Abstract: | For a submanifoldM
n of a Riemannian manifoldM
q, the concept of a torsion bivector at the point x M
n for given one- and two-dimensional directions fromT
x
M
n is introduced using only the first and second fundamental forms ofM
n. Its relation to the concept of Gaussian torsion is then established. It is proved that: 1) equality to zero of the torsion bivector is necessary and, whenM
n is a nondevelopable surface of a space of constant curvature with nonzero second fundamental form, is also sufficient for the "flattening" ofM
n into some totally geodesicM
n+1 inM
q; 2) when n = 2, the independence of the nonzero torsion bivector of direction characterizes a minimalM
2 inM
q.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 39–42, 1991. |