$\mathbf{g}_i\in T\mathcal{B}$ and $\mathbf{g}^i\in T^*\mathcal{B}$
The covariant derivative associated with the Levi-Civita connection on $\mathcal{B}_0$ is written as
$$ \stackrel{G}{\nabla}_A(\cdot)=(\cdot) ; A . $$Contravariant reference basis vectors can be written symbolically as gradients of coordinates in Euclidean space:
$$ \stackrel{G}{\nabla} X^A=\partial_B X^A \mathbf{G}^B=\delta_B^A \mathbf{G}^B=\mathbf{G}^A . $$ $$ \nabla=\stackrel{G}{\nabla} \Rightarrow \nabla_{\partial_A} \partial_B=\nabla_{\partial_B} \partial_A=\partial_B \partial_A=\partial_A \bm{\partial}_B, $$$$ \mathbf{G}_A\triangleq\frac{\partial}{\partial X^A}\equiv\partial_A, \qquad \mathbf{G}^A \equiv \bm{d} X^A $$ $$ \mathbf{a}_a\triangleq\frac{\partial}{\partial x^a}\equiv\partial_a, \qquad \mathbf{a}^a \equiv \bm{d} x^a $$ $$ \frac{\partial(\cdot)}{\partial X^A}=\partial_A(\cdot)=(\cdot)_{, A}, \quad \frac{\partial(\cdot)}{\partial x^a}=\partial_a(\cdot)=(\cdot)_{, a} $$ $$ \begin{aligned} \mathbf{g}^{A}(x, t)&=(F^{-1})_{~~a}^{A} \, \mathbf{a}^a \\ \end{aligned} \quad \mathbf{g}_A = F_{~~ A}^a \mathbf{a}_a $$ $$ \mathbf{G}_A(X)= \partial_A \bm{X}, \quad \mathbf{a}_a(x)= \partial_a \bm{x} \qquad\text{BAD, bold in }\partial(~\cdot~) $$ $$ \begin{array}{lcrlcrlcrlcrl} & & & \mathbf{G}_I & & & \mathbf{a}_i & & & \mathbf{g}_I \\ \hline \mathbf{G}_I & = & & \bm{\partial}_I & = & & & & \bm{F}^{-1} & \mathbf{g}_I \\ \mathbf{a}_i & = & & & = & & \bm{\partial}_i \\ \mathbf{g}_I & = & \bm{F} & \mathbf{G}_I & = & \partial_I x^i &\bm{\partial}_i \\ \mathbf{A}_i & = & \partial_i X^I & \mathbf{G}_I & = & \\[0.3cm] \hline \mathbf{G}^I & = & & \bm{d} X^I & = & & & = & \bm{F}^T & \mathbf{g}^I \\ \mathbf{a}^i & = & & & = & & \bm{d} x^i \\ \mathbf{g}^I & = & \mathbf{F}^{-\mathrm{T}} & \mathbf{G}^I & = & \partial_i X^I &\bm{d} x^i \\ \mathbf{A}^i & = & \partial_I ~ x^i & \mathbf{G}^I \\ \end{array} $$SOURCE: Clayton 2015
Let
$$ \mathbf{A}^b \triangleq \partial_B x^b ~ \mathbf{G}^B \quad\text{ and }\quad \mathbf{A}_b \triangleq \frac{\partial X^A}{\partial x^b}\frac{\partial}{\partial X^A} =\frac{\partial X^A}{\partial x^b}\mathbf{G}_A $$The covariant derivative associated with the Levi-Civita connection on $\mathcal{B}$ is written as
$$ \stackrel{\mathrm{a}}{\nabla}_a(~\cdot~)=(~\cdot~)_{; a} $$Contravariant spatial basis vectors can be written symbolically as gradients of coordinates:
$$ \stackrel{\mathrm{a}}{\nabla} x^a = \partial_b x^a \mathbf{a}^b = \delta_b^a \mathbf{a}^b = \mathbf{a}^a $$The tangent at $x \in M$ of a mapping $\chi \in \mathscr{C}^k(M, N)$ is a linear mapping $T_x \chi \in$ $\mathscr{L}\left(T_x M, T_{\chi(x)} N\right)$. The section $D \chi \in \mathscr{C}^{k-1}\left(T^* M \otimes \chi^* T N\right)$, defined at each $x \in M$ by
$$ D \chi(x) \cdot \xi(x):=\left(T_x \chi\right)(\xi(x)) \text { for all } \xi \in T M, $$is the differential of $\chi$ at $x$. In local charts,
