I am learning about differential and Riemannian geometry and I am having trouble understanding the Riemannian gradient of a parameterized submanifold of the Lie group. I am using https://arxiv.org/pdf/0802.4195.pdf to understand the Riemmanian gradient (the function in the example below is from section III D).
We have a group $G=\text{SU}(2)$, $U\in G$ and a function $f:G\to\mathbb{R}$, $f=\text{Tr}\{C^\dagger U A U^\dagger\}$, where $A,C\in\mathbb{C}^{2\times 2}$. The tangent space of $G$ at $U$ is $T_U G := \{V\in\mathbb{C}^{n\times n}| V^\dagger U + U^\dagger V=0\}$, or given the Lie algebra $\mathfrak{g}$, $T_U G := \{V=\Omega U| \Omega \in \mathfrak{g}\}$. By equipping the group with a bilinear form $\langle .,. \rangle$ and checking the compatibility condition, we can derive the Riemannian gradient to be \begin{align} \text{grad}f (U) = -[UAU^\dagger,C^\dagger]_S U \end{align} where the sub $S$ indicates the skew-hermitian part of the commutator. Now consider the case where $W(x,y)$ is a parameterized unitary, where $(x,y)\in\mathbb{R}^2$. We can understand $W(x,y)$ as a map $W: \mathbb{R}^2 \to G'$ where $G'\subset G$. The gradient of $f=\text{Tr}\{C^\dagger W(x,y) A W^\dagger(x,y)\}$ with respect to $x$ and $y$ is \begin{align} \nabla f = \begin{pmatrix} \frac{\partial f}{\partial x}\\ \frac{\partial f}{\partial y} \end{pmatrix} \end{align} which lives in $T_{(x,y)} \mathbb{R}^2$.
My question is now: how do I get the Riemannian gradient on $G'$ starting from this gradient?
What I have tried
Since $W$ is a smooth map between two manifolds, there is a natural way to transform differential operators between the two spaces. For $SU(2)$ we can choose a chart $\varphi: G \to \mathbb{R}^3$, which allows us to calculate the push-forward map between the tangent spaces $\varphi_*: T_{(x,y)} \mathbb{R}^2 \to T_{W(x,y)}G'$ as \begin{align} \varphi_*(\frac{\partial}{\partial x_i}) = \frac{\partial \varphi(W(x,y))^j}{\partial x_i} \frac{\partial}{\partial \phi_j} \end{align} I tried calculating this explicitly for some simple example, but I am unable to connect the two descriptions.
Is this the correct approach? My main point of confusion is when the additional structure of the Lie group becomes relevant.