I would like to know wether the derivation of the following formula is wrong (I'm a bit confused because we have to show a different formula in a exercise sheet [however I don't want a solution of the exercise, I only want to know where my mistake is or wether the statement of the exercise is wrong]).
Suppose that the Riemannian Manifold $M$ has constant sectional curvature $\kappa.$ Let $p\in M$, $e_1,...,e_n$ be a orthonormal basis of $T_pM$ and $(U,\varphi)$ be the corresponding normal chart, where $U$ is a normal ball around $p.$ Then $$g_{ij}(exp_p(v))=\frac{S_{\kappa}^2(\Vert v \Vert)}{\Vert v \Vert^2}\delta_{ij},$$ where $S_{\kappa}(t)=\frac{\sin(\sqrt{\kappa}t)}{\sqrt{\kappa}}$ if $\kappa > 0$, $S_{\kappa}(t)=t$ if $\kappa=0$ and $S_{\kappa}(t)=\frac{\sinh(\sqrt{-\kappa}t)}{\sqrt{-\kappa}}$ if $\kappa < 0.$
My attempt was the following: If $w(t)$ is the parallel transport of some $w_0$ along the curve $t\mapsto exp_p(t\frac{v}{\Vert v \Vert})$, then $$J(t)=S_{\kappa}(t)w(t)$$ is the unique Jacobi field along this curve with $J(0)=0,J'(0)=w_0.$ On the other hand we know that this Jacobi field can be written as $$J(t)=(d exp_p)_{t\frac{v}{\Vert v \Vert}}(tw_0).$$ Then because parallel transport maintains inner products and because of the definition of the normal chart we get (where $J_k$ is the Jacobi field with $J_k(0)=0$ and $J_k'(0)=e_k$) $$S_{\kappa}^2(\Vert v \Vert)\delta_{ij}=S_{\kappa}^2(\Vert v \Vert)\langle e_i,e_j\rangle =S_{\kappa}^2(\Vert v \Vert) \langle w_i(\Vert v \Vert), w_j(\Vert v \Vert)\rangle=\langle J_i(\Vert v \Vert),J_j(\Vert v \Vert)\rangle = \Vert v \Vert ^2 \langle (d exp_p)_{v}(e_i),( d exp_p)_{v}(e_j)\rangle=\Vert v \Vert ^2 g_{ij}(exp_p(v)),$$ which proves the claim.
However we should prove $$g_{ij}(exp_p(v))=\frac{v^iv^j}{\Vert v \Vert ^2}+\frac{S_{\kappa}^2(\Vert v \Vert)}{\Vert v \Vert^2}\left(\delta_{ij} -\frac{v^iv^j}{\Vert v \Vert ^2}\right)$$ for $v=v^ie_i.$
I would be vers thankful if someone could clarify whethet it's my mistake (and in that case where it is) or wether the statement of the exercise is wrong.