Abstract
Full Text
Ya. P. Blank
On Surfaces Carrying (\infty^2) Conical Nets
(Presented by Academician P. S. Aleksandrov, 1 XII 1959)
A Peterson surface, i.e. a surface carrying a conical net, admits the parametrization
[
x_i=A_i(u)+B_i(v)\qquad (i=1,2,3,4).
\tag{1}
]
A conical net is generated by the pair of curves
[
x_i=\frac{dA_i}{du},\qquad x_i=\frac{dB_i}{dv},
\tag{2}
]
formed by the vertices of the cones of the net.
Translation surfaces constitute the special case of Peterson surfaces when the curves generating the net are improper ((A'_4=B'_4=0)). S. Lie ((^1)) proved that translation surfaces carrying (\infty^1) nets of translation are generated by a pencil of curves of the second order.
The purpose of the present note is to prove the following analogue of Lie’s theorem.
Theorem. Peterson surfaces carrying (\infty^2) conical nets are generated by pairs of curves lying on surfaces of the second order belonging to one pencil; moreover, on each surface of the pencil there are (\infty^1) pairs of these curves.
Proof. In the papers ((^{2,3})) it was established that the surfaces carrying (\infty^2) conical nets are exhausted, up to collineations (real or imaginary), by the following types:
[
1^0.\ zx=ty^n,\qquad
2^0.\ \frac{z}{t}=\operatorname{arc\,tg}\frac{y}{x}.
\tag{3}
]
[
3^0.\ x^3+yt^2-xzt=0,\qquad
4^0.\ \frac{z}{t}=\frac{y}{x}+\lg\frac{x}{t},
]
Substituting in the equations of these surfaces, in place of the coordinates, their parametric expressions according to formulas (1), we obtain equations containing the unknown functions (A_i(u), B_i(v)). The solution of these functional equations may be represented by the following table:
| Type | Canonical representations of the Peterson surface |
|---|---|
| (1^0) | (\alpha u-\beta v,\quad \dfrac{\alpha}{v}-\dfrac{\beta}{u},\quad \dfrac{a}{v^n}-\dfrac{b}{u^n},\quad au^n-bv^n) |
| (2^0) | (a(\sin v-\sin u)-b(\cos v+\cos u),\quad a(\cos u-\cos v)-b(\sin u+\sin v),) (\alpha(u+v)+\beta(u^2-v^2),\quad 2\beta(u-v)+2\alpha) |
| (3^0) | (a(u^2-v^2)+b(u+v),\quad a(u^4-v^4)+2b(u^3+v^3)+\alpha(u^2-v^2)+\beta(u+v),) (2a(u^3-v^3)+3b(u^2+v^2)+\alpha(u-v)+\beta,\quad a(u-v)+b) |
| (4^0) | (\dfrac{a}{v}-\dfrac{b}{u},\quad \dfrac{\beta}{u}-\dfrac{\alpha}{v}+\dfrac{a}{v}\lg v-\dfrac{b}{u}\lg u,) (\beta v-\alpha u+bv\lg v-au\lg u,\quad au-bv) |
In case (1^\circ), the curves generating the net:
[
\begin{gathered}
\alpha,\quad \frac{\beta}{u^2},\quad \frac{nb}{u^{n+1}},\quad nau^{n-1},\
\beta,\quad \frac{\alpha}{v^2},\quad \frac{na}{v^{n+1}},\quad nbv^{n-1},
\end{gathered}
]
are situated on the surfaces of the second order of the pencil
[
\alpha\beta zt=abn^2xy
]
and are determined by the intersection of these surfaces with the surfaces
[
\begin{gathered}
a\beta^n zx^n=b\alpha^n ty^n,\
b\alpha^n zx^n=a\beta^n ty^n.
\end{gathered}
]
For (ba^n=a\beta^n), the curves lie on the surface (1^\circ) itself and serve as its asymptotic lines (non-rectilinear).
