# Connection of the curvature tensor on a geodesic and a Jacobi field

Let $$M$$ be a Riemannian manifold equipped with the Levi-Civita connection $$\nabla$$, $$\gamma : I \rightarrow M$$ a geodesic of $$M$$ such that $$0 \in I$$. Let $$Y,Z$$ two Jacobi fields along $$\gamma$$ vanishing at $$0$$, that is $$\left\{ \begin{array}{ll} \nabla^2_{\dot{\gamma}} Y + R(\dot{\gamma}, Y)\dot{\gamma} = 0,\quad Y(0) = 0 \\ \nabla^2_{\dot{\gamma}} Z + R(\dot{\gamma}, Z)\dot{\gamma} = 0, \quad Z(0) = 0 \end{array} \right.$$ where $$R$$ is the curvature tensor of $$\nabla$$.

Now I would like to show that $$\big\langle \nabla_{\dot{\gamma}}(R(\dot{\gamma}, Y)\dot{\gamma}), \nabla_{\dot{\gamma}} Z \big\rangle = \big\langle R(\dot{\gamma}, \nabla_{\dot{\gamma}}Y)\dot{\gamma}, \nabla_{\dot{\gamma}} Z \big\rangle$$ I tried to go back to the definition of $$R$$, or used the fact that $$\partial_t\langle Y,Z \rangle = \langle \nabla_{\dot{\gamma}}Y,Z \rangle + \langle Y,\nabla_{\dot{\gamma}}Z \rangle$$ for every vector fields along $$\gamma$$, but I didn't succeed. Do you have any idea how to proceed ?

Note that this equation holds at $$t=0$$. For any vector $$V$$, we have at $$t=0$$ \begin{align} \langle \nabla_{\dot\gamma} (R(\dot\gamma, Y)\dot\gamma), V\rangle &= \frac{d}{dt}\langle R(\dot\gamma, Y)\dot\gamma, V\rangle - \langle R(\dot\gamma, Y)\dot\gamma, \nabla_{\dot\gamma}V\rangle \tag{1}\\ &=\frac{d}{dt}\langle R(\dot\gamma, V)\dot\gamma, Y\rangle \tag{2}\\ &=\langle \nabla_{\dot\gamma} (R(\dot\gamma, V)\dot\gamma), Y\rangle+ \langle R(\dot\gamma, V)\dot\gamma, \nabla_{\dot\gamma}Y\rangle \tag{3}\\ &= \langle R(\dot\gamma, \nabla_{\dot\gamma}Y)\dot\gamma, V\rangle .\tag{4} \end{align} In $$(1)$$ we used $$\langle R(\dot\gamma, Y)\dot\gamma, \nabla_{\dot\gamma}V\rangle = 0$$ since $$Y(0)=0$$. For the same reason, we know that $$\langle \nabla_{\dot\gamma} (R(\dot\gamma, V)\dot\gamma), Y\rangle$$ vanishes in $$(3)$$. In equations $$(2)$$ and $$(4)$$ we used the symmetry property $$\langle R(X,Y)Z,W\rangle = \langle R(Z,W)X,Y\rangle.$$ Setting $$V = \nabla_{\dot\gamma}Z$$ gives the equation.