Variation vector field ($V$)

For a variation $F(s,t)$ of a curve $\gamma (t)=F(0,t)$, the variation vector field is $V(t)={\partial }_{s}F(0,t)$. A fixed-endpoint variation has $V=0$ at both endpoints.

Along a geodesic $\gamma$ with tangent $T$, the index form on endpoint-vanishing vector fields is

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

On continuous piecewise smooth vector field along a curve with zero endpoints, the radical of a bilinear form of the Riemannian index form is exactly the space of endpoint-vanishing Jacobi fields. Integration by parts against tests supported inside each smooth piece gives the Jacobi equation; tests with arbitrary values at breakpoints force continuity of the covariant derivative, hence a global smooth Jacobi field. This characterization uses $I(V,W)=0$ for every test field $W$, not merely $I(V,V)=0$, which is insufficient for an indefinite form.

// Target: riemannian-geometry.bigb

On continuous piecewise smooth perpendicular fields with zero endpoints, the Riemannian index form is nonnegative exactly when no point strictly inside the geodesic segment is conjugate to its initial point. It is positive on every nonzero field exactly when no conjugate point occurs up to and including the final endpoint. At a first conjugate endpoint its nullspace consists of the endpoint-vanishing Jacobi fields. These statements distinguish nonnegative quadratic forms from strictly positive ones and do not assume global minimization.

Let a nonzero Jacobi field $J$ vanish at times zero and $a$. On a longer geodesic segment ending at $b>a$, extend it by zero to a continuous piecewise smooth field $Z$. Then $I(Z,Z)=0$. Choose an endpoint-vanishing field $P$ with $P(a)=-D_{t}J(a)$; integration by parts gives $I(Z,P)=-|D_{t}J(a){|}^2<0$. Thus $I(Z+\epsilon P,Z+\epsilon P)<0$ for small positive $\epsilon$. Smooth approximation preserves negativity. The second variation of geodesic energy supplies a shorter path, so a minimizing geodesic cannot extend beyond its first conjugate point.

// Target: differential-geometry.bigb

Along a unit-speed minimizing geodesic of positive length $L$, take perpendicular parallel orthonormal fields $E_i$ and set $V_i=\sin (\pi t/L)E_i$. The second variation of geodesic energy makes each index form nonnegative, while $\mathrm{Ric}\ge (n-1)\kappa g$ bounds their sum as displayed. For $n\ge 2$ and $\kappa >0$, lengths greater than $\pi /\sqrt{\kappa }$ are impossible. Completeness is what supplies a minimizing geodesic between arbitrary points in the Bonnet-Myers theorem.

For $E=\frac{1}{2}\int |\dot{\gamma }{|}^2$ and a geodesic base curve, a variation field $V$ satisfiesThe endpoint term vanishes for fixed endpoints and for periodic variations of closed geodesics. The integral is the Riemannian index form. This uses the Levi-Civita connection and the convention $R(X,Y)={\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]}$.

On an oriented even-dimensional Riemannian manifold of strictly positive sectional curvature, a nonconstant closed geodesic has a length-decreasing smooth variation. Its parallel transport fixes the tangent and acts on the odd-dimensional normal space by a special orthogonal group element. An odd-dimensional special orthogonal transformation has a fixed vector, so there is a nonzero periodic parallel normal field $V$. The second variation of geodesic energy is then strictly negative. The Cauchy-Schwarz inequality converts lower energy into strictly lower length because the original geodesic has constant speed. For an embedded geodesic the small variation is a smooth isotopy; otherwise it is a deformation through immersed loops.

For a unit-speed geodesic and a fixed-endpoint variation, the second variation of length is the index form of the normal component of its variation vector field.

For a variation of a unit-speed geodesic, let $T$ be its tangent, $J$ its variation vector field and $A={\nabla }_{s}J$ its variation acceleration. The second variation of Riemannian arc length has the displayed endpoint term in addition to the Riemannian index form of $J^{\perp }=J-⟨J,T⟩T$. Differentiating the speed twice gives $|D_{t}J{|}^2-⟨D_{t}J,T{⟩}^2+⟨D_s{D}_{t}J,T⟩$. Commuting derivatives and integrating $D_{t}A$ by parts gives the formula. The term cannot be discarded unless the endpoints are fixed or an appropriate endpoint condition makes it zero.