Trough

A doubly-connected surface of annular type, containing a planar closed curve the plane of which is tangent to at all points of . Along the Gaussian curvature of vanishes. If, in these circumstances, divides into two parts on each of which is of constant sign, then the corresponding parts of 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 , it cannot be arbitrarily small, while its dimensions in directions transversal to 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 is convex, while the trough itself is positioned along one side of , with which it has a contact of the first order) and troughs of rotation (when 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 . From the point of view of differential equations, research into troughs reduces to the study of equations of mixed type.

References

[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

Comments

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