Abstract
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MATHEMATICS
Yu. G. LUMISTE
ON THE GEOMETRIC STRUCTURE OF A COMPLEX-ANALYTIC SURFACE \(V_{2n}\) IN THE SPACE \(R_{2N}\)
(Presented by Academician P. S. Aleksandrov, 3 XII 1956)
- A complex-analytic surface \(V_{2n}\) in Euclidean space \(R_{2N}\) is a surface which, in orthogonal coordinates \(x^J, x^{\bar J}\) \((J=1,\ldots,N;\ \bar J=N+J)\), can be represented by the equations
\[ x^J+ix^{\bar J}=f^J(u^k+iu^{\bar k})\qquad (k=1,\ldots,n), \]
where \(f(w^k)\) are analytic functions of \(n\) complex variables \(w^1,\ldots,w^n\).
In the case \(n=1\), such surfaces were considered by Kommerell and Eisenhart \((^1)\) for \(N=2\), and by Borůvka \((^2)\) for arbitrary \(N\). They proved that \(V_2\) in \(R_{2n}\) is complex-analytic if and only if all its indicatrices of normal curvature are circles. In this case the surface is a surface of translation of its isotropic curves.
In the present article a geometric characterization is given of a complex-analytic \(V_{2n}\) in \(R_{2N}\).
- The complex-analytic \(V_{2n}\) in \(R_{2N}\) is considered as the image of an analytic surface \(W_n\) of the unitary space \(U_N(i)\) under an isometric mapping \(U_N(i)\to R_{2N}\), under which the vector \(\xi^J e_J\in U_N(i)\) is mapped into the real vector \(x^J e_J+x^{\bar J}e_{\bar J}\in R_{2N}\) with \(x^J=x^{\bar J}=\xi^J\), where \(e_J, e_{\bar J}=e_J\) are vectors in \(R_{2N}\) lying, respectively, in two imaginary complex-conjugate completely isotropic directions \(I_N,\overline{I}_N\), so that \((e_J,e_{\bar K})=(\varepsilon_J\varepsilon_K)\). The image of the line \(\theta(\xi^J e_J)\in U_N(i)\), passing through the initial point of the frame, is the two-dimensional complex-analytic plane \((\theta\xi^J)e_J+(\overline{\theta\xi^J})e_{\bar J}\in R_{2N}\), determined by two complex-conjugate vectors in the directions \(I_N,\overline{I}_N\).
An analytic moving frame in \(U_N(i)\), whose infinitesimal displacement formulas are
\[ dM=\pi^J e_J,\qquad de_J=\pi^K_J e_K, \]
is mapped into a frame whose displacement is determined by the formulas
\[ dM=\omega^J e_J+\omega^{\bar J}e_{\bar J},\qquad de_J=\omega^K_J e_K,\qquad de_{\bar J}=\omega^{\bar K}_{\bar J}e_{\bar K}, \]
where
\[ \omega^{\bar J}=\overline{\omega^J}=\overline{\pi^J},\qquad \omega^{\bar K}_{\bar J}=\overline{\omega^K_J}=\overline{\pi^K_J}, \tag{1} \]
i.e., the image of the frame moves in such a way that
\[ \omega^{\bar K}_{J}=\omega^{K}_{\bar J}=0. \tag{2} \]
- Theorem. A real non-isotropic surface \(V_{2n}\) in the Euclidean space \(R_{2N}\) is complex-analytic if and only if:
1°. It is the transfer surface of two imaginary complex-conjugate completely isotropic analytic surfaces \(X_n,\overline{X}_n\).
2°. The surfaces \(X_n,\overline{X}_n\) lie respectively in two flat generators \(I_N,\overline{I}_N\) of the isotropic cone, intersecting only at a point of the surface.
A complex-analytic surface \(V_{2n}\) belongs, therefore, to the class of minimal surfaces with two isotropic conjugate directions \(I_n,\overline{I}_n\).
Necessity. Let the complex-analytic \(V_{2n}\subset R_{2N}\) be the image of the analytic surface \(W_n\subset U_N(i)\) under the mapping \(U_N(i)\to R_{2N}\). An analytic moving frame is attached to the point \(M\) of the surface \(W_n\) so that the vectors \(e_i\) \((i,j,\ldots=1,\ldots,n)\) lie in the tangent plane to \(W_n\) at the point \(M\). Then \(\pi^\alpha=0\) \((\alpha,\beta,\ldots=n+1,\ldots,N)\). Prolongation of these equations leads to the equations \(\pi_i^\alpha=\Lambda_{ij}^\alpha\pi^j\).
