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MATHEMATICS
Ya. P. BLANK
ON TRANSLATION SURFACES OF CONSTANT CURVATURE
(Presented by Academician P. S. Aleksandrov on 30 III 1961)
The aim of the present note is to prove the following propositions.
Theorem 1. A translation surface of constant curvature is a cylinder or a plane.
Theorem 2. In elliptic space, a surface of constant curvature \(K\) is simultaneously a sliding surface only when \(K=0\).
Theorem 3. On a surface of constant curvature \(K \ne 0\) there exists no Voss net.
Proof. Refer the translation surface of constant curvature to the conjugate Chebyshev net of translation curves. The coefficients of the differential forms of the surface satisfy the conditions
\[ E=G=1,\qquad F=\cos\omega,\qquad M=0. \tag{1} \]
From the Gauss equations
\[ LN=K\sin^2\omega, \tag{2} \]
\[ \omega_{uv}+K\sin\omega=0 \tag{3} \]
and the Peterson—Codazzi equations
\[ \sin\omega\,L_v=N\omega_u,\qquad \sin\omega\,N_u=L\omega_v. \tag{4} \]
it follows that
\[ L^2+\omega_u^2=U'^2,\qquad N^2+\omega_v^2=V'^2; \tag{5} \]
\[ (U'^2-\omega_u^2)(V'^2-\omega_v^2)=K^2\sin^4\omega. \tag{6} \]
\(U'\) and \(V'\) are the curvatures of the coordinate lines. If at least one of these functions is zero, the translation surface is cylindrical. Excluding this case, we take \(U,V\) as new independent variables. Excluding also the plane, we may assume \(K\ne0\).
Denote the partial derivatives of the function \(\omega\) with respect to the variables \(U,V\) by \(p,q,r,s,t\). In the new variables equations (3), (6) take the form
\[ sU'V' + K\sin\omega=0, \tag{7} \]
\[ \varepsilon\sqrt{1-p^2}\sqrt{1-q^2}=s\sin\omega,\qquad \varepsilon^2=1. \]
Eliminating the functions \(U'\), \(V'\) from system (7) leads to the system
\[ s=\varphi(\omega,p,q), \tag{8} \]
\[ r(1-q^2)+t(1-p^2)=\psi(\omega,p,q), \tag{9} \]
where
\[ \varphi=\frac{\varepsilon}{\sin\omega}(1-p^2)^{1/2}(1-q^2)^{1/2}; \tag{10} \]
\[ \psi=\operatorname{ctg}\omega(1-p^2)(1-q^2)(3p^2+3q^2-4p^2q^2-2) +4\frac{\varepsilon pq}{\sin\omega}(1-p^2)^{3/2}(1-q^2)^{3/2}. \tag{11} \]
The integrability condition of the system (8), (9)
\[ \frac{\partial^2 r}{\partial U\,\partial V}=\frac{\partial^2 s}{\partial U^2} \tag{12} \]
has the following form:
\[ Ar^2+Br+C=0, \tag{13} \]
where
\[ A=10\varphi(1-p^2)^{-2}; \tag{14} \]
\[ B=20\varphi\,\operatorname{ctg}\omega\,\frac{1-2p^2}{1-p^2} -40\frac{pq}{\sin^2\omega}; \tag{15} \]
\[ C=8pq(1-p^2)(4p^2+1)\frac{\cos\omega}{\sin^3\omega} +\frac{8\varphi}{\sin^2\omega}(4p^2+4q^2-2p^2q^2-1)\frac{1-p^2}{1-q^2} +2\varphi\,\operatorname{ctg}^2\omega(24p^4-28p^2+9). \tag{16} \]
From (9) and (13), \(r\) and \(t\) are determined as functions of \(\omega,p,q\), namely:
\[ r=\frac{2}{\sin\omega}(1-p^2)^{3/2}(1-q^2)^{-1/2}(\varepsilon pq+p\sqrt{\Delta}) +\operatorname{ctg}\omega(1-p^2)(2p^2-1); \tag{17} \]
\[ t=\frac{2}{\sin\omega}(1-q^2)^{3/2}(1-p^2)^{-1/2}(\varepsilon pq-p\sqrt{\Delta}) +\operatorname{ctg}\omega(1-q^2)(2q^2-1), \tag{18} \]
where
\[ \Delta=(1-4p^2)(1-4q^2)-(\cos\omega\sqrt{1-p^2}\sqrt{1-q^2}+3\varepsilon pq)^2; \tag{19} \]
\[ \rho^2=1/5. \tag{20} \]
Finally, the integrability conditions
\[ \frac{\partial r}{\partial V}=\frac{\partial s}{\partial U},\qquad \frac{\partial t}{\partial U}=\frac{\partial s}{\partial V} \tag{21} \]
are reduced to the form
\[ \begin{aligned} 4\varepsilon p\sqrt{\Delta}\,[q(1-p^2)+\varepsilon\theta]&=p\sigma(p,q,\theta),\\ -4\varepsilon p\sqrt{\Delta}\,[p(1-q^2)+\varepsilon\theta]&=q\sigma(p,q,\theta), \end{aligned} \tag{22} \]
where
\[ \theta=\cos\omega\sqrt{1-p^2}\sqrt{1-q^2}, \tag{23} \]
\[ \sigma=2-5p^2-5q^2+8p^2q^2-6\varepsilon pq\,\theta-2\theta^2. \tag{24} \]
It is easy to see that \(\Delta\ne0\). Indeed, in the case \(\Delta=0\), from equations (22) it follows that \(\sigma=0\), and from equations (19), (24) that \(\cos^2\omega=1\), which is impossible. Consequently, system (22) is equivalent to the system
\[ p^2+q^2-2p^2q^2+\varepsilon\theta(p+q)=0; \tag{25} \]
\[ 4\varepsilon p\sqrt{\Delta}(q-p)(1+pq)=(p+q)\sigma. \tag{26} \]
From this system \(p,q\) are determined as functions of \(\omega\):
\[ p=p(\omega),\qquad q=q(\omega). \tag{27} \]
But
\[ s=\frac{dp}{d\omega}\,q=\frac{dq}{d\omega}\,p, \tag{28} \]
therefore
\[ p = Cq, \qquad C = \mathrm{const}. \tag{29} \]
From equations (25), (26), (29) it follows that \(p\) and \(q\) are constant, which contradicts the assumption \(K \ne 0\). Theorem 1 is proved.
By the well-known theorem of Bianchi \((^1)\), surfaces of zero curvature in elliptic space are a special case of displacement surfaces, i.e., surfaces admitting a representation \(x = a(u)b(v)\), where \(a, b\) are quaternions.
In \((^2)\) it was proved that from the existence in elliptic space of displacement surfaces of constant curvature \(K \ne 0\) there follows the existence in Euclidean space of translation surfaces of constant curvature \(K\), and conversely. Hence Theorem 2 follows from Theorem 1.
In \((^2)\) it was proved that a Fossa surface of constant curvature \(K \ne 0\) is at the same time a translation surface, and conversely; hence Theorem 3 follows from Theorem 1.
Kharkov State University
named after A. M. Gorky
Received
24 III 1961
CITED LITERATURE
- L. Bianchi, Lezioni di geometria differenziale, 2, Bologna, 1924.
- Ya. P. Blank, Zap. Inst. matem. i mekh. Kharkovsk. gos. univ. i Kharkovsk. matem. obshch., 20, 61 (1950).