  We study the convergence issue for inexact descent algorithm (employing
general step sizes) for multiobjective optimizations on general Riemannian
manifolds (without curvature constraints). Under the assumption of the local
convexity/quasi-convexity, local/global convergence results are established. On
the other hand, without the assumption of the local convexity/quasi-convexity,
but under a Kurdyka-{\L}ojasiewicz-like condition, local/global linear
convergence results are presented, which seem new even in Euclidean spaces
setting and improve sharply the corresponding results in [24] in the case when
the multiobjective optimization is reduced to the scalar case. Finally, for the
special case when the inexact descent algorithm employing Armijo rule, our
results improve sharply/extend the corresponding ones in [3,2,38].
