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# Article

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Keywords:
systems of linear spaces; dual manifold; four-parametric manifolds; projective space; characteristics; differential geometry of systems of linear spaces
Summary:
Some results in the geometry of four-parametric manifolds of three-dimensional spaces in the projective space \$P_7\$ are found. The properties of such a manifold \$V^4_3\$ with characteristics consisting of a quadric and two planes are studied. The properties of the manifold dual to \$V^4_3\$ are found. Some results in the geometry of linear spaces from [1],[2],[3],[4] are used. The notation of the quantities is the same as in [4].
References:
[1] C. П. Фиников: Теория пар конгруенций. Mocквa, 1956. Zbl 0995.90522
[2] Л. З. Крузляков: Основы проективно-дифференциальной геометрии семейства многомерных плоскостей. Toмсk, 1980. Zbl 1138.81416
[3] A. Švec: Projective differential geometry of line congruences. Praha, 1956.
[4] J. Vala: Special Grassman manifold \$V_3^4\$ in the projective space \$P_7\$. Časopis pěst. mat. 113 (1988), 36-51. MR 0930804

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