  In this paper we present some results for a connected infinite graph $G$ with
finite degrees where the properties of balls of small radii guarantee the
existence of some Hamiltonian and connectivity properties of $G$. (For a vertex
$w$ of a graph $G$ the ball of radius $r$ centered at $w$ is the subgraph of
$G$ induced by the set $M_r(w)$ of vertices whose distance from $w$ does not
exceed $r$). In particular, we prove that if every ball of radius 2 in $G$ is
2-connected and $G$ satisfies the condition $d_G(u)+d_G(v)\geq |M_2(w)|-1$ for
each path $uwv$ in $G$, where $u$ and $v$ are non-adjacent vertices, then $G$
has a Hamiltonian curve, introduced by K\"undgen, Li and Thomassen (2017).
Furthermore, we prove that if every ball of radius 1 in $G$ satisfies Ore's
condition (1960) then all balls of any radius in $G$ are Hamiltonian.
