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Difference between revisions of "Translation surface"

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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Shulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Shulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>
 
 
 
====Comments====
 
 
 
====References====
 
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Chelsea, reprint  (1972)  pp. Sects. 81–84; 218</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''2''' , Springer  (1923)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''3''' , Springer  (1930)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.J. Struik,  "Lectures on classical differential geometry" , Dover, reprint  (1988)  pp. 103; 109; 184</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1–4''' , Chelsea, reprint  (1972)  pp. Sects. 81–84; 218</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''2''' , Springer  (1923)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''3''' , Springer  (1930)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.J. Struik,  "Lectures on classical differential geometry" , Dover, reprint  (1988)  pp. 103; 109; 184</TD></TR></table>

Latest revision as of 10:00, 5 July 2014

A surface formed by parallel displacement of a curve $L_1$ in such a way that some point $M_0\in L_1$ on it slides along another curve $L_2$. If $r_1(u)$ and $r_2(v)$ are the position vectors of $L_1$ and $L_2$, respectively, then the position vector of the translation surface is

$$r=r_1(u)+r_2(v)-r_1(u_0),$$

where $r_1(u_0)=r_2(v_0)$ is the position vector of $M_0$. The lines $u=\text{const}$ and $v=\text{const}$ form a transport net. Each ruled surface has $\infty^1$ transport nets (Reidemeister's theorem), while an enveloping translation surface can be only a cylinder or a plane. If a surface has two transport nets, then the non-singular points of the tangents of the lines in these nets lie on an algebraic curve of order four. An invariant feature of a translation surface is the existence of a conjugate Chebyshev net (a transport net). For example, an isotropic net on a minimal surface is a transport net, thus that surface is a translation surface. One may also characterize a translation surface by the fact that one of its curves (transport lines) passes into a line lying on the same surface as a result of the action of a one-parameter group of parallel displacements. Replacing this group by an arbitrary one-parameter group $G$ leads to generalized translation surfaces [1].

References

[1] V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)
[a1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1–4 , Chelsea, reprint (1972) pp. Sects. 81–84; 218
[a2] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , 2 , Springer (1923)
[a3] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , 3 , Springer (1930)
[a4] D.J. Struik, "Lectures on classical differential geometry" , Dover, reprint (1988) pp. 103; 109; 184
How to Cite This Entry:
Translation surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Translation_surface&oldid=32363
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article