Let \(M\) be a von Neumann algebra, \(\varphi\) be a faithful, normal semifinite weight on \(M\) and \(M^\varphi\) its centralizer. We characterize the conditional expectations \(E_\varphi: M\to M^\varphi\) of finite index for a faithful normal strictly semifinite weight \(\varphi\) on a semifinite von Neumann algebra \(M\) with finite-dimensional center. This result is used to characterize weights \(\varphi\) such that the orbit \({\mathcal U}_\varphi= \{\varphi\circ \text{Ad}(u):u\) unitary in \(M\}\) can be represented as a submanifold of \(M_1\) (= basic extension of \(E_\varphi: M\to M_\varphi\)).