\), consisting of \(n-1\) vectors, is equal to \(n_1\), then the surface \(V_n\) lies entirely in its osculating plane \(E_{n+n_1}\). In this case \(N=n+n_1\).
Theorem 1D. If, for \(n_1<n\), the conditions of Theorem 1C are not satisfied on the surface \(V_n\), then its matrix \(A_n^{n_1}\) can be brought to the form
\[ \left\| \begin{array}{cc} A_m^{m_1} & 0\\ 0 & A_{n-m}^{\,n-m} \end{array} \right\|, \tag{3} \]
where \(m_1=n_1-(n-m)\), and the matrix \(A_m^{m_1}\) satisfies the condition of Theorem 1C. The surface \(V_n\) then decomposes into the direct sum of surfaces \(V_m\), lying entirely in their osculating planes \(E_{m+m_1}\), and Cartan surfaces \(V_{n-m}\), the dimension of whose osculating planes is equal to \(2(n-m)\). The dimension of the second osculating plane of the surface \(V_n\) is equal to \(n+n_1+n_2\), where \(n_2\leq n-m\). In this case the planes \(E_{m+m_1}\) describe a surface of rank \(n-m\) of the special projective type \((^2)\), the dimension of whose tangent planes is \(m+n_1\), and the osculating planes coincide with the osculating planes of the surface \(V_n\).
- We say that a conjugate net of a surface \(V_n\) is completely holonomic if each of the equations
\[ \omega^p=0, \]
which determine the sub-surfaces of the net, is completely integrable. If, however, no subsystem consisting of fewer than \(n-1\) such equations is completely integrable, then we call the conjugate net of the surface \(V_n\) irreducible. The condition for complete holonomicity of the conjugate net of the surface \(V_n\) is the vanishing of the coefficients \(l^q_{ps}\) in the expansion (1) of the forms \(\omega^q_p\) for \(p\ne q,\ p\ne s,\ q\ne s\).
The following theorems hold:
Theorem 2A. If no three of the vectors \(\mathbf a_1,\ldots,\mathbf a_n\) of the surface \(V_n\) lie in one plane, then the net of conjugate lines of this surface will be completely holonomic.
Theorem 2B. An irreducible net of conjugate lines can be carried only by surfaces \(V_n\) of class 2 whose system of vectors \(\mathbf a_1,\ldots,\mathbf a_n\) is such that no two of them are collinear, and by surfaces of class 1.
Theorem 2V. The surface \(V_n\) may decompose into an \((n-m)\)-parameter family of surfaces \(V_m\) carrying an irreducible net of conjugate lines in two cases:
a) when, in the system of vectors \(\mathbf a_1,\ldots,\mathbf a_n\) of this surface, \(m\) vectors \((m\ge 3)\) lie in one plane which contains none of the remaining vectors of the system, while no two of them are collinear, and the focal planes of the plane generating surfaces \(S_{n-m,n}\) of rank \(m\) \((^3)\), tangent to the surface \(V_n\) at the points of its sub-surfaces \(V_m\), belong to one pencil; the class of the surfaces \(V_m\) in this case is equal to two;
b) when, in the system of vectors \(\mathbf a_1,\ldots,\mathbf a_n\) of the surface \(V_n\), \(m\) vectors are collinear; in this case the class of the surfaces \(V_m\) is equal to one.
- Let \(E_{n_1}\) be a normalizer of the surface \(V_n\), passing through its point \(A\) and lying in its osculating plane \(E_{n+n_1}\). We shall call this normalizer a Foss normal of the surface \(V_n\) if it is intersected by all two-dimensional osculating planes of the conjugate lines of this surface passing through the point \(A\). With respect to the affine connection induced on \(V_n\) by its Foss normals, the lines of the conjugate net of this surface will be geodesic lines.
The following theorems show how the question of the existence and construction of a Foss normal of the surface \(V_n\) is connected with the structure of its matrix \(A^{n_1}_n\).
