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Reports of the Academy of Sciences of the USSR
- Volume 140, No. 5
MATHEMATICS
G. S. BARKHIN and V. T. FOMENKO
ON THE BENDING OF SURFACES OF POSITIVE CURVATURE UNDER CERTAIN BOUNDARY CONDITIONS
(Presented by Academician P. S. Aleksandrov, 18 V 1961)
The question of the existence of surfaces isometric to a given one reduces, as is well known, to the problem of the existence of solutions of the system of Gauss and Peterson–Codazzi equations, where, for a prescribed metric, the role of the unknown functions is played by the coefficients of the second quadratic form. On a surface with boundary, by prescribing some characteristic of the contour (curvature, geodesic torsion), we subject the sought functions to an additional boundary condition. If the resulting boundary-value problem admits a solution with a known arbitrariness, then one may speak of a bending of the surface isometric to the original one. The latter, in this case, need not belong to a continuous family of isometric surfaces.
In the present note we consider certain conditions for the existence of solutions of the indicated boundary-value problem and the resulting criteria for the nonbendability of pieces of surfaces of positive curvature.
Let a piece of a surface \(S\) of positive Gaussian curvature, belonging to the class \(C_\alpha^3\) \((\alpha < 1)\), satisfy the following conditions:
1) An isometrically conjugate parametrization of \(S\) maps it homeomorphically onto a domain \(D\) (generally speaking, \((m+1)\)-connected) with boundary
\(L = L_1 + \ldots + L_{m+1}\)
of the parametric plane.
2) The domain \(D\), without loss of generality, will henceforth be assumed canonical.
Let \(\Delta L, \Delta M, \Delta N\) be the increments of the coefficients of the second form, and \(\Delta k_n, \Delta \tau_g\), respectively, those of the normal curvature and the geodesic torsion of the boundary under passage to an isometric surface. Introduce new unknown functions \(U, V, \Pi\), related to \(\Delta L, \Delta M, \Delta N\) by the formulas
\[ \Delta L = \Pi + V, \qquad \Delta M = U, \qquad \Delta N = \Pi - V. \tag{1} \]
The Codazzi equations will then be equivalent to the following system:
\[ U_x - V_y = \alpha U + \beta V + L_0 \left(\frac{\Pi}{L_0}\right)_y, \]
\[ U_y + V_x = \gamma U + \delta V + L_0 \left(\frac{\Pi}{L_0}\right)_x; \tag{2} \]
where \(\alpha, \beta, \gamma, \delta\) are known functions, invariant under bending, and \(L_0\) is the coefficient of the second form of the original surface.
As a consequence of the Gauss equation, the functions \(U, V, \Pi\) must satisfy the algebraic relation
\[ U^2 + V^2 - \Pi^2 = 2L_0\Pi. \tag{3} \]
Introduce on the contour two functions \(\lambda(r,\varphi)\) and \(\mu(r,\varphi)\), belonging to the class \(C_\alpha^1(L)\), with the aid of which, using the contour relation of the sought functions
with the quantities \(\Delta k_n\) and \(\Delta \tau_g\), we obtain the following boundary condition:
\[
U\left(-\mu \frac{g}{e}+\lambda \sin 2\varphi+\mu \cos 2\varphi\right)
+V\left(\mu \frac{f}{e}+\lambda \cos 2\varphi-\mu \sin 2\varphi\right)
\]
\[
=\Pi\left(\mu \frac{f\cos 2\varphi-g\sin 2\varphi}{e}+\lambda\right)
-p\left(\frac{\sqrt a}{e}\mu\Delta\tau_g+\lambda\Delta k_n\right).
