Second variation of geodesic energy ($E^{\prime \prime }(0)=I(V,V)$)

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.