Semi-parallel and parallel symplectic surfaces in the four-dimensional symplectic space (Q2711470)

Let \({\text Sp}_4\) be an affine space whose direction space is a 4-dimensional symplectic vector space \(\overarrow{\text Sp}_4\). If \(\{X,\overarrow {e_I}\}\), \(I=1,2,3,4\), is a moving frame in \({\text Sp}_4\) and \(\overarrow x\) is a position vector of a point \(X\), then the motion of the frame is described by the relations: \(d\overarrow x=\omega^L\overarrow {e_L}\), \(d\overarrow {e_I}=\omega^L_I\overarrow {e_L}\). Let \(M_2\) be a symplectic surface of \({\text Sp}_4\). The vectors \(\overarrow {e_i}\), \(i=1,2\), form a basis for \(T_XM_2\) and \(\overarrow {e_\alpha}\), \(\alpha=3,4\), form a basis for \(T_X^\perp M_2\). The differential equations of the symplectic surface \(M_2\) in the symplectic space \({\text Sp}_4\) are NEWLINE\[NEWLINE\omega ^\alpha_i=h_{ij}^\alpha\omega^j,\;h_{ij}^\alpha=h_{ji}^\alpha,\;dh_{ij}^\alpha-h_{kj}^\alpha\omega^k_i-h_{ik}^\alpha\omega^k_j+h_{ij}^\beta\omega^\alpha_\beta=h _{ijk}^\alpha\omega^k,\;h_{ijk}^\alpha=h_{ikj}^\alpha.NEWLINE\]NEWLINE Let \(\overline\nabla=\nabla \oplus\nabla ^\perp\) be the van der Waerden-Bortolotti connection. Then \(\overline\nabla h^\alpha_{ij}=h^\alpha_{ijk} \omega^k\) and \(\overline\nabla h^\alpha_{ijk}=h^\alpha_{ijkl}\omega^l\). A symplectic surface \(M_2\) in the symplectic space \({\text Sp}_4\) is parallel if \(\overline\nabla h^\alpha_{ij}=0\) and is semi-parallel if \(\overline\nabla h^\alpha_{ijk}\wedge \omega^k=0\). Every parallel symplectic surface is also semi-parallel. NEWLINENEWLINENEWLINEIn this paper, the author shows that a symplectic surface \(M_2\) in the symplectic space \({\text Sp}_4\) is semi-parallel if and only if its tangent connection \(\nabla\) is flat. It is also shown that a parallel symplectic surface \(M_2\) in the symplectic space \({\text Sp}_4\) is either an elliptic paraboloid, a hyperbolic paraboloid, or a parabolic cylinder in the 3-dimensional space. Finally, the author shows that in the symplectic space \({\text Sp}_4\) there exist semi-parallel symplectic surfaces that are not parallel.