The moving frame attached to \(V_{2n}\) under the mapping \(U_N(i)\to R_{2N}\) moves, by virtue of (1) and (2), in such a way that
\[ \omega_i^{\overline{j}}=\omega_{\overline{i}}^j=\omega_i^{\overline{\alpha}}=\omega_{\overline{i}}^\alpha =\omega_\alpha^{\overline{\beta}}=\omega_{\overline{\alpha}}^\beta=0, \]
\[ \omega_i^\alpha=\Lambda_{ij}^\alpha\omega^j,\qquad \omega_{\overline{i}}^{\overline{\alpha}}=\Lambda_{\overline{i}\overline{j}}^{\overline{\alpha}}\omega^{\overline{j}}. \]
The systems \(\omega^i=0,\ \omega^{\overline{i}}=0\) will both be completely integrable, and the directions \(I_n,\overline{I}_n\), constructed on the vectors \(e_i,e_{\overline{i}}\), envelop families of completely isotropic sub-surfaces \(X_n\) and \(\overline{X}_n\).
The necessity of conditions 1° and 2° now follows from the formulas
\[ de_i=\omega_i^j e_j+\Lambda_{ij}^\alpha\omega^j e_\alpha,\qquad de_\alpha=\omega_\alpha^i e_i+\omega_\alpha^\beta e_\beta, \]
\[ de_{\overline{i}}=\omega_{\overline{i}}^{\overline{j}}e_{\overline{j}} +\Lambda_{\overline{i}\overline{j}}^{\overline{\alpha}}\omega^{\overline{j}}e_{\overline{\alpha}},\qquad de_{\overline{\alpha}}=\omega_{\overline{\alpha}}^{\overline{i}}e_{\overline{i}} +\omega_{\overline{\alpha}}^{\overline{\beta}}e_{\overline{\beta}}. \]
Sufficiency. The moving frame at the point \(M\) of the surface \(V_{2n}\subset R_{2N}\) is attached so that the vectors \(e_i,\ e_{\overline{j}}=\overline{e}_j\) are tangent respectively to the sub-surfaces \(X_n,\overline{X}_n\), while \(e_\alpha,\ e_{\overline{\alpha}}\) lie in the planes \(I_N,\overline{I}_N\) and are orthogonal respectively to \(e_{\overline{i}},e_i\). Then
\(g_{ij}=g_{\overline{i}\overline{j}}=g_{\alpha\beta}=g_{\overline{\alpha}\overline{\beta}}
=g_{i\alpha}=g_{\overline{i}\overline{\alpha}}=g_{i\overline{\alpha}}=g_{\overline{i}\alpha}=0\), and
\[ \omega^\alpha=\omega^{\overline{\alpha}}=0. \tag{3} \]
From condition 1° it follows that, in the formulas of the infinitesimal displacement of the frame, the forms
\(\omega_i^{\overline{j}},\omega_i^{\overline{\alpha}},\omega_\alpha^{\overline{i}},\omega_\alpha^{\overline{\beta}}\) and
\(\omega_{\overline{i}}^j,\omega_{\overline{i}}^\alpha,\omega_{\overline{\alpha}}^i,\omega_{\overline{\alpha}}^\beta\)
are expressed only, respectively, in terms of \(\omega^{\overline{k}}\) and \(\omega^k\). From condition 2° it follows that the same forms are expressed only, respectively, in terms of \(\omega^k\) and \(\omega^{\overline{k}}\). Consequently,
\[ \omega_i^{\overline{j}}=\omega_i^{\overline{\alpha}} =\omega_\alpha^{\overline{i}}=\omega_\alpha^{\overline{\beta}} =\omega_{\overline{i}}^j=\omega_{\overline{i}}^\alpha =\omega_{\overline{\alpha}}^i=\omega_{\overline{\alpha}}^\beta=0. \tag{4} \]
Exterior differentiation of equations (3) leads to the equations
\[ [\omega^i\omega_i^\alpha]=0,\qquad [\omega^{\overline{i}}\omega_{\overline{i}}^{\overline{\alpha}}]=0. \tag{5} \]
The system of Pfaffian equations
\[ dM=\omega^i e_i,\qquad de_i=\omega_i^j e_j+\omega_i^\alpha e_\alpha,\qquad de_\alpha=\omega_\alpha^i e_i+\omega_\alpha^\beta e_\beta, \]
where \(\omega^i,\omega^j_i,\omega^\alpha_i,\omega^\alpha,\omega^\beta_\alpha\) are prescribed forms of an infinitesimal displacement of the frame attached to \(V_{2n}\), now turns out, by virtue of (4), (5) and the structural equations of the space \(R_{2N}\), to be completely integrable and defines a certain \(n\)-dimensional surface \(W_n\) in the space \(U_N(i)\) with metric tensor
\[ \gamma_{ij}=\overline{\gamma}_{ji}=g_{ij},\qquad \gamma_{i\alpha}=0,\qquad \gamma_{\alpha\beta}=\overline{\gamma}_{\beta\alpha}=g_{\alpha\bar\beta}. \]
Since \(D\omega^i=[\omega^j\omega^i_j]\equiv0\pmod{\omega^k}\), we have \(\omega^i=a^i_k\,dw^k\), and \(dM=M_k\,dw^k\), where \(M_k=a^i_k e_i\). Consequently, \(\partial M/\partial\overline{w}^k=0\), and the radius vector of a point of the surface \(W_n\) is an analytic function of \(n\) complex parameters \(w^k\). The image of the surface \(W_n\) under the mapping \(U_N(i)\to R_{2N}\) is the surface \(V_{2n}\) under consideration.
