= Instability of a closed geodesic in positive even-dimensional curvature
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.
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