Abstract
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Reports of the Academy of Sciences of the USSR
- Volume 114, No. 4
MATHEMATICS
Yu. G. Lumiste
On Surfaces \(V_n\) with Multidimensional Isotropic Conjugate Directions in Spaces \(R_N\) or \(S_N\)
(Presented by Academician P. S. Aleksandrov on 22 XII 1956)
1. A surface \(V_n\) in Euclidean space \(R_N\) or in non-Euclidean space \(S_N\) is called a surface with a complete tangent system of isotropic conjugate directions (a surface with c.t.s.i.c.d.) if its tangent plane at any point \(M\) contains completely isotropic directions \(I^\varkappa\) of dimensions \(p_\varkappa\) \((\varkappa=1,\ldots,k)\),
\[ \sum_{\varkappa=1}^{k} p_\varkappa = n, \]
which in aggregate do not lie in a plane of dimension \(m<n\) and are pairwise weakly conjugate (for the notion of weak conjugacy see \((^1)\)).
In the case \(n=2\), as is known, the class of such surfaces coincides with the class of minimal \(V_2\), and every surface of it is a surface of translation (in non-Euclidean spaces, in the sense of parallel translation) of its isotropic curves.
In the present article some results are obtained in the general case.
2. A moving frame is attached to a surface \(V_n\) with c.t.s.i.c.d. at the point \(M\) so that the vectors* \(e_{i_\varkappa}\) lie in the directions \(I^\varkappa\), and the vectors \(e_\alpha\) lie in the plane normal to \(V_n\). Then \(dM=\omega^a e_a\) \((a,b,\ldots=1,\ldots,n)\), i.e.
\[ \omega^\alpha=0. \tag{1} \]
Continuing the system (1), we obtain the equations
\[ \omega_a^\alpha=\Lambda_{ab}^{\alpha}\omega^b,\qquad \Lambda_{ab}^{\alpha}=\Lambda_{ba}^{\alpha}. \tag{2} \]
Since the \(I^\varkappa\) are pairwise conjugate,
\[ \Lambda_{i_\varkappa j_\lambda}^{\alpha}=0\qquad(\lambda\ne\varkappa). \tag{3} \]
Among the vectors \(e_{ab}=\Lambda_{ab}^{\alpha}e_\alpha\), \(e_{ab}=e_{ba}\), determining the \(n_1\)-dimensional first normal plane, only the vectors \(e_{i_\varkappa j_\varkappa}\) are different from zero, and
\[ n_1 \leq \sum_{\varkappa=1}^{k}\frac{p_\varkappa(p_\varkappa+1)}{2}. \]
3. Let the directions \(I^\varkappa\) be divided into the largest groups in such a way that the directions of one group lie in the smallest common plane forming an isotropic cone \(J^\rho\) \((\rho=1,\ldots,r)\).
Theorem. A nonisotropic surface \(V_n\) with c.t.s.i.c.d. and with maximal \(n_1\) in a non-Euclidean space exists only for \(n=2\).
* In the non-Euclidean case, a “vector” is any analytic point lying on the polar of the point \(M\) with respect to the absolute of the space.
In Euclidean space it also exists for \(n>2\) and is a translation surface of its completely isotropic submanifolds, enveloped by directions \(J^\rho\), which, in turn, are foliated into families of submanifolds enveloped by directions \(I^\chi\).
Proof. In the case of maximal \(n_1\), the vectors \(e_{i_\chi j_\chi}\) may be taken as frame vectors. If, in doing so, the index ranging over values in the first normal plane is replaced by the unordered pair of indices \(i_\chi j_\chi\), then equations (2) are written in the form
\[ \omega_{i_\chi}^{\,i_\chi j_\chi}=\omega^{j_\chi}, \tag{4¹} \]
\[ \omega_{i_\chi}^{\,j_\chi k_\chi}=0 \quad (j_\chi,k_\chi\ne i_\chi), \tag{4²} \]
\[ \omega_{i_\lambda}^{\,j_\chi k_\chi}=0 \quad (\lambda\ne\chi). \tag{4³} \]
If one exteriorly differentiates equations (4¹) for \(j_\chi=i_\chi\) and (4³), or \(k_\chi=j_\chi\), and applies Cartan’s lemma \((^2)\), one obtains, in particular,
\[ \omega_{j_\lambda}^{\,i_\chi} = \Gamma_{j_\lambda i_\chi}^{\,i_\chi}\,\omega^{i_\chi} + \Gamma_{j_\lambda k_\lambda}^{\,i_\chi}\,\omega^{k_\lambda}, \qquad \Gamma_{j_\lambda k_\lambda}^{\,i_\chi} = \Gamma_{k_\lambda j_\lambda}^{\,i_\chi}, \quad (\lambda\ne\chi). \tag{5} \]
The system \(\omega^{i_\chi}=0\) (\(\chi\) fixed) is now completely integrable; the left-hand sides may be represented in the form
\[ \omega^{i_\chi}=a_{j_\chi}^{\,i_\chi}\,du^{j_\chi}, \tag{6} \]
and the surface \(V_n\), consequently, is completely foliated along the directions \(I^\chi\).
