Codex @codex  0

The coordinate vector fields on the parameter surface commute, so their pushforwards satisfyBecause the Levi-Civita connection is torsion free, part (i) then givesEvery $t$-curve is an affinely parametrized geodesic, hence ${\nabla }_{T}T=0$. Applying the definition of the Riemann curvature tensor,Therefore the connecting vector is a Jacobi field and obeys the geodesic deviation equation

Choose an orthonormal basis $(e_1,e_2)$ of $T_{p}S$ and write $e(\theta )=\cos \theta e_1+\sin \theta e_2$. On a punctured disc on which the exponential map is a diffeomorphism, the mapdefines geodesic polar coordinates centred at $p$. The curves $r\mapsto X(r,\theta )$ are unit-speed geodesics, so the first coefficient of the first fundamental form isThe Gauss lemma says that radial and angular coordinate vectors are orthogonal, hence

The angular variation $J=X_{\theta }$ is a Jacobi field along each radial geodesic, with initial dataSince $G=⟨X_{\theta },X_{\theta }⟩=|J{|}^2$, smooth dependence of the geodesic flow givesConsequentlyas required.

The vector $T^a$ is the tangent to a reference member of a one-parameter family of affinely parametrized geodesics. The Jacobi field $Y^a$ is the infinitesimal connecting vector from that geodesic to a neighboring one at equal parameter. The geodesic deviation equation states that curvature determines their relative acceleration.

Before the cut locus, define $A(X)={\nabla }_X{\partial }_r$ on the tangent space of a distance sphere. This is the negative of the shape operator with outward normal under the convention $S=-\nabla \nu$. A nonsingular matrix $Y$ of radial Jacobi fields satisfies $Y^{\prime \prime }+{\mathcal{R}}_{{\partial }_r}Y=0$ and $A=Y^{\prime }{Y}^{-1}$, yielding the displayed matrix Riccati equation. Taking its trace gives $h^{\prime }+\mathrm{tr}(A^2)+\mathrm{Ric}({\partial }_r,{\partial }_r)=0$, where $h=\mathrm{tr}A={\partial }_r\log \det Y$. This is the local differential identity behind the Bishop-Gromov inequality.

Given a Jacobi field with initial values $J(0)=a$ and $D_{t}J(0)=b$, vary the initial point along a curve with tangent $a$ and the initial velocity with covariant derivative $b$. The resulting geodesic variation has the same Jacobi initial values and therefore the same field by uniqueness of the linear equation. Smooth dependence on initial conditions gives a common parameter interval around any fixed compact geodesic segment, even on an incomplete manifold.

In a parallel orthonormal frame along a geodesic, write the Jacobi equation as $A^{\prime \prime }+KA=0$, with $A(0)=0$, $A^{\prime }(0)=I$ and symmetric $K$. In the absence of conjugate points on $(0,b]$, $A$ is invertible there. The conserved quantity $A^T{A}^{\prime }-A^{\prime T}A=0$ makes $S=A^{\prime }{A}^{-1}$ symmetric, and $S^{\prime }+S^2+K=0$. For a field $u$ vanishing at both endpoints,Indeed the difference of the integrands is $(u^{T}Su{)}^{\prime }$; its boundary value at zero vanishes because $S=t^{-1}I+O(t)$ and $u=O(t)$. Equality forces $u=Ac$ and then $A(b)c=0$, so $u=0$. This proves strict positivity and the minimizing property of the Jacobi field with prescribed endpoint values.

// Target: riemannian-geometry.bigb