$$ D \chi(\xi)=\frac{\partial \chi^\alpha(\xi)}{\partial x^i} \, d x^i({\xi}) \otimes \left(\frac{\partial}{\partial y^\alpha}\circ \chi(\xi)\right). $$Given a $C^1$ map $\phi: \mathcal{M} \rightarrow \mathcal{N}$, the tangent map $T \phi: T \mathcal{M} \rightarrow T \mathcal{N}$ is defined. Papadapoulos:
$$ \bm{F}: T_x^* \mathcal{R} \times T_X \mathcal{R}_0 \rightarrow \mathbb{R} \qquad\text { or }\qquad \bm{F}: T_X \mathcal{R}_0 \rightarrow T_x \mathcal{R} $$The tangent map $\bm{F}$ is represented by the two-point tensor
$$ \bm{F}=F_{~~A}^a \, \mathbf{a}_a \otimes \mathbf{G}^A \equiv \mathbf{g}_{A}\otimes \mathbf{G}^{A} \qquad\text{with}\qquad \mathbf{g}_A\triangleq F^a_A \mathbf{a}_a =\frac{\partial x^a}{\partial X^A} \bm{\partial}_a=\frac{\partial x^a}{\partial X^A} \frac{\partial}{\partial x^a}%=\partial_A \bm{x} $$with components
$$ F_{~~A}^a(X, t)=\frac{\partial \varphi^a(X, t)}{\partial X^A}=\frac{\partial x^a(X, t)}{\partial X^A}=x_{, A}^a(X, t)=\partial_A x^a(X, t) . $$ $$ \begin{gathered} \bm{F}=\frac{\partial x^a}{\partial X^A} \partial_a \bm{x} \otimes \mathbf{G}^A=\partial_A \bm{x} \otimes \mathbf{G}^A \end{gathered} $$From Simo and Marsden:
$$ \bm{F}=F_A^a \frac{\partial}{\partial x^a} \otimes \bm{d} X^A = F^a_A\bm{\partial}_a \otimes \bm{d}X^A $$with
$$ F_A^a=\frac{\partial \chi^a}{\partial X^A} $$Similarly, the inverse deformation gradient and its components are
$$ \begin{gathered} \bm{F}^{-1}=(F^{-1})_{~~ a}^{A} \mathbf{G}_A \otimes \mathbf{a}^a, \\ \left(F^{-1}\right)_{~. a}^{A}(x, t)=\frac{\partial \Phi^A(x, t)}{\partial x^a}=\frac{\partial X^A(x, t)}{\partial x^a}=X_{, a}^A(x, t)=\partial_a X^A(x, t) . \end{gathered} $$It follows that
$$ \begin{gathered} \bm{F}^{-1}=\frac{\partial X^A}{\partial x^a} \partial_A \bm{X} \otimes \mathbf{a}^a=\partial_a \bm{X} \otimes \mathbf{a}^a . \end{gathered} $$ $$ \begin{aligned} & \bm{F}=\mathbf{g}_{i} \otimes \mathbf{G}^i \\ & \bm{F}^{\mathrm{t}}=\mathbf{G}^i \otimes \mathbf{g}_{i} \\ & \bm{F}^{-1}=\mathbf{G}_{i} \otimes \mathbf{g}^{i} \\ & \bm{F}^{-\mathrm{t}}=\mathbf{g}^{i} \otimes \mathbf{G}_{i} \end{aligned} $$The adjoint $(T \phi)^{\star}: T^* \mathcal{N} \rightarrow T^* \mathcal{M}$ of $T \phi$ is such that
$$ \left<(T \phi)^{\star}\left(\bm{z}^*\right), \bm{Z}\right\rangle_{\mathcal{M}}=\left<\bm{z}^*, T \phi(\bm{Z})\right\rangle_{\mathcal{N}}, $$for all vector fields $\bm{Z}$ on $\mathcal{M}$ and $\bm{z}$ on $\mathcal{N}$.
$$ (T \phi)^{\star}=(T \phi)_A^i \mathbf{E}^{* A} \otimes \mathbf{e}_i $$Define $(T \phi)^{\mathrm{T}}: T \mathcal{N} \rightarrow T \mathcal{M}$ by
$$ \left<\mathbf{G} \bm{Z},(T \phi)^{\mathrm{t}} \bm{z}\right\rangle_{\mathcal{M}}=\left<\bm{g} \bm{z},(T \phi) \bm{Z}\right>_{\mathcal{N}} $$for all vector fields $\bm{Z}$ on $\mathcal{M}$ and $\bm{z}$ on $\mathcal{N}$.