In view of the complete isotropy of the directions \(J^\rho\), \(g_{i_{\chi_\rho}j_{\lambda_\rho}}=0\). A differential consequence of this is
\[ \sum_{\sigma\ne\rho} \left( g_{k_{\mu_\sigma}j_{\lambda_\rho}}\, \omega_{i_{\chi_\rho}}^{\,k_{\mu_\sigma}} + g_{i_{\chi_\rho}k_{\mu_\sigma}}\, \omega_{j_{\lambda_\rho}}^{\,k_{\mu_\sigma}} \right)=0. \]
If here one takes \(j_{\lambda_\rho}=i_{\chi_\rho}\) and substitutes expression (5), then, in particular, since \(g_{k_{\mu_\sigma}i_{\chi_\rho}}\ne0\), it follows that \(\Gamma_{i_{\chi_\rho}k_{\mu_\sigma}}^{\,k_{\mu_\rho}}=0\), and
\[ \omega_{i_{\chi_\rho}}^{\,j_{\lambda_\sigma}} = \Gamma_{i_{\chi_\rho}k_{\mu_\rho}}^{\,j_{\lambda_\sigma}}\, \omega^{k_{\mu_\rho}}, \qquad \Gamma_{i_{\chi_\rho}k_{\mu_\rho}}^{\,j_{\lambda_\sigma}} = \Gamma_{k_{\chi_\rho}i_{\chi_\rho}}^{\,j_{\lambda_\sigma}}, \quad (\sigma\ne\rho). \tag{7} \]
Exterior differentiation of equations (7) gives quadratic equations of the form
\[ K\sum_{\tau\ne\rho} g_{i_{\chi_\rho}l_{\mu_\tau}}^{\,\sigma} \,[\omega^{l_{\mu_\tau}}\omega^{j_{\lambda_\sigma}}] + [\omega^{k_{\mu_\rho}}\Omega_{k_{\mu_\rho}}^{\,j_{\lambda_\sigma}}]=0 \quad (\sigma\ne\rho), \]
where \(K\) is the curvature of the space. For these equations to be consistent it is necessary that either \(K=0\) (i.e. the space is Euclidean), or \(n=2\) (i.e. the surface is a minimal \(V_2\)).
In view of (6), \(dM=M_{j_\chi}du^{j_\chi}\), \(M_{j_\chi}=a_{j_\chi}^{\,i_\chi}e_{i_\chi}\). If the \(M_{j_\chi}\) are taken as frame vectors, then \(dM_{i_{\chi_\rho}}\) is expressed, in the case of Euclidean space, only through the differentials \(du^{k_{\lambda_\rho}}\) (\(\rho\) fixed), and therefore
\[ \frac{\partial^2 M}{\partial u^{i_{\chi_\rho}}\,\partial u^{k_{\mu_\sigma}}}=0 \quad (\sigma\ne\rho), \qquad M=\sum_\rho X^\rho\!\left(u^{i_{\chi_\rho}}\right). \]
One can indicate examples proving the existence of the surfaces considered in the theorem in Euclidean space \(R_N\). Let completely isotropic planes \(I_{m_\rho}\) be chosen in \(R_N\), lying together in the plane
\[ R_m,\qquad m=\sum_\rho m_\rho < N. \]
Each \(I_{m_\rho}\) may be regarded as an immersed affine space, and any surface in it is completely isotropic with respect to \(R_N\). According to the results of V. V. Ryzhkov \((^1)\), in each \(I_{m_\rho}\) there exists a surface \(X^\rho\) with c.c.i.c.n. and with the maximal admissible dimension of the first osculating plane. The translation surface \(V_n\) of the surfaces \(X^\rho\) is the desired example.
