Cheeger and Gromoll proved the well known splitting theorem:
If $(M,g)$ is a complete manifold with $Ric\geq 0$ that contains a line, then $M$ splits isometrically as $M' \times \mathbb{R}$.
Here, line means that for any $R>0$, $\gamma|_{[-R,R]}$ has minimal length when compared to any geodesic with endpoints $\gamma(-R)$ and $\gamma(R)$.
What happens if we only assume that $\gamma$ is homotopically length minimizing?
In other words, assume that for any $R>0$, $\gamma|_{[-R,R]}$ has minimal length when compared to any geodesic with endpoints $\gamma(-R)$ and $\gamma(R)$ that is homotopic to $\gamma|_{[-R,R]}$ relative to the endpoints?
Is the splitting theorem still true under this weaker hypothesis? What if one replaces homotopy by homology? Isotopy? Essentially, I am asking for any counterexample or result where "line" is replaced by something more restrictive in the statement.
