Definition of a branched 1-manifold.

Question

i'm studying a papper which has this term "branched 1-manifolds", but the papper does not explain this, according to Wikipédia:

"A finite graph whose edges are smoothly embedded arcs in a surface, such that all edges incident to a given vertex v have the same tangent line at v, is a branched one-manifold. "

But i do not understand this definition. Someone could help me?

Answer 1

A branched n-manifold is covered by n-dimensional "coordinate charts", each of which involves one or several "branches" homeomorphically projecting into the same differentiable n-disk in $\mathbb{R}^n$.

See https://en.wikipedia.org/wiki/Branched_manifold

The finite graph case is just an example of a branched 1-manifold, not the definition.

Furthermore, they add a more specific example of such a graph, namely the figure of the number $8$ drawn on the plane.At the point where both circle intersect to give the "$8$", the 1-d curve describing it has one single tangent, yet in a neighborhood of it the second derivatives for each circle differ giving rise to two different branches -as see from the point of intersection.