= Solution
Parametrize the nonconstant <closed geodesic> at unit speed on $[0,L]$. Its <parallel transport> around the loop fixes the tangent $\dot\gamma(0)$. Since the manifold is orientable, it preserves orientation on the full <tangent space>, and hence has determinant one on the normal space. This normal space has odd dimension because the manifold has even dimension.
An <odd-dimensional special orthogonal transformation has a fixed vector>. Indeed, nonreal <eigenvalues> of a real orthogonal transformation occur in conjugate pairs with product one; the real <eigenvalues> are $\pm1$, and determinant one makes the number of $-1$ <eigenvalues> even. Odd dimension therefore requires a $+1$ <eigenvalue>. Transport a unit fixed normal vector around the <geodesic> to obtain a periodic parallel field $V$, with $D_tV=0$ and $V\perp\dot\gamma$.
The variation $\gamma_s(t)=\exp_{\gamma(t)}(sV(t))$ is defined uniformly for small $s$, because the image of the original loop is <compact>. Periodicity gives closed curves and a <homotopy> through loops. With the standard positive-sectional-curvature <Riemannian index form>, the <second variation of geodesic energy> is
$$
E''(0)=I(V,V)=\int_0^L\bigl(|D_tV|^2-K(\dot\gamma,V)\bigr)dt
=-\int_0^LK(\dot\gamma,V)\,dt<0.
$$
The first variation vanishes for a <closed geodesic>, since the endpoint terms cancel. Therefore $E(\gamma_s)<E(\gamma_0)=L/2$ for sufficiently small nonzero $s$. The <Cauchy-Schwarz inequality> gives
$$
\operatorname{length}(\gamma_s)^2\leq L\int_0^L|\dot\gamma_s|^2dt
=2LE(\gamma_s)<L^2.
$$
Thus \b[the <closed geodesic> is homotopic to a strictly shorter closed curve]. This proves the <instability of a closed geodesic in positive even-dimensional curvature>; completeness of the manifold is unnecessary for this compact-loop variation. Constant loops are excluded by the usual nonconstant meaning of a <closed geodesic> in this assertion.
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