I am trying to understand the Cartan decomposition in the case of a compact group $G$ with subgroup $K$, such that their respective Lie algebras $(\mathfrak{g},\mathfrak{k})$ correspond to a Cartan pair for an involution $\theta$.
In this setting, we have a decomposition $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ where $\mathfrak{p}$ is the -1 eigenspace of the involution $\theta$ on $\mathfrak{g}$. When I read https://en.wikipedia.org/wiki/Cartan_decomposition, it seems that $G$ must be diffeomorphic to $K\times \mathfrak{p}$, and this diffeomorphism is there called "Global Cartan Decomposition". [Edit: More precisely, I am refering to the statement :
"The mapping $K×\mathfrak{p}→G$ given by $(k,X)↦k⋅exp(X)$ is a diffeomorphism" in the section "Cartan decomposition on the Lie group level". ]
What I don't understand is that $K\times \mathfrak{p}$ is not compact (right?), but I want to consider $G$ as a compact group. Is there a mistake in the wikipedia page? Or is it only true for non-compact groups ? If so, what is then known about the Cartan decomposition at the level of groups for compact groups ?
The statement is simply false. Cartan's theorem says says if $G$ is semi-simple, and $K$ a maximal compact subgroup, then $(G,K)$ is a symmetric pair, and if $\cal p$ is the orthogonal in $\cal G$ of the Lie algebra of $ K$ $\exp :{\cal P}\to G/K$ is a diffeomrophism. This is a special case of the Cartan-Hadamard theorem which says that for a complete simply connected manifold of non positive curvature the exponential map is a diffeomorphism.