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A doubly-connected surface $ S $ of annular type, containing a planar closed curve $ L $ the plane $ P $ of which is tangent to $ S $ at all points of $ L $. Along $ L $ the Gaussian curvature $ K $ of $ S $ vanishes. If, in these circumstances, $ L $ divides $ S $ into two parts on each of which $ K $ is of constant sign, then the corresponding parts of $ S $ are called the positive and negative semi-trough. An example of a trough is a narrow band of a torus along one of its closed parabolic parallels.

A trough occupies an intermediate position between the objects of "global" and "local" geometry, since, because it contains a specific closed curve $ L $, it cannot be arbitrarily small, while its dimensions in directions transversal to $ L $ may be arbitrary small. Interest in the study of troughs is aroused by the fact that a sufficiently narrow band of surfaces of alternating curvature along a closed parabolic line is often a trough, and for this reason, knowing the properties of troughs under various deformations sometimes enables one to obtain information on the corresponding properties of the surface "in the large" .

The most detailed research has been carried out into the so-called planar troughs (for which the curve $ L $ is convex, while the trough $ S $ itself is positioned along one side of $ P $, with which it has a contact of the first order) and troughs of rotation (when $ L $ is a parallel of the surface of rotation). For analytic planar troughs, their rigidity relative to analytic infinitesimal deformations of the second order has been proved. For troughs of rotation, the study of their infinitesimal deformations of the first and second orders has been extended to the regularity class $ C ^ {1} $. From the point of view of differential equations, research into troughs reduces to the study of equations of mixed type.


[1] N.V. Efimov, "Quantitative problems of the theory of deformation of surfaces" Transl. Amer. Math. Soc. (1) , 6 (1962) pp. 274–423 Uspekhi Mat. Nauk , 3 : 2 (1948) pp. 47–158
[2a] S.E. Cohn-Vossen, "Bending of surfaces in the large" Uspekhi Mat. Nauk , 1 (1936) pp. 33–76 (In Russian)
[2b] S.E. Cohn-Vossen, "Die parabolische Kurve" Math. Ann. , 99 (1928) pp. 273–308
[3a] E. Rembs, "Ueber die Verbiegbarkeit der Rinnen" Math. Z. , 71 : 1 (1959) pp. 89–93
[3b] E. Rembs, "Ueber die Verbiegbarkeit der Rinnen II" Mat. Z. , 73 : 4 (1960) pp. 330–332
[4a] I.Kh. Sabitov, "On infinitesimal bending of troughs of revolution I" Math. USSR Sb. , 27 (1975) pp. 103–117 Mat. Sb. , 98 : 1 (1975) pp. 113–29
[4b] I.Kh. Sabitov, "On infinitesimal bending of troughs of revolution II" Math. USSR Sb. , 28 (1976) pp. 41–48 Mat. Sb. , 99 : 1 (1976) pp. 49–57


The original (German) term for "trough" is "Flächenrinne" . In the translated Russian literature it also occurs as belt.

How to Cite This Entry:
Trough. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article