$$
\begin{array}{l} \tilde {\boldsymbol {H}} _ {i j} = \frac {\partial^ {2} (\mathcal {L} \circ \varphi^ {- 1})}{\partial \psi^ {i} \partial \psi^ {j}} - \tilde {\boldsymbol {\Gamma}} _ {i j} ^ {k} \frac {\partial (\mathcal {L} \circ \varphi^ {- 1})}{\partial \psi^ {k}} \\ = \frac {\partial^ {2} \mathcal {L}}{\partial \boldsymbol {\theta} ^ {m} \partial \boldsymbol {\theta} ^ {n}} \frac {\partial \boldsymbol {\theta} ^ {m}}{\partial \psi^ {i}} \frac {\partial \boldsymbol {\theta} ^ {n}}{\partial \psi^ {j}} + \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial^ {2} \boldsymbol {\theta} ^ {o}}{\partial \psi^ {i} \partial \psi^ {j}} - \left(\Gamma_ {m n} ^ {o} \frac {\partial \psi^ {k}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial \boldsymbol {\theta} ^ {m}}{\partial \psi^ {i}} \frac {\partial \boldsymbol {\theta} ^ {n}}{\partial \psi^ {j}} + \frac {\partial^ {2} \boldsymbol {\theta} ^ {o}}{\partial \psi^ {i} \partial \psi^ {j}} \frac {\partial \psi^ {k}}{\partial \boldsymbol {\theta} ^ {o}}\right) \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial \boldsymbol {\theta} ^ {o}}{\partial \psi^ {k}} \\ = \frac {\partial^ {2} \mathcal {L}}{\partial \boldsymbol {\theta} ^ {m} \partial \boldsymbol {\theta} ^ {n}} \frac {\partial \boldsymbol {\theta} ^ {m}}{\partial \psi^ {i}} \frac {\partial \boldsymbol {\theta} ^ {n}}{\partial \psi^ {j}} + \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial^ {2} \boldsymbol {\theta} ^ {o}}{\partial \psi^ {i} \partial \psi^ {j}} - \Gamma_ {m n} ^ {o} \frac {\partial \psi^ {k}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial \boldsymbol {\theta} ^ {m}}{\partial \psi^ {i}} \frac {\partial \boldsymbol {\theta} ^ {n}}{\partial \psi^ {j}} \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial \boldsymbol {\theta} ^ {o}}{\partial \psi^ {k}} - \frac {\partial^ {2} \boldsymbol {\theta} ^ {o}}{\partial \psi^ {i} \partial \psi^ {j}} \frac {\partial \psi^ {k}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial \boldsymbol {\theta} ^ {o}}{\partial \psi^ {k}} \\ = \frac {\partial^ {2} \mathcal {L}}{\partial \boldsymbol {\theta} ^ {m} \partial \boldsymbol {\theta} ^ {n}} \frac {\partial \boldsymbol {\theta} ^ {m}}{\partial \psi^ {i}} \frac {\partial \boldsymbol {\theta} ^ {n}}{\partial \psi^ {j}} + \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}} \frac {\partial^ {2} \boldsymbol {\theta} ^ {o}}{\partial \psi^ {i} \partial \psi^ {j}} - \Gamma_ {m n} ^ {o} \frac {\partial \boldsymbol {\theta} ^ {m}}{\partial \psi^ {i}} \frac {\partial \boldsymbol {\theta} ^ {n}}{\partial \psi^ {j}} \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}} - \frac {\partial^ {2} \boldsymbol {\theta} ^ {o}}{\partial \psi^ {i} \partial \psi^ {j}} \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}} \\ = \frac {\partial \boldsymbol {\theta} ^ {m}}{\partial \psi^ {i}} \frac {\partial \boldsymbol {\theta} ^ {n}}{\partial \psi^ {j}} \left(\frac {\partial^ {2} \mathcal {L}}{\partial \boldsymbol {\theta} ^ {m} \partial \boldsymbol {\theta} ^ {n}} - \Gamma_ {m n} ^ {o} \frac {\partial \mathcal {L}}{\partial \boldsymbol {\theta} ^ {o}}\right) \\ = \frac {\partial \boldsymbol {\theta} ^ {m}}{\partial \psi^ {i}} \frac {\partial \boldsymbol {\theta} ^ {n}}{\partial \psi^ {j}} \boldsymbol {H} _ {m n}. \tag {8} \\ \end{array}
$$