$$ \begin{aligned} & \bm{F}^{\mathrm{t}}=F^{\mathrm{t} A}{ }_{. a} \mathbf{G}_A \otimes \mathbf{a}^a \\ & =G^{A B}(T \phi)_B^a g_{a b} \mathbf{E}_A \otimes \mathbf{e}^{* b} \\ & =F_{. B}^b g_{a b} G^{A B} \mathbf{G}_A \otimes \mathbf{a}^a \\ & =\partial_B x^b g_{a b} G^{A B} \mathbf{G}_A \otimes \mathbf{a}^a \\ & =\partial_B x^b \mathbf{G}^B \otimes \mathbf{a}_b \\ & = \mathbf{A}^b \otimes \mathbf{a}_b \\ \end{aligned} $$with $\mathbf{A}^i \triangleq \partial_J x^i \mathbf{G}^J$
$$ \begin{aligned} \bm{F}^{-\mathrm{t}} & =\left(\bm{F}^{-1}\right)^{\mathrm{t}} \\ & =\left(\bm{F}^{\mathrm{t}}\right)^{-1} \\ & =F^{-\mathrm{t}}{ }^a_{~~A} \mathbf{a}_a \otimes \mathbf{G}^A \\ & =F^{-1}{ }_{~~b} G_{A B} g^{a b} \mathbf{a}_a \otimes \mathbf{G}^A \\ & =\partial_b X^B G_{A B} g^{a b} \mathbf{a}_a \otimes \mathbf{G}^A . \end{aligned} $$ $$ \begin{array}{llll} \bm{F} & = \mathbf{g}_I\otimes\mathbf{G}^I \\ & = \mathbf{a}_i\otimes\mathbf{A}^i \\ \bm{F}^{-1} & = \mathbf{A}_i\otimes\mathbf{a}^i \\ \bm{F}^{-1} & \stackrel{?}{=} \mathbf{G}_I\otimes\mathbf{g}^I \\ \bm{F}^* & \stackrel{~}{=} \mathbf{G}^I\otimes\mathbf{g}_I \\ \bm{F}^t & \stackrel{?}{=} \mathbf{A}^i\otimes\mathbf{a}_i \\ & = \mathbf{a}^i\otimes\mathbf{A}_i \\ & = \mathbf{g}^I\otimes\mathbf{G}_I \\ \hline \bm{1} & = \mathbf{g}^I\otimes\mathbf{g}_I \\ \bm{1} & = \mathbf{g}_I\otimes\mathbf{g}^I \\ \bm{1} & = \mathbf{a}^i\otimes\mathbf{a}_i \\ \bm{1} & = \mathbf{G}_I\otimes\mathbf{G}^I \\ \bm{1} & = \mathbf{A}_i\otimes\mathbf{A}^i \\ \bm{1} & = \mathbf{A}^i\otimes\mathbf{A}_i \\ \bm{1} & = \mathbf{a}_i\otimes\mathbf{a}^i \\ \bm{1} & = \mathbf{G}^I\otimes\mathbf{G}_I \\ \hline & = \mathbf{A}^i\otimes\mathbf{G} \\ & = \mathbf{A}^i\otimes\mathbf{g} \\ & = \mathbf{a}_i\otimes\mathbf{G} \\ & = \mathbf{a}_i\otimes\mathbf{g} \\ & = \mathbf{a}^i\otimes\mathbf{G} \\ & = \mathbf{a}^i\otimes\mathbf{g} \\ & = \mathbf{g}^I\otimes\mathbf{a} \\ & = \mathbf{g}^I\otimes\mathbf{A} \\ & = \mathbf{A}_i\otimes\mathbf{G} \\ & = \mathbf{A}_i\otimes\mathbf{g} \\ & = \mathbf{g}_I\otimes\mathbf{a} \\ & = \mathbf{g}_I\otimes\mathbf{A} \\ & = \mathbf{G}_I\otimes\mathbf{a} \\ & = \mathbf{G}_I\otimes\mathbf{A} \\ & = \mathbf{G}^I\otimes\mathbf{a} \\ & = \mathbf{G}^I\otimes\mathbf{A} \\ \end{array} $$