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On special 4-planar mappings of almost Hermitian quaternionic spaces. (English) Zbl 1032.53008

Marchiafava, S. (ed.) et al., Proceedings of the 2nd meeting on quaternionic structures in mathematics and physics, Roma, Italy, September 6-10, 1999. Dedicated to the memory of André Lichnerowicz and Enzo Martinelli. Rome: Dipartimento di Matematica “Guido Castelnuovo”, Università di Roma “La Sapienza”, 265-271, electronic only (2001).
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: \[ \begin{matrix} & \overset {1} F^\alpha_i & \overset {2} F^\alpha_i & \overset {3} F^\alpha_i\\ \overset {1} F^h_\alpha & -\delta^h_i & -\overset {3} F^h_i & \overset {2} F^h_i\\ \overset {2} F^h_\alpha & \overset {3} F^h_i & -\delta^h_i & -\overset {1} F^h_i\\ \overset {3} F^h_\alpha & -\overset {2} F^h_i & \overset {1} F^h_i & -\delta^h_i\end{matrix}. \] 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.
For the entire collection see [Zbl 0958.00032].

MSC:

53B25 Local submanifolds
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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