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An odd-dimensional projective space  $  P _ {2n + 1 }  $
over a field  $  k $
endowed with an involutory relation which is a null polarity; it is denoted by  $   \mathop{\rm Sp} _ {2n + 1 }  $.

Let  $   \mathop{\rm char}  k \neq 2 $. 
The absolute null polarity in  $   \mathop{\rm Sp} _ {2n + 1 }  $
can always be written in the form  $  u _ {i} = a _ {ij} x  ^ {j} $, 
where  $  \| a _ {ij} \| $
is a skew-symmetric matrix  $  ( a _ {ij} = - a _ {ji} ) $. 
In vector form, the absolute null polarity can be written in the form  $  \mathbf u = A \mathbf x $, 
where  $  A $
is a skew-symmetric operator whose matrix, in a suitable basis, reduces to the form

$$ 
\| A \|  = \ 
\left \|

\begin{array}{rrrrrr}
 0  &+ 1  &{}  &{}  &{}  &{}  \\
- 1  & 0  &{}  &{}  &{}  &{}  \\
{}  &{}  &\cdot  &{}  &{}  &{}  \\
{}  &{}  &{}  & 0  &+ 1  &{}  \\
{}  &{}  &{}  &- 1  & 0  &+ 1  \\
{}  &{}  &{}  &{}  &- 1  & 0  \\
\end{array}
 \ 
\right \| .
$$

In this case the absolute null polarity takes the canonical form

$$ 
u _ {2i}  =  x ^ {2i + 1 } ,\ \ 
u _ {2i + 1 }   =  - x  ^ {2i} .
$$

The absolute null polarity induces a bilinear form, written in canonical form as follows:

$$ 
\mathbf x A \mathbf y  = \ 
\sum _ { i }
( x  ^ {2i} y ^ {2i + 1 } - x ^ {2i + 1 } y  ^ {2i} ).
$$

Collineations of  $   \mathop{\rm Sp} _ {2n + 1 }  $
that commute with its null polarity are called symplectic transformations; the operators defining these collineations are called symplectic. The above canonical form of  $  \| A \| $
defines the square matrix of order  $  2n + 2 $
of a symplectic operator  $  U $
whose elements satisfy the conditions

$$ 
\sum _ { i }
( U _ {j}  ^ {2i} U _ {k} ^ {2i + 1 } -
U _ {j} ^ {2i + 1 } U _ {k}  ^ {2i} )  = \ 
\delta _ {j, k - 1 }  -
\delta _ {j, k + 1 }  ,
$$

where  $  \delta _ {a,b} $
is the Kronecker delta. Such a matrix is called symplectic; its determinant is equal to one. The symplectic transformations form a group, which is a Lie group.

Every point of the space  $   \mathop{\rm Sp} _ {2n + 1 }  $
lies in its polar hyperplane with respect to the absolute null polarity. One can also define polar subspaces in  $   \mathop{\rm Sp} _ {2n + 1 }  $. 
The manifold of self-polar  $  n $-
spaces of  $   \mathop{\rm Sp} _ {2n + 1 }  $
is called its absolute linear complex. In this context, a symplectic group is also called a (linear) complex group.

Every pair of straight lines, and their polar  $  ( 2n - 1) $-
spaces in the null polarity, define a unique symplectic invariant in  $   \mathop{\rm Sp} _ {2n + 1 }  $
with respect to the group of symplectic transformations of this space. Through every point of each line there passes a transversal of this line and  $  ( 2n - 1) $-
spaces, which defines a projective quadruple of points. This is the geometrical interpretation of a symplectic invariant, which asserts the equality of the cross ratios of these quadruples of points.

The symplectic  $  3 $-
dimensional space admits an interpretation in hyperbolic space and this indicates, among other things, a connection between symplectic and hyperbolic spaces. Thus, the group of symplectic transformations of  $   \mathop{\rm Sp} _ {3} $
is isomorphic to the group of motions of the hyperbolic space  $  {}  ^ {2} \textrm{ S } _ {4} $. 
In this interpretation, the symplectic invariant is related to the distance between points in hyperbolic space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian) {{MR|}} {{ZBL|0767.53035}} {{ZBL|0713.53012}} {{ZBL|0702.53009}} </TD></TR></table>

====Comments====
The notation  $   \mathop{\rm Sp} _ {2n+} 1 $
for the symplectic geometry in  $  P _ {2n+} 1 $
is not usual. By  $   \mathop{\rm Sp} _ {2n} ( k) $
one denotes the symplectic group in the linear space  $  k  ^ {2n} $
provided with an alternating (i.e. skew-symmetric) bilinear form. The corresponding group of projectivities in  $  P _ {2n-} 1 ( k) $
is denoted by  $   \mathop{\rm PSp} _ {2n} ( k) $; 
this is the group referred to in the main article above, and it is called the projective symplectic group.

The polar subspaces, or isotropic subspaces, as they are also called, in a projective space with a null polarity form an example of what is called a polar geometry (cf. also [[Polar space|Polar space]]; see ). In J. Tits' theory of buildings, the symplectic spaces interpreted as polar geometries are buildings of type  $  C _ {n} $(
see [[#References|[a2]]] and [[Tits building|Tits building]]).

====References====
<table><TR><TD valign="top">[a1a]</TD> <TD valign="top"> F.D. Veldkamp, "Polar geometry" ''Indag. Math.'' , '''21''' (1959) pp. 512–551 {{MR|0125472}} {{ZBL|0090.11902}} </TD></TR><TR><TD valign="top">[a1b]</TD> <TD valign="top"> F.D. Veldkamp, "Polar geometry" ''Indag. Math.'' , '''22''' (1960) pp. 207–212 {{MR|0158301}} {{MR|0125472}} {{ZBL|0094.15602}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Tits, "Buildings of spherical type and finite BN-pairs" , ''Lect. notes in math.'' , '''386''' , Springer (1974) {{MR|0470099}} {{ZBL|0295.20047}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Baer, "Linear algebra and projective geometry" , Acad. Press (1952) {{MR|0052795}} {{ZBL|0049.38103}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1963) {{MR|}} {{ZBL|0221.20056}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E. Artin, "Geometric algebra" , Interscience (1957) pp. Chapt. II {{MR|1529733}} {{MR|0082463}} {{ZBL|0077.02101}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> O.T. O'Meara, "Symplectic groups" , Amer. Math. Soc. (1978) {{MR|}} {{ZBL|0422.15014}} {{ZBL|0414.20037}} {{ZBL|0393.47029}} {{ZBL|0383.20001}} {{ZBL|0378.22015}} {{ZBL|0366.22014}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> H. Weyl, "The classical groups" , Princeton Univ. Press (1946) pp. 120 {{MR|1488158}} {{MR|0000255}} {{ZBL|1024.20502}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 69–70 {{MR|1628013}} {{MR|1531918}} {{MR|0087965}} {{MR|0006835}} {{ZBL|0909.51003}} {{ZBL|0077.13903}} {{ZBL|0060.32807}} {{ZBL|68.0322.02}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> V. Guillemin, S. Sternberg, "Symplectic techniques in physics" , Cambridge Univ. Press (1984) {{MR|0770935}} {{ZBL|0576.58012}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> A.J. Hahn, O.T. O'Meara, "The classical groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091850/s09185037.png" />-theory" , Springer (1989) {{MR|1007302}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> P. Libermann, C.-M. Marle, "Symplectic geometry and analytical mechanics" , Reidel (1987) (Translated from French) {{MR|0882548}} {{ZBL|0643.53002}} </TD></TR></table>