Let $(M,g)$ be a connected complete Riemannian manifold with $\operatorname{Ric}_g \geq 0$. Suppose that the Cheeger-Gromoll splitting procedure can be performed for $k \in \mathbb{N}$ successive steps in the following sense: there exist connected Riemannian manifolds $(M_j,g_j)$ for $0 \leq j \leq k$, with $(M_0,g_0)=(M,g)$, and for each $1 \leq j \leq k$ an isometry

\begin{align*} \Phi_j : (M_{j-1},g_{j-1}) \longrightarrow (M_j \times \mathbb{R},\, g_j + d t_j^2), \end{align*}

obtained by applying the splitting theorem to a line contained in the residual factor $(M_{j-1},g_{j-1})$ rather than in any previously extracted Euclidean factor. Suppose moreover that the procedure is maximal, meaning that the final residual factor $(M_k,g_k)$ contains no line.

Then there is an isometry

\begin{align*} (M,g) \cong (N \times \mathbb{R}^k,\, h + g_{\mathrm{Euc}}) \end{align*}

where $(N,h)=(M_k,g_k)$ is complete, satisfies $\operatorname{Ric}_h \geq 0$, and contains no line.