Let $k$ be a local field (if necessary, assume characteristic zero). In general, if $X$ is a smooth variety of finite type over $k$ of dimension $n$, then the set of $k$-rational points $X(k)$ is an analytic manifold over $k$ of dimension $n$. I was thinking about the passage $X \mapsto X(k)$ from smooth varieties to manifolds in the context of coset spaces, and had a couple of questions.


Let $H$ be a closed subgroup of a linear algebraic group $G$ over $k$. Assume $H$ is defined over $k$. Then the coset space $G/H$ has the structure of a smooth quasi-projective variety over $k$, which tells us that $G/H(k)$ is an analytic manifold over $k$ of dimension $n = \operatorname{Dim}G - \operatorname{Dim}H$.


On the other hand, $H(k)$ is a closed subgroup of the analytic Lie group $G(k)$, and the quotient $G(k)/H(k)$ has the structure of an $n$ dimensional analytic manifold (the quotient map $G(k) \rightarrow G(k)/H(k)$ is a principal fiber bundle with $H(k)$ as a fiber). This just follows from the general theory of Lie groups.


Of course, $G(k)/H(k)$ injects into $G/H(k)$, but they don't have to be equal. They are the same if $H^1(\operatorname{Gal}(k\_s/k),H) $ is trivial.


What can we say about the inclusion $G(k)/H(k) \rightarrow G/H(k)$ from the perspective of manifolds? Is the inclusion an analytic map? Does it make $G(k)/H(k)$ into an open submanifold? How different can these two manifolds be?



$G(k)/H(k)$ as a submanifold of $G/H(k)$
If $k$ has characteristic zero, then $H^1(k,H)$ is finite (Borel-Serre: 
<https://mathscinet.ams.org/mathscinet-getitem?mr=181643>)
and the map $G(k)/H(k) \rightarrow (G/H)(k)$ has open image (can also be proved by the implicit function theorem). Moreover, a Baire category argument shows that this map is a homeomorphism onto the image. In positive characteristic, there is an old paper by Bernstein Zelevinsky (<https://mathscinet.ams.org/mathscinet-getitem?mr=425030>; see the appendix) which says (I think) that the map is a homeo onto the image, and that the image is closed.


