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Given a function $\Phi:\Omega^{\Phi}\subset \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\Omega^{\phi}\subset \mathbb{R}^2\rightarrow\mathbb{R}$, 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 manifold $F$ of "slices" where the manifold structure is a result of the group action of $SE(3)$.

I was wondering, if tangent spaces of this manifold have the same dimension, as that of tangent spaces to the $SE(3)$ group (=6)?

Please forgive the imprecise question formulation. I would welcome any suggestion / references where a representation for the tangent space is constructed using the generators of the "underlying" transformation lie-algebra.

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    $\begingroup$ Welcome to mathoverflow. What do we know about $\Omega^\Phi$? Do you mean by "slice" the set $\Omega_s=s^{-1}\cap\mathbb R^2\times\{0\}$? And by "manifold of slices" the subset of $SE(3)$ such that $\Omega_s\ne\emptyset$? Please clarify this, and think about posting this question on math.stackexchange, as this does not really look like a research level question. $\endgroup$
    – Sebastian Goette
    Commented Mar 8, 2016 at 18:55
  • $\begingroup$ I 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. $\endgroup$
    – Peter
    Commented Mar 9, 2016 at 15:45
  • $\begingroup$ 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 $SE(3)$? $\endgroup$
    – Peter
    Commented Mar 9, 2016 at 15:45
  • $\begingroup$ Following the suggestions of Sebastian Goette I have posted the question on math.stackexchange ( math.stackexchange.com/questions/1690221/…) $\endgroup$
    – Peter
    Commented Mar 9, 2016 at 16:06

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