Distortion of Volume along Geodesic Curves

Question

Let ${M}$ be a smooth manifold. Let ${X}_{m}\in\Gamma^{\infty}({M},{T}{M}),{1}\leqslant{m}\leqslant\mathrm{dim}({M})$ be a collection of vector fields. The volume of the shape spanned by these vectors is given by \begin{align*} \sqrt{\mathrm{det}({g}\langle{X}_{m},{X}_{n}\rangle)}{\,}{.} \end{align*} Given a smooth geodesic curve $\gamma:\mathbb{R}\to{M}$, consider the parallel transport of these vector fields along the geodesic with the initial condition \begin{align*} \frac{\mathcal{D}{X}_{m}(\gamma({s}))}{\mathcal{D}{s}}\bigg\vert_{{s}={0}}={0}{\,}{.} \end{align*} A taylor expansion of this scalar product gives \begin{align*} {g}(\gamma({s}))\langle{X}_{m}(\gamma({s})),{X}_{n}(\gamma({s}))\rangle&={g}(\gamma({0}))\langle{X}_{m}(\gamma({0})),{X}_{n}(\gamma({0}))\rangle+\frac{\mathrm{d}{g}(\gamma({s}))\langle{X}_{m}(\gamma({s})),{X}_{n}(\gamma({s}))\rangle}{\mathrm{d}{s}}\bigg\vert_{{s}={0}}{s}\\ &+\frac{1}{2}\frac{\mathrm{d}^{2}{g}(\gamma({s}))\langle{X}_{m}(\gamma({s})),{X}_{n}(\gamma({s}))\rangle}{\mathrm{d}{s}^{2}}\bigg\vert_{{s}={0}}{s}^{2}+\mathcal{O}({s}^{3}){\,}{.} \end{align*} From looking at the first derivative term \begin{align*} \frac{\mathrm{d}{g}(\gamma({s}))\langle{X}_{m}(\gamma({s})),{X}_{n}(\gamma({s}))\rangle}{\mathrm{d}{s}}={g}(\gamma({s}))\bigg\langle\frac{\mathcal{D}{X}_{m}(\gamma({s}))}{\mathcal{D}{s}},{X}_{n}(\gamma({s}))\bigg\rangle+{g}(\gamma({s}))\bigg\langle{X}_{m}(\gamma({s})),\frac{\mathcal{D}{X}_{n}(\gamma({s}))}{\mathcal{D}{s}}\bigg\rangle{\,}{,} \end{align*} one can follow that \begin{align*} \frac{\mathrm{d}{g}(\gamma({s}))\langle{X}_{m}(\gamma({s})),{X}_{n}(\gamma({s}))\rangle}{\mathrm{d}{s}}\bigg\vert_{{s}={0}}={0}{\,}{,} \end{align*} from the initial condition. From looking at the second derivative term \begin{align*} \frac{\mathrm{d}^{2}{g}(\gamma({s}))\langle{X}_{m}(\gamma({s})),{X}_{n}(\gamma({s}))\rangle}{\mathrm{d}{s}^{2}}&={g}(\gamma({s}))\bigg\langle\frac{\mathcal{D}^{2}{X}_{m}(\gamma({s}))}{\mathcal{D}{s}^{2}},{X}_{n}(\gamma({s}))\bigg\rangle+{2}{g}(\gamma({s}))\bigg\langle\frac{\mathcal{D}{X}_{m}(\gamma({s}))}{\mathcal{D}{s}},\frac{\mathcal{D}{X}_{n}(\gamma({s}))}{\mathcal{D}{s}}\bigg\rangle\\ &+{g}(\gamma({s}))\bigg\langle{X}_{m}(\gamma({s})),\frac{\mathcal{D}^{2}{X}_{n}(\gamma({s}))}{\mathcal{D}{s}^{2}}\bigg\rangle{\,}{,} \end{align*} and the deviation equation \begin{align*} \frac{\mathcal{D}^{2}({X}_{m})^{\varrho}(\gamma({s}))}{\mathcal{D}{s}^{2}}=-\mathrm{Riem}^{\varrho}{}_{\alpha\mu\nu}(\gamma({s}))\frac{\mathrm{d}{x}^{\alpha}(\gamma({s}))}{\mathrm{d}{s}}\frac{\mathrm{d}{x}^{\nu}(\gamma({s}))}{\mathrm{d}{s}}({X}_{m})^{\mu}(\gamma({s})){\,}{,} \end{align*} one can follow \begin{align*} \mathrm{vol}({s})=\mathrm{vol}({0}){\,}\sqrt{\mathrm{det}(\delta^{\varrho}_{\mu}-{s}^{2}\mathrm{Riem}^{\varrho}{}_{\alpha\mu\nu}(\gamma({0}))({x}^{\alpha}\circ\gamma)^{\prime}({0})({x}^{\nu}\circ\gamma)^{\prime}({0}))}{\,}{,} \end{align*} where $\mathrm{vol}({s}):=\mathrm{det}({g}(\gamma({s}))\langle{X}_{m}(\gamma({s})),{X}_{n}(\gamma({s}))\rangle)$. But according to the literature It should be \begin{align*} \mathrm{vol}({s})=\mathrm{vol}({0}){\,}\bigg({1}-\frac{{s}^{2}}{6}\mathrm{Ric}_{\mu\nu}(\gamma({0}))\bigg(\frac{\mathrm{d}{x}^{\mu}(\gamma({s}))}{\mathrm{d}{s}}\frac{\mathrm{d}{x}^{\nu}(\gamma({s}))}{\mathrm{d}{s}}\bigg)\bigg\vert_{{s}={0}}\bigg){\,}{.} \end{align*} Have I made a mistake along the way, or how can I arrive at that final equation?