There is a family of comparison theorems in Riemannian geometry (Rauch, Günther-Bishop, Gromov, Toponogov-Cheng) that all rely on two hypotheses: some boundedness of the sectional or Ricci curvature, and completeness.

All the places that I have seen assume completeness, while only Chavel ("Riemannian Geometry: A Modern Introduction", 2nd ed., page 128) briefly says "for convenience, we assume that $M$ is complete". Since I am working with manifolds not necessarily complete, could anyone please firmly clarify whether completeness is necessary, or just convenient? I wouldn't want to base my work on flawed results. (I am interested mostly in the Bishop-Gromov volume comparison theorem.)

Thank you.

** Later edit **

After further reflection, I realize that I may have misunderstood Chavel's words: I have interpreted them as "completeness is convenient for the proof, but we can drop it and not lose too much", but now I realize that it might also mean "completeness is convenient for the proof, but we can replace it with other assumptions and still get the results, but through uglier proofs".

As such, I would like to ask whether requiring bounded Ricci curvature (both above and below) would still allow us to get the aforementioned theorems, in particular Bishop-Gromov.