Let $G = SU(n)$ and $G_c = SL(n, \mathbb{C})$. Let $g$ be a right-invariant metric on $G$ and let $g_k$ be the Killing metric on $G_c$.
Define the map $p$ from $G_c$ to $G$ which maps $h \in G_c$ to $$p(h) = (\sqrt{h h^*})^{-1} h$$
Note that $p$ is equivariant under right multiplication by an element in $G$. It is the projection induced by the Cartan decomposition of $h$ as a product of an hermitian positive definite matrix times a unitary matrix.
- Does there exist a smooth isometric embedding $f$ from $(G, g)$ to $(G_c, g_k)$ such that $p \circ f$ is the identity on $G$?
- If so, what are all such $f$'s?
These questions occurred to me while thinking about something else. Something I thought of, but I am not sure if it leads to an answer or not is this. Embed the Lie algebra of $G$ with its inner product coming from g as an inner product subspace $W$ of the Lie algebra of $G_c$ (the latter endowed with the Killing form). Assume that $W$ intersected with the orthogonal complement $\mathfrak{g}^{\perp}$ of $\mathfrak{g}$ inside $\mathfrak{g}_c$ with respect to the Killing form is $0$, i.e.
$$ W \cap \mathfrak{g}^{\perp} = \mathbf{0}. $$
One may then write $W$ uniquely as the graph of a linear map $f$ from $\mathfrak{g}$ to $\mathfrak{g}^{\perp}$. Let $F(v) = (v, f(v))$ for $v \in \mathfrak{g}$. Assume that $f$ is such that the pullback of the Killing form using $F$ is the inner product on $\mathfrak{g}$ induced by $g$.
Then consider the image of the exponential map restricted to the subspace $W$. However, unless the subspace is itself closed under the Lie bracket, this may only give some local construction. Well, I am not sure if this would give a local solution of question 1 or not.
Motivation: I would like to think of a right-invariant metric on $G$ as the local data for some deformation of $G$ inside $G_c$, so that the Killing metric on $G$ would just correspond to the standard inclusion of $G$ inside $G_c$. This is the goal, though I may be on a "wrong path" or something (which I kind of suspect).