\tag{4}
\]
Here
\[ a=EG-F^2,\qquad p(r,\varphi)=E\sin^2\varphi-2F\sin\varphi\cos\varphi+G\cos^2\varphi, \]
\[ e=\frac{1}{2}(E+G),\qquad g=\frac{1}{2}(E-G),\qquad f=F. \]
Theorem 1. Let \(D\) be a canonical domain; \(L\) its boundary; let \(\lambda(r,\varphi)\), \(\mu(r,\varphi)\), \(\sigma(r,\varphi)\) be functions prescribed on \(L\), belonging to \(C_\alpha^1(L)\) and satisfying the conditions:
\[
\varkappa=\operatorname{Ind}(\mu+i\lambda)>-(m-1),\qquad
C_\alpha^1(\sigma)<\varepsilon
\]
(\(\varepsilon\) is a sufficiently small positive constant, the norm being defined as in \((^1)\)).
Then the system of equations (2), (3), (4) has precisely a \((2\varkappa-3+3m)\)-parameter family of solutions belonging to \(\overline C_\alpha^1(D+L)\) such that
\[ \frac{\sqrt a}{e}\mu(r,\varphi)\Delta\tau_g(r,\varphi) +\lambda(r,\varphi)\Delta k_n(r,\varphi)=\sigma(r,\varphi). \tag{5} \]
Proof. By the methods of \((^1)\) one can obtain the general solution
\[
w(x,y,\Pi)=U(x,y,\Pi)+iV(x,y,\Pi)
\]
of the system (2), (4), in which the function \(\Pi(x,y)\in C_\alpha^1(D+L)\) is regarded as given. The latter must be such that the integro-differential equation, obtained by substituting \(w(x,y)\) into (3), with respect to the now unknown function \(\Pi(x,y)\),
\[ |w|^2-\Pi^2=2L_0\Pi \tag{6} \]
is solvable. The solvability of (6) is proved by the method of successive approximations under the assumption \(\Pi^0=0\). The solution of (6) is unique.
Corollary. For a simply connected piece of a surface, under the conditions of Theorem 1, there exists a \((2\varkappa-3)\)-parameter family of isometric surfaces satisfying the boundary condition (5).
Theorem 2. Let \(\lambda(r,\varphi)\), \(\mu(r,\varphi)\subset C_\alpha^1(L)\) satisfy the condition
\[
\operatorname{Ind}(\mu,\lambda)<-2(m-1).
\]
For the piece \(S\) to be non-bendable, it is sufficient that the increment of some linear combination of \(k_n\) and \(\tau_g\), prescribed on the contour, be equal to zero:
\[ \frac{\sqrt a}{e}\mu(r,\varphi)\Delta\tau_g(r,\varphi) +\lambda(r,\varphi)\Delta k_n(r,\varphi)=0. \]
The assertion of the theorem follows from the fact that, for
\[
\varkappa<-2(m-1)
\]
and \(\sigma(r,\varphi)=0\), equation (6) has the unique solution \(\Pi\equiv 0\), and the system (2), (4) admits no solutions other than the trivial ones.
In particular, for \(m=0\) one obtains sufficient conditions for non-bendability:
a) \(\Delta\tau_g=0\) (K. M. Belov \((^3)\)),
b) \(\Delta k_n=\Delta\tau_g\),
c) \(\Delta k_n=0\). The last case we formulate in the form of the following theorem:
Theorem 3. If between two pieces of surfaces of positive curvature of class \(C_\alpha^3\) an isometric correspondence is established, under which the curvature of corresponding boundary points is the same, then such surfaces are congruent or symmetric.
The condition \(\Delta k=0\) can be realized by means of a construction consisting of two surfaces of positive curvature glued along a common edge in such a way that the gluing angle, different from zero at every point, does not change under continuous isometric deformations. Such a construction is uniquely determined. Moreover, the theorem on ova-
to people with a hemmed edge for \(m=0\) \((^1)\) carries over to continuous bendings.
