While studying differential geometry I often come across propositions with $M$ being a connected surface as their hypothesis. They then often take paths between arbitrary points, which to me suggests that they actually mean that $M$ is path-connected.
Is this true? Is it some sort of convention in differential geometry to not distinguish between normal- and path-connectedness?
(The question is basically answered in the comment)
As a topological manifold is locally path connected, a connected manifold is automatically path connected, as a connected locally path connected topological space is path connected