[proofplan] Argue by contradiction. If $M$ is not simply connected, then $\pi_1(M) \ne 0$, so there is a non-trivial free homotopy class of closed loops, and by [Minimal Geodesic in a Non-Trivial Homotopy Class](/theorems/2731) it contains a length-minimising closed geodesic $\gamma$. Parallel transport once around $\gamma$ defines a linear isometry $P : T_p M \to T_p M$ that fixes $\dot\gamma(0)$, hence restricts to an isometry $P : N \to N$ on the normal bundle $N := \dot\gamma(0)^\perp$, an oriented Euclidean space of even codimension (because $\dim M$ is even, $\dim N = \dim M - 1$ is odd, and orientability of $M$ plus a chosen orientation of $\dot\gamma(0)$ orients $N$). An orientation-preserving isometry of an odd-dimensional oriented Euclidean space has $+1$ as an eigenvalue, so $P$ fixes a unit vector $w \perp \dot\gamma(0)$. Parallel-translate $w$ along $\gamma$ to a parallel normal vector field $W$, and consider the variation $H(t, s) := \exp_{\gamma(t)}(s W(t))$. The second variation formula plus $K > 0$ on the section $(W, \dot\gamma)$ yields a strictly negative second variation of length, contradicting the length-minimising property of $\gamma$. [/proofplan] [step:Reduce to producing a length-minimising closed geodesic in a non-trivial homotopy class] Suppose, for contradiction, that $\pi_1(M, p_0) \ne 0$ for some basepoint $p_0$. Then there is a continuous loop $\alpha : S^1 \to M$ that is not null-homotopic, and the free homotopy class $[\alpha]$ of $\alpha$ (forgetting basepoints) is non-trivial: a free null-homotopy of $\alpha$ would, combined with the path-connectedness of $M$, produce a basepointed null-homotopy. By [Minimal Geodesic in a Non-Trivial Homotopy Class](/theorems/2731) — whose hypotheses are met since $M$ is compact and $[\alpha]$ is non-trivial — there exists a closed geodesic $\gamma : [0, T] \to M$ with $\gamma \in [\alpha]$ and \begin{align*} \ell(\gamma) = \inf_{\sigma \in [\alpha]} \ell(\sigma) > 0. \end{align*} The infimum is strictly positive because, by $K > 0$ and compactness, $M$ has a positive injectivity radius (this is standard but in any case: a curve of length below the injectivity radius lifts to a contractible neighbourhood and is null-homotopic). Reparametrise $\gamma$ so that it has unit speed, $|\dot\gamma(t)|_g = 1$ for all $t \in [0, T]$, with $T = \ell(\gamma)$. The closed-geodesic condition reads $\gamma(T) = \gamma(0) =: p$ and, smoothness of $\gamma$ at the joining of the two ends in $M$, $\dot\gamma(T) = \dot\gamma(0)$ as elements of $T_p M$. [/step] [step:Define parallel transport once around $\gamma$ and identify $+1$ as an eigenvalue] Let $P_\gamma : T_p M \to T_p M$ be the parallel transport along $\gamma$ from $\gamma(0) = p$ to $\gamma(T) = p$ via the Levi-Civita connection. Parallel transport is a linear isometry of inner product spaces: \begin{align*} P_\gamma : T_p M &\to T_p M, \\ g_p(P_\gamma u, P_\gamma v) &= g_p(u, v) \quad \text{for all } u, v \in T_p M. \end{align*} The velocity $\dot\gamma$ is parallel along $\gamma$ (geodesic equation $\nabla_t \dot\gamma = 0$), so $P_\gamma \dot\gamma(0) = \dot\gamma(T) = \dot\gamma(0)$. Set $N := \{w \in T_p M : g_p(w, \dot\gamma(0)) = 0\}$, the orthogonal complement of $\dot\gamma(0)$ inside $T_p M$. Since $P_\gamma$ is a linear isometry that fixes $\dot\gamma(0)$, it preserves $N$: \begin{align*} g_p(P_\gamma w, \dot\gamma(0)) = g_p(P_\gamma w, P_\gamma \dot\gamma(0)) = g_p(w, \dot\gamma(0)) = 0 \quad \text{for } w \in N. \end{align*} Hence $P_\gamma|_N : N \to N$ is a linear isometry of the inner product space $(N, g_p|_N)$, which has dimension $\dim M - 1$. We now use orientability of $M$ to give $N$ a canonical orientation. Fix an orientation of $T_p M$ (induced from a chosen orientation of $M$). Together with a choice of orientation on the line $\mathbb{R} \cdot \dot\gamma(0)$ — say, the orientation determined by $\dot\gamma(0)$ as a positive basis — this induces a unique orientation on the complementary subspace $N$ via