The Bianchi transformation of \(n\)-surfaces in \(E^{2n-1}\) (Q1387286)

A Bianchi transformation \(f:M\to \overline M\) of smooth \(n\)-surfaces in the Euclidean space \(E^{2n-1}\) is defined by the properties \(f(r)= r+\rho V\) where \(r=r(p)\) is the radius vector of \(p\in M\), \(V\in E^{2n-1}\) is a unit vector, \(\rho= \text{const.}\), \(V_p\in T_p M\), \(T_pM\) is the span of \(V_p\) and \(T^\perp_{f(p)} \overline M\), and \(T_{f(p)} \overline M\) is the span of \(V_p\) and \(T_p^\perp M\). For such a Bianchi transformation \(f\), the author proves two interesting formulae; we state the shorter one: \[ R^\perp (X,Y) \alpha (Z,V)= \alpha \left(R(X,Y)Z + {1\over \rho^2} \overline g(X,Y)- {1\over \rho^2} \overline g(X,Z)Y,V \right), \] where \(X, Y,Z,V \in TM\). Concerning the notation, \(R^\perp\) is the curvature tensor of the normal connection of \(M\), \(\overline R\) is the curvature tensor of the metric \(\overline g(X,Y)= \langle df X,\;df Y\rangle\) induced by \(f\) on \(M\), and \(\alpha\) is the second fundamental form of \(M\).