The Bianchi identities in an explicit form (Q1108590)

Let v(x) be a given vector field in Euclidean 3-space \(E_ 3\), while s(x) is the unit vector tangent to the vector-lines of v, and n is the unit principal normal to this vector-line. The unit bi-normal is \(b=s\times n\). The components of grad s, grad n, grad b on the basis (s,n,b) are connected by nine compatibility conditions contained in the relations grad\(\times \text{grad} s=0\), grad\(\times \text{grad} n=0\) grad\(\times \text{grad} b=0\), which imply the vanishing of the anholonomic components of the curvature tensor. For \(E_ 3\) these compatibility conditions satisfy three Bianchi identities. In the rather technical note under review the three Bianchi identities are presented in explicit form for the basis (s,n,b), and three different paths leading to each identity are provided.