Semi-symplectic space

A projective $ ( 2n + 1) $-space in which there is given a $ ( 2n - 2m _ {0} - 1) $-plane $ T _ {0} $, in this a $ ( 2n - 2m _ {1} - 1) $-plane $ T _ {1} $, etc., up to a $ ( 2n - 2m _ {r - 1 } - 1) $-plane $ T _ {r - 1 } $, where in the space a null-system is given, taking all the points of the space to planes passing through the plane $ T _ {0} $; the plane $ T _ {0} $ is given an absolute null-system taking all its points to $ ( 2n - 2m _ {0} - 2) $-planes lying in it and passing through the $ ( 2n - 2m _ {1} - 1) $-plane $ T _ {1} $, etc., up to an absolute null-system of the $ ( 2n - 2m _ {r - 1 } - 1) $-plane $ T _ {r - 1 } $, taking all its points to $ ( 2n - 2m _ {r - 1 } - 2) $-planes lying in it, $ 0 \leq m _ {0} < m _ {1} < \dots < m _ {r - 1 } < n $. This semi-symplectic space is denoted by $ \mathop{\rm Sp} _ {2n + 1 } ^ {2m _ {0} + 1 \dots 2m _ {r - 1 } + 1 } $.

A semi-symplectic space is obtained by a method analogous to the transition from elliptic and hyperbolic spaces to semi-elliptic and semi-hyperbolic spaces, and is more general than a quasi-symplectic space.

The collineations of a semi-symplectic space that take the planes $ T _ {i} $ to themselves and that commute with the null-systems are called semi-symplectic transformations of the semi-symplectic space.

There exist invariants of semi-symplectic transformations analogous to the symplectic invariants of symplectic spaces. The semi-symplectic transformations form a Lie group.

References

[1]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

References

[a1]	B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian)