On special 4-planar mappings of almost Hermitian quaternionic spaces (Q2715744)

The space \(A_n= (M_n,\Gamma,\overset {1} F,\overset {2} F,\overset {3} F)\) is called an almost quaternionic space with torsionfree affine connection if \(\overset {1} F\), \(\overset {2} F\), \(\overset {3} F\) are almost complex structures which satisfy the following product rule: NEWLINE\[NEWLINE\begin{matrix} & \overset {1} F^\alpha_i & \overset {2} F^\alpha_i & \overset {3} F^\alpha_i\\ \overset {1} F^h_\alpha & -\delta^h_i & -\overset {3} F^h_i & \overset {2} F^h_i\\ \overset {2} F^h_\alpha & \overset {3} F^h_i & -\delta^h_i & -\overset {1} F^h_i\\ \overset {3} F^h_\alpha & -\overset {2} F^h_i & \overset {1} F^h_i & -\delta^h_i\end{matrix}.NEWLINE\]NEWLINE A curve \(x^h= x^h(t)\) in \(A_n\) is called 4-planar if \(\lambda^h={dx^h\over dt}\) under parallel transports along this curve remain in 4-dimensional space generated by \(\lambda^h\), \(\overset {1} F^h_\alpha\lambda^h\), \(\overset {2} F^h_\alpha\lambda^\alpha\), \(\overset {3} F^h_\alpha\lambda^\alpha\). If \(A_n= (M_n,\Gamma, \overset {2} F,\overset {2} F,\overset {3} F)\), and \(\overline A_n= (M_n,\overline\Gamma, \overset {1} F, \overset {2} F,\overset {3} F)\), where \(\Gamma\) and \(\overline\Gamma\) are torsionfree affine connections, then a diffeomorphism \(f: A_n\to\overline A_n\) is called a 4-planar mapping if it maps any geodesic of \(A_n\) to a 4-planar curve of \(\overline A_n\). Conditions for the diffeomorphism \(f\) to be 4-planar are given. Fundamental equations of these mappings are expressed in linear Cauchy form. These results improve the equations of I. N. Kurbakova.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].