Theorem 3A. If the condition of Theorem 1A is fulfilled on the surface \(V_n\) and it has a Foss normal, then this normal will be the direct sum of the Foss normals of the surfaces \(V_m\) and \(V_{n-m}\) into which the surface \(V_n\) decomposes.
Theorem 3B. If the surface \(V_n\) is a Cartan surface, then at each of its points it has a family of Foss normals depending on \(n\) parameters.
Theorem 3V. If, for \(n_1=n-1\), the matrix \(A^{n_1}_n\) satisfies the condition of Theorem 1B, then the surface \(V_n\) has at each of its points a unique Foss normal.
Theorem 3G. If, for \(n_1=n-1\), the matrix \(A^{n_1}_n\) has the form (3), then the surface \(V_n\) has at each of its points a family of Foss normals depending on \(n-m\) parameters.
Theorem 3D. For \(n_1<n-1\), the surface \(V_n\), generally speaking, has no Foss normal. It has a Foss normal only in the case when the coefficients \(l^p_{qq}\) in the expansion (1) of the forms \(\omega^p_q\) of this surface can be made zero.
- Consider a surface \(V_n\) for \(n_1<n-1\). Suppose that the net of its conjugate lines is completely holonomic and that at each of its points it has a Foss normal. Then the following theorems hold:
Theorem 4A. If, on the surface \(V_n\), the rank of each subsystem of the system of vectors \(a_1,\ldots,a_n\) consisting of \(n-2\) vectors is equal to \(n_1\), then the congruence of Fosse normals of this surface decomposes into \(n\) families of developable surfaces, and these developable surfaces correspond to the conjugate lines of the surface \(V_n\). Each conjugate line of the surface \(V_n\) and the corresponding developable surface of the congruence of Fosse normals lie entirely in one \((n_1+1)\)-dimensional plane. Such surfaces \(V_n\) exist with arbitrariness
\(s_1=(2n_1+n-1)n\) functions of one argument.
An example of such a surface is the translation surface
\[ r(u^1,\ldots,u^n)=r_1(u^1)+\ldots+r_n(u^n), \]
the conjugate lines \(r_p(u^p)\) of which belong entirely to their own \((n_1+1)\)-dimensional osculating planes, while all these osculating planes belong to one pencil with an \(n_1\)-dimensional axis.
Theorem 4B. If, on the surface \(V_n\), the condition of Theorem 4A is not satisfied, but the condition of Theorem 1B is satisfied, then its matrix \(A_n^{n_1}\) can be reduced to the form
\[ \left\| \begin{matrix} A_m^{m_1} & A_{n-m}^{m_1}\\ 0 & A_{n-m}^{\,n_1-m_1} \end{matrix} \right\|, \]
where \(n_1-m_1=n-m-1\), and the matrix \(A_{n-m}^{\,n_1-m_1}\) satisfies the condition of Theorem 3B. In this case the following cases may occur:
a) the matrix \(A_n^{m_1}=\|A_m^{m_1}, A_{n-m}^{m_1}\|\) satisfies the conditions of Theorem 4A; such surfaces exist with arbitrariness
\(s_2=(n-m)(n-m-1)\) functions of two arguments;
b) \(A_{n-m}^{m_1}=0\), \(m_1=m-1\), and the matrix \(A_m^{m_1}\) satisfies the condition of Theorem 3B; such a surface \(V_n\) decomposes into the direct sum of surfaces \(V_m\) and \(V_{n-m}\) of the type of Theorem 3B and exists with arbitrariness
\(s_2=m(m-1)+(n-m)(n-m-1)\) functions of two arguments;
c) if the conditions of the two preceding items are not satisfied, then the process of reducing the matrix \(A_n^{n_1}\) should be continued.
Moscow Institute of Steel
named after I. V. Stalin
Received
31 III 1961
REFERENCES CITED
- V. T. Bazylev, Izv. Vyssh. uchebn. zaved., Mathematics, No. 1 (20), 27 (1961).
- E. Cartan, Bull. Soc. Math. de France, 47 (1919); 48 (1920).
- M. A. Akivis, Izv. Vyssh. uchebn. zaved., Mathematics, No. 1, 9 (1957).