  This paper proves that the remote state preparation (RSP) scheme in real
Hilbert space can only be implemented when the dimension of the space is 2,4 or
8. This fact is shown to be related to the parallelazablity of the $n$-1
dimensional sphere $S^{n-1}$. When the dimension is 4 and 8 the generalized
scheme is explicitly presented. It is also shown that for a given state with
components having the same norm, RSP can be generalized to arbitrary dimension
case.
