Assume we are in a pseudo-Riemannian manifold.
Let's denote a smooth 1-parameter family of geodesics by $\gamma(t, s)$, where $t$ is the parameter along each geodesic and $s$ is the parameter that distinguishes different geodesics in the family. We assume that $\gamma(t, s)$ is twice continuously differentiable with respect to both $t$ and $s$.
The tangent vector to each geodesic is given by $U(t, s) = \frac{\partial \gamma}{\partial t}(t, s)$. Because each curve $\gamma(t, s)$ is a geodesic, $U$ satisfies the geodesic equation:
$$ \frac{D}{dt} U = 0, $$where $\frac{D}{dt}$ denotes the covariant derivative along a curve for the curve $\gamma(t, s)$.
Now, consider the variation vector field $J(t, s) = \frac{\partial \gamma}{\partial s}(t, s)$. This vector field describes how the geodesics in the family vary with $s$, and it satisfies the equation:
$$ \frac{D^2}{dt^2} J + R(J, U)U = 0, $$where $\frac{D}{dt}$ is the covariant derivative along $\gamma(t, 0)$ and $R$ is the Riemann curvature tensor. The expression above is called the Jacobi equation.
Definition
A vector field $X$ defined along a geodesic $\gamma$ is called a Jacobi field if satisfies Jacobi equation, otherwise written
$$ \nabla_{U} \nabla_{U} X+R(X, U) U=0 $$$\blacksquare$
So the variation field of a geodesic variation is a Jacobi field. Here it is shown that any Jacobi field can be realized as the variation field of some geodesic variation of $\gamma$.
I.e., there exist a smooth map $F:I\times (-\delta,\delta) \to \mathcal{S}$ with
See @lee2006riemannian lemma 10.8.
Lemma
Suppose $(M, g)$ is a Riemannian manifold with constant sectional curvature $C$, and $\gamma$ is a unit speed geodesic in $M$. The normal Jacobi fields along $\gamma$ vanishing at $t = 0$ are precisely the vector fields
$$ J(t) = u(t)E(t), $$
where $E$ is any parallel normal vector field along $\gamma$, and $u(t)$ is given by
$$ u(t) = \begin{cases} t & \text{if } C = 0; \\ R\sin(\frac{t}{R}) & \text{if } C = R^2 > 0; \\ \sinh(\frac{1}{R}) & \text{if } C = -R^2 < 0. \end{cases} $$In the particular case of a surface, the Riemann curvature tensor satisfies
$$ R(J,U)U = K g(U,U)J - K g(J,U)U $$where $K$ is the Gaussian curvature, so if the geodesic is of unit length the Jacobi equation becomes
$$ \nabla_{U} \nabla_{U} J+K\left(J - g(J,U)U\right)=0, $$and if $J$ is normal to $U$ then we have
$$ \nabla_{U} \nabla_{U} J+KJ=0. $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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