Put the following sentences as a new paragraph after "rather than that of f".
More geometrically speaking, when we regard~$\obj$ as a function on the
surface~$\mathcal{S}{\bar{\iter}}$ and approximate it in the tangent space~$\mathcal{T}{\bar{\iter}}$, the
second-order Taylor expansion of~$\obj$ does not attain the same accuracy as it does in~$\R^n$, because
neither this Taylor expansion nor~$\mathcal{T}{\bar{\iter}}$ can reflect the curvature information
of~$\mathcal{S}{\bar{\iter}}$. In contrast, the objective function of the \gls{sqp} subproblem achieves
a better accuracy by including such curvature information via~$\nabla_{x,x}^2\lag(\bar{\iter}, \bar{\lm})$.
https://github.com/ragonneau/phd-thesis/blob/f5e9029df3534f5d076922ead00db04702703cfb/content/sqp.tex#L374
Put the following sentences as a new paragraph after "rather than that of f".
More geometrically speaking, when we regard~$\obj$ as a function on the surface~$\mathcal{S}{\bar{\iter}}$ and approximate it in the tangent space~$\mathcal{T}{\bar{\iter}}$, the second-order Taylor expansion of~$\obj$ does not attain the same accuracy as it does in~$\R^n$, because neither this Taylor expansion nor~$\mathcal{T}{\bar{\iter}}$ can reflect the curvature information of~$\mathcal{S}{\bar{\iter}}$. In contrast, the objective function of the \gls{sqp} subproblem achieves a better accuracy by including such curvature information via~$\nabla_{x,x}^2\lag(\bar{\iter}, \bar{\lm})$.