- A question arises as to the relation between the theorem and the result of Kommerell and Eisenhart.
Let a one-dimensional direction tangent to the complex-analytic \(V_{2n}\) in \(R_{2N}\) rotate in a complex-analytic two-dimensional direction. It follows from (3) that the endpoint of the corresponding vector of normal curvature describes a circle lying in a complex-analytic two-dimensional direction of the normal plane.
Thus the complex-analytic surface \(V_{2n}\) satisfies a certain condition which is a generalization of the Kommerell–Eisenhart condition. However, this condition is not sufficient in the general case.
In the particular case when \(N=n+1\) and the normal plane of the surface \(V_{2n}\) has a positive-definite metric, condition \(2^\circ\) of the theorem can nevertheless be replaced by the following condition:
\(2'\). When the tangent direction is rotated in the two-dimensional direction determined by two complex-conjugate directions tangent, respectively, to \(X_n,\overline{X}_n\), the endpoint of the corresponding vector of normal curvature describes a circle in the two-dimensional normal plane.
Let the vectors \(e_i,e_{\bar i}=\overline{e}_i,e_N,e_{\bar N}\) of the moving frame attached to a point of the surface \(V_{2n}\) be directed so that
\[ g_{ij}=g_{\bar i\bar j}=g_{NN}=g_{\bar N\bar N} =g_{iN}=g_{i\bar N}=g_{\bar i N}=g_{\bar i\bar N}=0, \tag{6} \]
\[ \Lambda^N_{\bar i j}=\Lambda^{\bar N}_{i\bar j}=0. \]
From condition \(1^\circ\) it follows that \(\omega^{\bar j}_i=\Gamma^{\bar j}_{ik}\omega^k,\ \Gamma^{\bar j}_{ik}=\Gamma^{\bar j}_{ki}\). If this expression is substituted in the differential consequence of \((6_1)\), one obtains the equation
\[ g_{i\bar l}\Gamma^{\bar l}_{jk}+g_{l\bar j}\Gamma^{\bar l}_{ik}=0. \]
Cyclic permutation of this equation in the indices \(i,j,k\) gives three equations, from which it follows that \(\Gamma^{\bar j}_{ik}=0\), i.e. \(\omega^{\bar j}_i=0\). Analogously it is proved that \(\omega^j_{\bar i}=0\). Differentiation of \((6_{3,4})\) leads to the results
\[ \omega^N_{\bar N}=\omega^{\bar N}_N=0. \]
Since the vector of normal curvature corresponding to the direction \(\omega^i=\theta\xi^i,\ \omega^{\bar i}=\theta\bar\xi^i\) is equal to \(\Phi/ds^2\), where
\[ \Phi=(\varphi^N\theta^2+\psi^N\bar\theta^2)e_{\bar N} +(\varphi^{\bar N}\bar\theta^2+\psi^{\bar N}\theta^2)e_N, \]
\[ \varphi^N=\Lambda^N_{ij}\xi^i\xi^j,\qquad \psi^N=\Lambda^N_{\bar i\bar j}\bar\xi^i\bar\xi^j,\qquad \varphi^{\bar N}=\Lambda^{\bar N}_{\bar i\bar j}\bar\xi^i\bar\xi^j,\qquad \psi^{\bar N}=\Lambda^{\bar N}_{ij}\xi^i\xi^j, \]
then condition \(2'\) is written in the form of the identity
\[ (\Phi,\Phi)\equiv C\,ds^{4} \tag{7} \]
with respect to \(\theta,\bar{\theta}\), where \(C\) does not depend on \(\theta,\bar{\theta}\).
From the reality of the vector \(\Phi\) and the identity (7) it follows that \(\psi^{N}\equiv \bar{\varphi}^{N}\equiv 0\) identically with respect to \(\xi^{i}\), i.e. \(\Lambda^{N}_{ij}=\Lambda^{\bar N}_{ij}=0\), and, consequently, \(\omega^{\,i}_{N}=\omega^{\,i}_{\bar N}=\omega^{\bar N}_{\,i}=\omega^{N}_{\,i}=0\).
It is now not difficult to verify that condition \(2^\circ\) is satisfied.
Moscow State University
named after M. V. Lomonosov and
Tartu State University
Received
30 XI 1956
CITED LITERATURE
- I. A. Schouten, D. J. Struik, Introduction to the New Methods of Differential Geometry, 2, 1948.
- O. Borůvka, Publ. Fac. Sci. Univ. Masaryk, 214, 1 (1935).
- M. Z. Osipova, Abstract of a Candidate’s dissertation, Moscow City Pedagogical Institute named after Potemkin, 1954.