  Tensor generalizations of affine vector fields called symmetric and
antisymmetric affine tensor fields are discussed as symmetry of spacetimes. We
review the properties of the symmetric ones, which have been studied in earlier
works, and investigate the properties of the antisymmetric ones, which are the
main theme in this paper. It is shown that antisymmetric affine tensor fields
are closely related to one-lower-rank antisymmetric tensor fields which are
parallelly transported along geodesics. It is also shown that the number of
linear independent rank-$p$ antisymmetric affine tensor fields in $n$
dimensions is bounded by $(n+1)!/p!(n-p)!$. We also derive the integrability
conditions for antisymmetric affine tensor fields. Using the integrability
conditions, we discuss the existence of antisymmetric affine tensor fields on
various spacetimes.
