The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields

(Chart computation of the commutator.) Let $V$ be a smooth field along $F$, fix a parameter pair $(r,t)$ and a chart around $F(r,t)$. By [L3] applied to the curves with the other parameter fixed, $D_r{V}^a={\mathrm{\partial }}_r{V}^a+{\sum }_{b,c}{\mathrm{\Gamma }}^a{}_{bc}{\mathrm{\partial }}_r{F}^b{V}^c$ and $D_t{V}^a={\mathrm{\partial }}_t{V}^a+{\sum }_{b,c}{\mathrm{\Gamma }}^a{}_{bc}{\mathrm{\partial }}_t{F}^b{V}^c$ (indices here are chart components of the fields). Applying $D_t$ to the first expression and $D_r$ to the second and subtracting, the second-order terms ${\mathrm{\partial }}_t{\mathrm{\partial }}_r{V}^a$ and ${\mathrm{\partial }}_r{\mathrm{\partial }}_t{V}^a$ cancel by [L2], and the remaining terms are exactly ${\sum }_{b,c,d}({\mathrm{\partial }}_c{\mathrm{\Gamma }}^a{}_{db}-{\mathrm{\partial }}_d{\mathrm{\Gamma }}^a{}_{cb}+{\mathrm{\Gamma }}^a{}_{ce}{\mathrm{\Gamma }}^e{}_{db}-{\mathrm{\Gamma }}^a{}_{de}{\mathrm{\Gamma }}^e{}_{cb})T^c(J^F{)}^d{V}^b$, with the Christoffel-product terms collected by relabelling summed indices; this is the coordinate expression whose coefficient is the curvature component of [L1].

(The two derivatives commute on the tangent.) If $D_{t}T=0$, then by the component expressions of [L3] with $V=T$ and by [L2], $D_t{J}^F-D_{r}T$ has components ${\mathrm{\partial }}_t{\mathrm{\partial }}_r{F}^a+{\mathrm{\Gamma }}^a{}_{bc}{T}^b(J^F{)}^c-{\mathrm{\partial }}_r{\mathrm{\partial }}_t{F}^a-{\mathrm{\Gamma }}^a{}_{bc}(J^F{)}^b{T}^c$, which vanishes because the mixed partials commute and ${\mathrm{\Gamma }}^a{}_{bc}$ is symmetric. Hence $D_t{J}^F=D_{r}T$.

(Jacobi fields with prescribed initial data.) Let $p:=\gamma (s_0)$ and extend a basis $E_1(s_0),\dots ,E_4(s_0)$ of $T_{p}M$ to a parallel frame $E_1,\dots ,E_4$ along $\gamma$ by [L5]. Writing a field along $\gamma$ as $J={\sum }_i{J}^i{E}_i$, [L3] and $D_s{E}_i=0$ give $D_s^{2}J={\sum }_i(J^i{)}^{\mathrm{\prime }\mathrm{\prime }}{E}_i$. For each $j$, define the smooth coefficient functions $C^i{}_j$ by the unique frame expansion $R({\gamma }^{\mathrm{\prime }},E_j){\gamma }^{\mathrm{\prime }}={\sum }_i{C}^i{}_j{E}_i$; then [L1] gives $R({\gamma }^{\mathrm{\prime }},J){\gamma }^{\mathrm{\prime }}={\sum }_{i,j}{J}^j{C}^i{}_j{E}_i$. The Jacobi equation is therefore equivalent to the second-order linear system $(J^i{)}^{\mathrm{\prime }\mathrm{\prime }}={\sum }_j{C}^i{}_j{J}^j$ for the coefficient functions.

(Identification with the curvature and conclusion of (a).) The coefficient of $T^c(J^F{)}^d{V}^b$ in step 1.1 is ${\mathrm{\partial }}_c{\mathrm{\Gamma }}^a{}_{db}-{\mathrm{\partial }}_d{\mathrm{\Gamma }}^a{}_{cb}+{\sum }_e({\mathrm{\Gamma }}^a{}_{ce}{\mathrm{\Gamma }}^e{}_{db}-{\mathrm{\Gamma }}^a{}_{de}{\mathrm{\Gamma }}^e{}_{cb})$, which is the coordinate formula for $R^a{}_{bcd}$ of [L1]. Therefore $(D_t{D}_r-D_r{D}_t)V=R(T,J^F)V$ at the chosen point; the point was arbitrary and the identity is between smooth fields, so (a) holds.