By the method presented above, relying on Nitsche’s theorem \((^4)\), one can prove the following theorem:
Theorem 4. Let \(\lambda(\varphi), \mu(\varphi) \subset C_\alpha^1(L)\) be prescribed on the contour, satisfying the condition \(\chi=\operatorname{Ind}(\mu+i\lambda)\ge 2\), and
\[ h_k(\varphi)=1,\cos\varphi,\sin\varphi,\cos 2\varphi,\sin 2\varphi,\ldots . \]
For a simply connected piece of a surface \(S\) of positive curvature of class \(C_\alpha^3\) to be rigid, it is necessary and sufficient that the following conditions be satisfied on its contour:
\[ \Delta\left[\frac{\sqrt a}{e}\mu(\varphi)\tau_g(\varphi)+\lambda(\varphi)k_n(\varphi)\right]=0, \]
\[ \int_0^{2\pi} \frac{\chi(\varphi)}{\sqrt{\lambda^2+\mu^2}} \left[ \mu(\varphi)\Delta k_n(\varphi) -\lambda(\varphi)\frac{\sqrt a}{e}\Delta\tau_g(\varphi) \right] h_k(\varphi)\,d\varphi=0 \]
\[ (k=1,\ldots,2\chi-3), \]
where
\[ \chi(\varphi)= \frac{E\sin^2\varphi-2F\sin\varphi\cos\varphi+G\cos^2\varphi} {\left[-\sin(\theta-2\varphi)+\frac{g}{e}\sin\theta\right]^2 +\left[\cos(\theta-2\varphi)-\frac{f}{e}\sin\theta\right]^2}, \]
\[ \sin\theta=\frac{\lambda}{\sqrt{\lambda^2+\mu^2}}, \qquad \cos\theta=\frac{\mu}{\sqrt{\lambda^2+\mu^2}}. \]
Finally, let us consider the bending of a belt \(\widetilde S\) of a surface of revolution of the second order of positive curvature, under the condition that the geodesic torsion is prescribed on the edge \((\lambda=0,\ \mu=1)\). The case of bending a spherical belt into a surface with a prescribed constant normal curvature of the edge \((\lambda=1,\ \mu=0)\) was investigated by Nitsche \((^5)\).
Theorem 5. Let the belt \(\widetilde S\), mapped onto the annulus \(D\) with boundary \(L=L_1+L_2\), satisfy the following requirements:
\[ \Delta\tau_g(\varphi)\in C_\alpha'(L),\qquad \Delta k_{n_2}(\varphi)\in C_\alpha'(L_2), \]
\[ \left|\int_{-\pi}^{\pi}\Delta k_{n_1}(\varphi)\,d\varphi\right|<\varepsilon, \qquad C_\alpha'(\Delta\tau_g)<\varepsilon, \]
where \(\varepsilon\) is a sufficiently small positive number. For there to exist no more than a one-parameter family of surfaces, isometric to \(\widetilde S\), for which the geodesic torsion of the edge assumes a prescribed value \(\tau_g(r,\varphi)\), it is necessary and sufficient that the contour condition be fulfilled
\[ \sqrt{f_1(1)f_2(1)} \int_{-\pi}^{\pi}\sqrt[4]{K_1}\,\Delta\tau_{g_1}(\varphi)\,d\varphi = q^4\sqrt[4]{f_1(q)f_2(q)} \int_{-\pi}^{\pi}\sqrt[4]{K_2}\,\Delta\tau_{g_2}(\varphi)\,d\varphi, \]
where \(f_i(r)\) are known functions of \(r=\sqrt{x^2+y^2}\), invariant under bending, \(r_1=1,\ r_2=q<1\), and \(K_i\) is the Gaussian curvature of the surface along the corresponding contour.
Rostov-on-Don
State University
Received
31 III 1961
CITED LITERATURE
- I. N. Vekua, Generalized Analytic Functions, 1959.
- Joachim Nitsche, Arch. Math., 4, No. 4, 331 (1953); 6, No. 1, 13 (1954); 6, No. 2, 143 (1955).
- K. M. Belov, DAN, 127, No. 2 (1959).
- Johannes Nitsche, Arch. Math., 6, 18 (1955).
- Joachim Nitsche, Math. Zs., 62, 386 (1955).