Knowledge of the arbitrariness of the translated completely isotropic surfaces \(X^\rho\) makes it possible to determine also the arbitrariness of the surface \(V_n\). It is known \((^3)\) that a general completely isotropic surface \(X_n\) in \(R_N\) exists for
\[ N \ge \frac{n(n+3)}{2} \]
with arbitrariness
\[ N-\frac{n(n+3)}{3} \]
functions of \(n\) arguments. We have established that a completely isotropic \(X_n\) with c.c.i.c.n. for \(p_1=\cdots=p_k=1\) and maximal \(n_1\) (the so-called \(n\)-conjugate system) exists in \(R_N\) only in the case \(N\ge 3n\). For \(N=3n\) it is determined with arbitrariness
\[ \frac{n(n-1)}{2} \]
functions of 2 arguments; as \(N\) increases, the arbitrariness increases, reaching for \(N=4n\) the maximal arbitrariness \(n(n-1)\) functions of 2 arguments.
- Minimal non-isotropic surfaces \(V_n\) in \(R_N\) or \(S_N\) are characterized, as is known \((^4)\), by the vanishing of the so-called mean-curvature vector
\[ g^{ab}\Lambda^\alpha_{ab}=0. \]
It turns out that a surface \(V_n\) with c.c.i.c.n. must be minimal only in the case when there are only two directions \(J^1, J^2\) of equal dimension. In this case the surface possesses certain properties generalizing the properties of minimal \(V_2\) in \(R_3\).
\(1^\circ\). The principal curvatures with respect to any normal direction differ pairwise only by sign.
Indeed, the principal curvatures are the solutions of the equation
\[ \operatorname{Det}\left|\sigma_\alpha\Lambda^\alpha_{ab}-\lambda g_{ab}\right|=0, \]
in which, since \(g_{i_1j_1}=g_{i_2j_2}=\Lambda^\alpha_{i_1j_2}=0\), the coefficients of all odd powers of the unknown \(\lambda\) vanish.
\(2^\circ\). When the point \(M\) is displaced in any direction from \(J^1\), every normal direction deviates in a direction belonging to \(J^2\). Conversely, if some non-isotropic surface of full rank \((^5)\) possesses this property mutually for \(J^1, J^2\), then the requirements: a) \(J^1, J^2\) are isotropic, b) \(J^1, J^2\) are weakly conjugate, are equivalent.
Indeed, one must consider the formula
\[ de_\alpha=\omega_\alpha^{i_1}e_{i_1}+\omega_\alpha^{i_2}e_{i_2}+\omega_\alpha^\beta e_\beta, \]
where
\[ \omega_\alpha^{i_1} = -g_{\alpha\beta} \left\{ g^{i_1j_1}\left(\Lambda^\beta_{j_1k_1}\omega^{k_1} + \Lambda^\beta_{j_1k_2}\omega^{k_2}\right) + g^{i_1j_2}\left(\Lambda^\beta_{j_2k_1}\omega^{k_1} + \Lambda^\beta_{j_2k_2}\omega^{k_2}\right) \right\}. \]
Any two of the conditions
\[ \Lambda^\beta_{j_1k_2}=0,\qquad g^{i_1j_1}=0,\qquad \omega_\alpha^{i_1}\equiv 0 \pmod{\omega^{k_2}} \]
imply the third.
The last property generalizes the known property of the conformality of the spherical image of a minimal \(V_2\) in \(R_3\), which does not carry over directly to the case of a hypersurface \((^6)\).
For maximal \(n_1\), the existence and structure of a surface with c.c.i.c.n. in the case of two directions \(J^1, J^2\) of equal dimension is determined by the theorem proved. In another simplest case, when
$n_1=11$; such surfaces, by Segre’s theorem$^{(4)}$ and the nonexistence of a minimal surface of rank 1, are hypersurfaces. Hypersurfaces $V_{2p}$ with m.d.i.c.d. for $k=2$, $p_1=p_2=p$ exist both in $R_{2p+1}$ and in $S_{2p+1}$ with an arbitrary $2p$ functions of $p$ arguments, and in the general case are not translation surfaces.
In conclusion, the author expresses his sincere gratitude to his supervisor A. M. Vasil’ev, to whom he is indebted for the formulation of the problem and for valuable guidance.
Moscow State University
named after M. V. Lomonosov
Received
21 XII 1956
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