Radial Jacobi tensor

Let $(M,g)$ be a Riemannian manifold of dimension $n\ge 2$ and let $\gamma :I\to M$ be a unit-speed geodesic on an interval $I\subseteq \mathbb{R}$ with nonempty interior and $0\in I$; write $p:=\gamma (0)$ and $T(t):=\dot{\gamma }(t)$, so that $g(T,T)\equiv 1$ and $D_{t}T=0$. The normal space at time $t$ is $N_t:=\{X\in T_{\gamma (t)}M:g_{\gamma (t)}(X,T(t))=0\},$ an $(n-1)$-dimensional subspace, and $N_0\subseteq T_{p}M$ is its initial model.

For $w\in N_0$, let $J_w$ be the unique Jacobi field along $\gamma$ with $J_w(0)=0,D_t{J}_w(0)=w,$ as supplied by Existence and uniqueness of jacobi fields from initial data; the derivative at the included endpoint $0$ is one-sided when $I$ has $0$ as an endpoint, and $J_w$ solves $D_t^{2}J+R(J,T)T=0$ in the conventions of Jacobi field. Then $J_w$ is normal, $g(J_w(t),T(t))=0$ for every $t\in I$: the function $u(t):=g(J_w(t),T(t))$ has $u^{\mathrm{\prime }}(t)=g(D_t{J}_w(t),T(t))$ and $u^{\mathrm{\prime }\mathrm{\prime }}(t)=g(D_t^2{J}_w(t),T(t))=-\mathrm{Rm}(J_w,T,T,T)=0$ by metric compatibility of the Levi-Civita connection (Levi civita connection, Covariant derivative along a curve), the curvature convention $\mathrm{Rm}(X,Y,Z,W)=g(R(X,Y)Z,W)$ (Riemann curvature four-tensor) and the following local metric-compatibility calculation. For smooth local fields $X,Y,Z,W$, expand $XYg(Z,W)-YXg(Z,W)-[X,Y]g(Z,W)=0$ twice by metric compatibility. The mixed first-derivative terms cancel, leaving $g(R(X,Y)Z,W)+g(Z,R(X,Y)W)=0,$ with the curvature commutator of Curvature of an affine connection. Taking $Z=W=T$ gives $g(R(X,Y)T,T)=0$, without invoking the countable-choice-dependent full curvature-symmetry theorem. Thus $u^{\mathrm{\prime }\mathrm{\prime }}=0$, while $u(0)=0$ and $u^{\mathrm{\prime }}(0)=g(w,T(0))=0$ because $w\perp T(0)$; hence $u\equiv 0$.

The radial Jacobi tensor of $\gamma$ is the family of linear maps $A(t):N_0⟶N_t,A(t)w:=J_w(t),$ defined for $t\in I$. It is well defined by the existence and uniqueness of $J_w$, it is linear in $w$ because $J_{\lambda w+\mu w^{\mathrm{\prime }}}-\lambda J_w-\mu J_{w^{\mathrm{\prime }}}$ is a Jacobi field with both initial data zero and therefore vanishes by uniqueness, and it is smooth in $t$ because the coefficients of the Jacobi equation depend smoothly on $t$. Its initial values are $A(0)=0$ and, in the parallel identification introduced next, ${\bar{A}}^{\mathrm{\prime }}(0)={\mathrm{id}}_{N_0}$.

Parallel identification. Let $E_1,\dots ,E_{n-1}$ be an orthonormal basis of $N_0$ and let $P_t:T_{p}M\to T_{\gamma (t)}M$ be parallel transport along $\gamma$, which exists uniquely for every $t\in I$ by Existence and uniqueness of parallel sections and maps $N_0$ onto $N_t$, because $g(P_{t}w,T(t))=g(w,T(0))=0$ for $w\in N_0$ is constant in $t$ by metric compatibility. Composing with $P_t$ represents $A$ by the smooth matrix family $\bar{A}(t):=P_t^{-1}\circ A(t)\in \mathrm{End}(N_0),$ with $\bar{A}(0)=0$ and ${\bar{A}}^{\mathrm{\prime }}(0)={\mathrm{id}}_{N_0}$: differentiating $\bar{A}(t)w=P_t^{-1}{J}_w(t)$ at $t=0$ gives $({\bar{A}}^{\mathrm{\prime }}(0))w=D_t{J}_w(0)=w$ for every $w\in N_0$. In the parallel orthonormal frame $(P_t{E}_j)$ the tensor $A$ is recovered from $\bar{A}$, and the matrix satisfies the Jacobi equation ${\bar{A}}^{\mathrm{\prime }\mathrm{\prime }}+R_{\gamma }\bar{A}=0$, where $R_{\gamma }(t)\in \mathrm{End}(N_0)$ is defined by $R_{\gamma }(t)w:=P_t^{-1}(R(P_{t}w,T(t))T(t))$. This matrix description is a representation of the same tensor, not a second definition: it depends on the chosen parallel frame only through conjugation by the (constant) change of orthonormal basis of $N_0$, so invariance under conjugation transports statements about invertibility, self-adjointness, positivity and trace of $\bar{A}$ to $A$.

The tensor is used only up to the first conjugate instant in the results that follow; no claim about $A(t)$ beyond such an instant is part of this definition.