In this paper, the author studies a natural deduction calculus for hybrid logic. The underlying language contains satisfaction operators \(\forall_a\) as well as binders \(\forall a\) and \(\downarrow a\), where \(a\) is any nominal. It turns out that certain first-order properties of the involved accessibility relation, originating from so-called \textit{geometric theories,} can be treated by suitable proof rules of the calculus at the same time. For the arising logical system, a soundness and completeness theorem is proved. Moreover, a normalization theorem for derivations and a certain subformula property for the resulting normal derivations are established. Finally, a Gentzen system corresponding to the above natural deduction system is introduced and proved to be sound and complete as well.