(Solution of the linear system.) The system of step 1.3 is equivalent to the first-order system $Y^{\mathrm{\prime }}=A(s)Y$ for $Y=(J^1,\dots ,J^4,(J^1{)}^{\mathrm{\prime }},\dots ,(J^4{)}^{\mathrm{\prime }})$ and the continuous $8\times 8$ block matrix $A=\left(\begin{array}{cc}0& I\\ C& 0\end{array}\right)$. By [L6], on each compact subinterval of $I$ there is a unique solution with prescribed initial value; two such solutions agree on the overlap of their intervals, so they patch to a unique solution on all of $I$. Every pair $(J_0,J_1)=(J^i(s_0),(J^i{)}^{\mathrm{\prime }}(s_0))$ is an admissible initial value, and since the frame is a basis, prescribing $(J_0,J_1)\in T_{p}M\oplus T_{p}M$ prescribes exactly these coefficients.

(Deviation equation.) By step 1.2 and $D_{t}T=0$, $D_t^2{J}^F=D_t{D}_{r}T$, and by step 2.1 with $V=T$, $D_t{D}_{r}T=D_r{D}_{t}T+R(T,J^F)T=0+R(T,J^F)T$. Hence $D_t^2{J}^F=R(T,J^F)T$; by [L7] the field $t\mapsto J^F(r,t)$ is a Jacobi field along the geodesic $t\mapsto F(r,t)$ for each $r$, which proves (b).

(Vector space and dimension.) By step 2.2 the map $J\mapsto (J(s_0),D_{s}J(s_0))$ from the set of smooth Jacobi fields along $\gamma$ to $T_{p}M\oplus T_{p}M$ is well defined, linear over $\mathbb{R}$, injective (two Jacobi fields with the same data agree by uniqueness of step 2.2) and surjective (existence in step 2.2). It is therefore a linear isomorphism, the space of Jacobi fields has dimension $\dim (T_{p}M\oplus T_{p}M)=8$, and (c) holds.

(Tangential component.) For a Jacobi field $J$ along a geodesic, put $a(s):=g(J,{\gamma }^{\mathrm{\prime }})$. Metric compatibility and $D_s{\gamma }^{\mathrm{\prime }}=0$ give $a^{\mathrm{\prime }\mathrm{\prime }}=g(D_s^{2}J,{\gamma }^{\mathrm{\prime }})=g(R({\gamma }^{\mathrm{\prime }},J){\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=0$ by [L4], so $a$ is affine. If in addition $g({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }})=-1$, the normal component defined in Jacobi fields, geodesic variations and conjugate instants along a geodesic is $J^{\perp }=J+a{\gamma }^{\mathrm{\prime }}$. Therefore $D_s^2{J}^{\perp }=D_s^{2}J+a^{\mathrm{\prime }\mathrm{\prime }}{\gamma }^{\mathrm{\prime }}+2a^{\mathrm{\prime }}{D}_s{\gamma }^{\mathrm{\prime }}+a{D}_s^2{\gamma }^{\mathrm{\prime }}=R({\gamma }^{\mathrm{\prime }},J){\gamma }^{\mathrm{\prime }}$. Since $J=J^{\perp }-a{\gamma }^{\mathrm{\prime }}$ and $R({\gamma }^{\mathrm{\prime }},{\gamma }^{\mathrm{\prime }}){\gamma }^{\mathrm{\prime }}=0$ by the first-slot skew-symmetry of [L4], this is $R({\gamma }^{\mathrm{\prime }},J^{\perp }){\gamma }^{\mathrm{\prime }}$. Thus $J^{\perp }$ is a Jacobi field, proving (d).