  In this note we extend integrability conditions for the symmetric stretch
tensor $U$ in the polar decomposition of the deformation gradient
$\nabla\varphi=F=R\,U$ to the non-symmetric case. In doing so we recover
integrability conditions for the first Cosserat deformation tensor. Let $F=\bar
R\,\bar U$ with $\bar R:\Omega\subset\mathbb{R}^3\longrightarrow\mathrm{SO}(3)$
and $\bar U:\Omega\subset\mathbb{R}^3\longrightarrow \mathrm{GL}(3)$. Then
$\mathfrak{K}:={\bar R}^T\mathrm{Grad}\,{\bar R}=\mathrm{Anti}\Big(
\frac{1}{\mathrm{det} \bar U}\Big[\bar U(\mathrm{Curl} \bar U)^T-\frac{1}{2}
\mathrm{tr}(\bar U(\mathrm{Curl} \bar U)^T) 1\!\!1 \Big]\bar U\Big),$ giving a
connection between the first Cosserat deformation tensor $\bar U$ and the
second Cosserat tensor ${\mathfrak{K}}$. (Here, Anti denotes an isomorphism
between $\mathbb{R}^{3\times 3}$ and
$\mathfrak{So}(3):=\{\,\mathfrak{A}\in\mathbb{R}^{3\times 3\times
3}\,|\,\mathfrak{A}.u\in\mathfrak{so}(3)\;\forall u\in \mathbb{R}^3\}$.) The
formula shows that it is not possible to prescribe $\bar U$ and $\mathfrak{K}$
independent from each other. We also propose a new energy formulation of
geometrically nonlinear Cosserat models which completely separate the effects
of nonsymmetric straining and curvature. For very weak constitutive assumptions
(no direct boundary condition on rotations, zero Cosserat couple modulus,
quadratic curvature energy) we show existence of minimizers in Sobolev-spaces.
