We know that a connection $\nabla$ in a manifold M hashas the purpose of performing the same role as the covariant derivative of vector fields of surfaces in $\mathbb{R}^3$. Such analogies are clearly perceived in the connection definition.
When M has a Riemannian metric,$~~ g~~$, to make things even more similar to the case in $\mathbb{R}^3$ we introduce the concept of Riemannian connection or Levi-Civita connection, which is a connection satisfying:
compatibility with the metric: $Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)$
torsion-free: $T(X,Y)=\nabla_X Y-\nabla_YX-[X,Y]\equiv 0$
where $ X, Y, Z $ are vector fields on $M.$
What I understand is the need / intuition of the second property, torsion-free,what would work well if not we assume this hypothesis?