= Solution
Use the <energy of a curve> normalization
$$
E(\gamma)=\frac12\int_a^b|\dot\gamma|_g^2\,dt.
$$
For a variation $F(s,t)$ put $T=\partial_tF$ and $V=\partial_sF$. The <Levi-Civita connection> is torsion free, so $\nabla_sT=\nabla_tV$. Differentiating energy once gives $E'=[\langle V,T\rangle]_a^b-\int\langle V,\nabla_tT\rangle$. Along a geodesic, $\nabla_tT=0$. Differentiating again, commuting covariant derivatives, and integrating by parts gives the <second variation of geodesic energy>:
$$
\boxed{E''(0)=
[\langle\nabla_sV,T\rangle]_a^b+
\int_a^b\bigl(|\nabla_tV|^2-\langle R(V,T)T,V\rangle\bigr)\,dt.}
$$
The curvature convention is $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$, consistent with the specified positive <sectional curvature>. Indeed $\nabla_s\nabla_tV=\nabla_t\nabla_sV+R(V,T)V$, and $\langle R(V,T)V,T\rangle=-\langle R(V,T)T,V\rangle$. The endpoint term vanishes for fixed endpoints or for periodic variations of a closed geodesic. The integral is the <Riemannian index form> $I(V,V)$.
Let $\dim M=2m$. <Parallel transport> around the closed geodesic preserves the metric and orientation, so lies in the <special orthogonal group>. It fixes the nonzero tangent $T$, and its restriction to $T^\perp$ is an orientation-preserving orthogonal map of odd dimension $2m-1$. An <odd-dimensional special orthogonal transformation has a fixed vector>: nonreal eigenvalues occur in conjugate pairs, while an odd-dimensional real orthogonal map with determinant one must have an eigenvalue $+1$. Choose a nonzero fixed vector $v$ normal to $T$ and parallel-transport it along the curve. It gives a nonzero smooth periodic normal field $V$ with $\nabla_tV=0$.
For the exponential variation $\gamma_s(t)=\exp_{\gamma(t)}(sV(t))$, the endpoints match periodically. Strictly positive <sectional curvature> gives
$$
E''(0)=-\int\langle R(V,T)T,V\rangle\,dt<0,\qquad E'(0)=0.
$$
Thus $E(\gamma_s)<E(\gamma)$ for small nonzero $s$. By the <Cauchy-Schwarz inequality>,
$$
L(\gamma_s)^2\leq2(b-a)E(\gamma_s)
<2(b-a)E(\gamma)=L(\gamma)^2,
$$
where the final equality uses the geodesic's constant speed. This proves the <instability of a closed geodesic in positive even-dimensional curvature>.
For an embedded closed geodesic, sufficiently small variations remain embeddings, hence give a <smooth isotopy> with strictly shorter curves. For a nonembedded closed geodesic the construction gives a smooth deformation through immersions; an isotopy class of embeddings is not literally defined for such a curve. The stated isotopy conclusion therefore uses the usual embedded-curve interpretation, while the shorter-loop variation holds without it.
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