Jacobi field

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:

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:

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

$◼$
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

• $F(t,0)=\gamma (t)$,
• each curve ${\gamma }_s:t\mapsto F(t,s)$ is a geodesic for every $s\in (-\delta ,\delta )$,
• $J(t)={\partial }_{s}F(t,0)$.

Constant curvature

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

where $E$ is any parallel normal vector field along $\gamma$, and $u(t)$ is given by

Surfaces

In the particular case of a surface, the Riemann curvature tensor satisfies

where $K$ is the Gaussian curvature, so if the geodesic is of unit length the Jacobi equation becomes

and if $J$ is normal to $U$ then we have