My major is mechanical engineering. Recently, I'm reading a paper about tangent spaces of rotation group SO(3). The following is a part of it.
The notations used in the figure are: $\mathbf I $ is the 3$\times$3 identity matrix, $\mathbf R=exp(\widetilde{\psi}) $ is an arbitrary rotation tensor on SO(3), and $\widetilde{\psi}$ and $\widetilde{\theta}$ represent the associated skew-symmetric tensors for vectors $\psi$ and $\theta$, respectively.
According to this paper, there are two definitions for tangent spaces of SO(3), given by Makinen and Simo et al. respectively.
Here are my questions:
1) Which definition is more precise and why?
2) Is an element $\widetilde{\theta}_{R}$ of any tangent space $T_{R}SO(3)$ a skew-symmetric tensor, as Makinen stated?
Thank you very much!
References:
Mäkinen, Jari, Rotation manifold $\mathrm{SO}(3)$ and its tangential vectors, Comput. Mech. 42, No. 6, 907-919 (2008). ZBL1163.74472.
Simo, J.C.; Vu-Quoc, L., On the dynamics in space of rods undergoing large motions - A geometrically exact approach, Comput. Methods Appl. Mech. Eng. 66, No.2, 125-161 (1988). ZBL0618.73100.
$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Lie}[1]{\mathfrak{#1}}$This is a mathematical answer, and may require post-processing (e.g., translation back into engineering language) to be useful to you as an engineer.
In general, if $G$ is a Lie group (such as the rotation group $SO(n)$) and $\Lie{g} = T_{e}G$ is its Lie algebra (such as $\Lie{so}(n)$, the space of skew-symmetric $n \times n$ real matrices), the tangent bundle is trivial via $$ G \times \Lie{g} \simeq TG,\quad (g, v) \mapsto (L_{g})_{*}(v). $$
If $G \subset \Reals^{n^{2}}$ is a matrix group, then $\Lie{g}$ may be identified with a vector subspace of $\Reals^{n^{2}}$, and the bundle isomorphism above amounts to $$ (R, \tilde{\Theta}) \mapsto (R, R\tilde{\Theta}). \tag{1} $$
I wouldn't say it's a matter of one definition being more precise than the other, more a matter of which identification in (1) is more useful in a specific situation.
Though $\tilde{\Theta} \in \Lie{so}(n) = T_{I} SO(n)$ is skew-symmetric, the product $R\tilde{\Theta}$ generally is not. Instead, $$ (R\tilde{\Theta})^{T} = \tilde{\Theta}^{T} R^{T} = -\tilde{\Theta} R^{-1}. $$
Finally, in case it's helpful, the way the definition of Simo et al. is written, it looks as if only the product $R\tilde{\Theta}$ is retained, with $R$, the "point of tangency", implicit. By analogy, the tangent bundle of the unit circle may be viewed in two standard ways:
As the cylinder $S^{1} \times \Reals \subset S^{1} \times \Reals^{2}$.
As the family of tangent lines to the circle in $\Reals^{2}$.
A point of $\Reals^{2}$ outside the circle (analogous to a product $R\tilde{\Theta}$) only represents a tangent vector once the point of tangency $R$ is specified.