In case (2^\circ), the curves generating the net:
[
\begin{gathered}
b\sin u-a\cos u,\quad -a\sin u-b\cos u,\quad \alpha+2\beta u,\quad 2\beta,\
a\cos v+b\sin v,\quad a\sin v-b\cos v,\quad \alpha-2\beta v,\quad -2\beta,
\end{gathered}
]
are situated on the surfaces of the second order of the pencil
[
4\beta^2(x^2+y^2)=(a^2+b^2)t^2
]
and are determined by the intersection of these surfaces with the surfaces
[
\frac{z}{t}=\operatorname{arc\,tg}\frac{y}{x}\pm\left(\frac{\alpha}{2\beta}-\operatorname{arc\,tg}\frac{b}{a}\right).
]
For
[
\frac{\alpha}{2\beta}=\operatorname{arc\,tg}\frac{b}{a},
]
the curves lie on the helicoid (2^\circ) itself and serve as its asymptotic lines (non-rectilinear).
The translation nets of the helicoid are obtained for (\beta=0).
In case (3^\circ) (Cayley surface of the 3rd order), the curves generating the net:
[
\begin{gathered}
2au+b,\quad 4au^3+6bu^2+2\alpha u+\beta,\quad 6au^2+6bu+\alpha,\quad a,\
2av-b,\quad 4av^3-6bv^2+2\alpha v-\beta,\quad 6av^2-6bv+\alpha,\quad a,
\end{gathered}
]
are situated on the surfaces of the second order of the pencil
[
a^2(3x^2-2zt)=(3b^2-2a\alpha)t^2
]
and are determined by the intersection of these surfaces with the surfaces
[
a^3(x^3+yt^2-xzt)\pm(ab\alpha-a^2\beta-b^2)t^2=0.
]
For
[
ab\alpha-a^2\beta-b^3=0,
]
the curves are situated on the surface (3^\circ) itself and serve as its asymptotic lines (non-rectilinear).
The translation nets of the Cayley surface are obtained for (a=0).
In case (4^\circ), the curves generating the net:
[
\begin{gathered}
\frac{a}{u^2},\quad \frac{b\lg u-b-\beta}{u^2},\quad -a\lg u-a-\alpha,\quad a,\
\frac{a}{v^2},\quad \frac{a\lg v-a-\alpha}{v^2},\quad -b\lg v-b-\beta,\quad b,
\end{gathered}
]
are located on the second-order surfaces of the pencil
[
xz+yt+\left(\frac{\alpha}{a}+\frac{\beta}{b}+2\right)xt=0
]
and are determined by the intersection of these surfaces with the surfaces
[
\frac{z}{t}=-\frac{y}{x}+\lg\frac{x}{t}\pm\left(\frac{\alpha}{a}-\frac{\beta}{b}+\lg\frac{b}{a}\right).
]
For
[
\frac{\alpha}{a}-\frac{\beta}{b}+\lg\frac{b}{a}=0
]
the curves are located on the surface (4^\circ) itself and serve as its asymptotic lines (non-rectilinear).
In cases (1^\circ) and (4^\circ) one of the second-order surfaces of the pencil may be taken as the absolute of Lobachevsky space; consequently, these surfaces realize the hyperbolic analogue of surfaces of translation with (\infty^1) nets of translation, in the sense that they carry (\infty^1) conjugate nets composed of cylindrical lines (lines of tangency of the surface with a ruled surface formed by Lobachevsky parallels).
In [4] these surfaces were determined by the coefficients of both differential forms. Here we have their equations in finite form.
Kharkov State University
named after A. M. Gorky
Received
20 XI 1959
REFERENCES
- S. Lie, Geometrie der Berührungstransformationen, Leipzig, 1896.
- Ya. P. Blank, DAN, 64, No. 6 (1949).
- Ya. P. Blank, Notes of the Department of Physics and Mathematics, Kharkov State University and Kharkov Mathematical Society, 23, 113 (1952).
- Ya. P. Blank, ibid., 25, 35 (1957).