 Solution

Write $T=\dot{\gamma }$ along an affinely parametrized geodesic and $D_t={\nabla }_T$ for the covariant derivative from the Levi-Civita connection. We use the Riemann curvature tensor convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, for which positive sectional curvature means $⟨R(X,Y)Y,X⟩>0$ on an orthonormal pair. A Jacobi field is a smooth vector field along $\gamma$ satisfyingSuch a field describes the first-order separation of nearby geodesics, as the geodesic variation construction in part (c) shows.

For $t_0>0$, $\gamma (t_0)$ is a conjugate point of $\gamma (0)$ along this geodesic if a nonzero Jacobi field satisfies $J(0)=J(t_0)=0$. The multiplicity of a conjugate point isIt is equivalently the dimension of the kernel of the differential of the Riemannian exponential map at $t_0\dot{\gamma }(0)$, by Jacobi fields vanishing at their initial point. Conjugacy concerns the specified geodesic and its two parameter values, even when its endpoints represent the same point of the Riemannian manifold.