A projective property of an embeddable polar space
is a property of the family of its projective embeddings, such as
the existence of an embedding of vector dimension twice the rank of
or the fact that
all embeddings of
have the same dimension, while properties such as the fact that all pairs of opposite
points are regular or all triads of points are centric, are synthetic properties.
We prove that all pairs of opposite points of an embeddable polar space
of rank
are regular if and
only if
admits
a
-dimensional
embedding; we also prove that all embeddings of
have the
same dimension if and only if the subspace spanned by two opposite singular subspaces is
closed under taking hyperbolic lines. Moreover, we characterize the fact that all triads of
points of
are centric by means of suitable properties of the universal embedding of
.
