Given a function $\Phi: \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\mathbb{R}^2\rightarrow\mathbb{R}$, using a rigid mapping $s \in SE(3)$ such that for $(x,y)\in\Omega^{\phi} \quad \phi(x,y)= \Phi(s(x,y,0))$.
We consider the slice to be a two-dimensional linear submanifold $\mathcal{S}$ of $\mathcal{M}=\mathbb{R}^3$. The action of the transformation group ($SE(3)$) maps points $x$ of $\mathcal{S}$ along curves, corresponding to flows of vector fields (diffeomorphisms), "induced" by the generators of the group. My question is if approximations to the function values $\Phi$ at the transformed points (jet) can be consistently provided with respect to arbitrary infinitesimal actions of the group, using a linear combination of basis vectors with coefficients of the Lie-algebra.
For the particular case of rotations, function values can be approximated as $\hat{\Phi}(x) = \Phi(x) - \nabla\Phi(x)^T\cdot[x]_{\times}\cdot\omega$ with $[.]_{\times}$ the skew symmetric cross product matrix, and $\omega$ the vector of exponential rotation coordinates. The term $[x]_{\times}\cdot\omega$ determines a tangent vector field for the curves under the group action. Therefore function values along the slightly perturbed slice are approximated in the form $\hat{\Phi} = \Phi + \sum_{i=1}^3 \omega_i\mathcal{L}_i$. I would like to know if this process can be replicated for $SE(3)$?