Geodesic
Sources of curvature of a vector field
Sbornik. Mathematics, Tome 9 (1969) no. 2, pp. 199-211 
Yu. A. Aminov. Sources of curvature of a vector field. Sbornik. Mathematics, Tome 9 (1969) no. 2, pp. 199-211. http://geodesic.mathdoc.fr/item/SM_1969_9_2_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is known that for a vector field in three-dimensional space we can introduce the concepts of curvature and mean curvature. In the present article we derive integral formulas for these concepts; these formulas allow us to decide whether a vector field has, for example, singularities in a domain. We explain the influence of the modulus of the curvature of a vector field on the magnitude of its nonholonomity. We also consider the question of the influence of the curvature of a family of surfaces on the distortion of the enveloping space for a given size of domain. Bibliography: 5 titles.

[1] D. M. Sintsov, “O sisteme integralnykh krivykh Pfafova uravneniya $Pdx+Qdy+Rdz=0$”, Nauk. zap. n-d. mat. kaf. Ukr., III, 1928, 107–148

[2] S. S. Byushgens, “Geometriya vektornogo polya”, Izv. AN SSSR, seriya matem., 10 (1946), 73–96

[3] N. V. Efimov, “Issledovanie odnoznachnoi proektsii poverkhnosti otritsatelnoi krivizny”, DAN SSSR, 93:4 (1953), 609–611 | MR | Zbl

[4] V. V. Vagner, “Geometricheskaya interpretatsiya vektora krivizny semeistva ploschadok v $\mathbf{R}^3$”, Matem. sb., 4(46) (1938), 339–356

[5] Yu. A. Aminov, “Divergentnye svoistva krivizn vektornogo polya i semeistva poverkhnostei”, Matem. zametki, 3:1 (1968), 103–111 | MR | Zbl