Energy of a curve ($E(\gamma )$)

The energy of a curve is one half the integral of its squared speed.

For $E(\gamma )=\frac{1}{2}{\int }_0^1|\dot{\gamma }{|}^{2}dt$ and variation vector field $V$, the first variation is $[⟨V,\dot{\gamma }⟩{]}_0^1-{\int }_0^1⟨V,D_t\dot{\gamma }⟩dt$. For fixed endpoints the boundary term vanishes, so the critical points of the energy are exactly affinely parametrized geodesics.

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

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.