1)By Cartan--Von Neumann theorem on closed subgroups, if $G$ is a Lie group and $H$ is a closed subgroup of $G$, then $H$ is an embedded Lie subgroup of $G$, so 1)its connected components w.r.t. the subspace topology and its connected components w.r.t. this Lie group structure are the same.

2)The connected components of a topological manifold are open.

Now 1) and 2) should be conclusive for your question, if I am not wrong.