The system of generators of the differential field of all G-invariant differential rational functions of a vector field in the n-dimensional Euclidean space R^n is described for groups G=M(n) and G=SM(n), where M(n) is the group of all isometries of R^n and SM(n) is the group of all euclidean motions of R^n. Using these results, vector field analogues of the first part of the Bonnet theorem for groups Aff(n), M(n), SM(n) in R^n are obtained, where Aff(n) is the group of all affine transformations of R^n. These analogues are given in terms of the first fundamental form and Christoffel symbols of a vector field.
Vector field; Christoffel symbol; Bonnet theorem; Differential invariant
"Complete systems of differential invariants of vector fields in a euclidean space,"
Turkish Journal of Mathematics: Vol. 34:
4, Article 11.
Available at: https://journals.tubitak.gov.tr/math/vol34/iss4/11