the splitting $T_p M = \mathbb{R}\cdot\dot\gamma(0) \oplus N$. We claim $P_\gamma|_N$ is orientation-preserving on $N$. To see this, note that parallel transport along $\gamma$ varies continuously (in fact smoothly) with the curve, so for each $\tau \in [0, T]$ the parallel-transport map $P_\gamma^{0, \tau} : T_{\gamma(0)} M \to T_{\gamma(\tau)} M$ is a continuous family of linear isometries. Composing with a smooth path of isometric identifications $T_{\gamma(\tau)} M \to T_p M$ obtained by parallel-transporting along a fixed path back to $p$ (or by working with $M$ orientable so that $\det dF = +1$ for any local isometry $F$ defined by parallel transport in a tubular neighbourhood), we obtain a continuous path from $\operatorname{id}_{T_p M}$ to $P_\gamma$ inside $\mathrm{O}(T_p M, g_p)$. This path lies in the identity component $\mathrm{SO}(T_p M, g_p)$, so $P_\gamma$ has determinant $+1$, hence is orientation-preserving on $T_p M$. Since $P_\gamma|_{\mathbb{R}\cdot\dot\gamma(0)} = \operatorname{id}$, we obtain $\det(P_\gamma|_N) = +1$, so $P_\gamma|_N \in \mathrm{SO}(N, g_p|_N)$. By hypothesis $\dim M$ is even, so $\dim N = \dim M - 1$ is odd. An orientation-preserving isometry of an odd-dimensional Euclidean space has $+1$ as an eigenvalue (its characteristic polynomial has odd degree and constant term equal to $\det = +1$, so by the intermediate value theorem evaluated at $\pm \infty$ on the real line and the product of complex-conjugate eigenvalue pairs giving positive real contributions, a real eigenvalue must occur, and given the constraint of being on the unit circle, this forces $+1$ — see the claim below). [claim:An element of $\mathrm{SO}(k)$ with $k$ odd has $+1$ as an eigenvalue] [proof] Let $A \in \mathrm{SO}(k)$ with $k$ odd. The complex eigenvalues of $A$ lie on the unit circle (because $A^\top A = I$ implies $|\lambda|^2 = 1$ for each eigenvalue). They come in complex-conjugate pairs $\lambda, \bar\lambda$ contributing $|\lambda|^2 = 1$ to $\det A$, and any real eigenvalue is $\pm 1$. Since $k$ is odd, the number of real eigenvalues counted with multiplicity is odd, so it is non-zero. Let $r_+$ and $r_-$ be the multiplicities of $+1$ and $-1$ respectively, so $r_+ + r_-$ is odd and $\det A = (-1)^{r_-} \cdot 1 = (-1)^{r_-}$. From $\det A = +1$ we get $r_-$ even. Combined with $r_+ + r_-$ odd, we obtain $r_+$ odd, in particular $r_+ \ge 1$. Hence $+1$ is an eigenvalue of $A$. [/proof] [/claim] Apply the claim with $A = P_\gamma|_N$ and $k = \dim M - 1$ (odd). There exists a unit vector $w \in N$ with $P_\gamma w = w$. [/step] [step:Construct a parallel normal variation field along $\gamma$] Define $W$ as the parallel transport of $w$ along $\gamma$: \begin{align*} W : [0, T] &\to TM \quad \text{(section of } \gamma^* TM\text{)} \\ t &\mapsto P_\gamma^{0, t} w. \end{align*} By construction, $W$ is parallel along $\gamma$: \begin{align*} \nabla_t W(t) = 0 \quad \text{for all } t \in [0, T]. \end{align*} Since parallel transport is an isometry, $|W(t)|_g = |w|_g = 1$ for all $t$, and $g(W(t), \dot\gamma(t)) = g_p(w, \dot\gamma(0)) = 0$ for all $t$ (orthogonality is preserved by parallel transport along $\gamma$, since $\dot\gamma$ is also parallel). Furthermore, $W(0) = w$ and $W(T) = P_\gamma w = w$, so $W$ is a closed parallel normal field: it descends to a smooth section of $\gamma^* TM$ where we identify the fibres at $0$ and $T$ via $\gamma(0) = \gamma(T) = p$. Define the variation \begin{align*} H : [0, T] \times (-\varepsilon, \varepsilon) &\to M \\ (t, s) &\mapsto \exp_{\gamma(t)}(s W(t)) \end{align*} for $\varepsilon > 0$ chosen small enough that $\exp_{\gamma(t)}(s W(t))$ is defined for all $(t, s) \in [0, T] \times (-\varepsilon, \varepsilon)$ — possible by compactness of $[0, T]$, smoothness of $W$, and the [Exponential Map as a Local Diffeomorphism](/theorems/2712) applied at each point of $\gamma$. Because $W(0) = W(T)$, the variation respects the closed condition: $H(0, s) = H(T, s) = \exp_p(s w)$ for every $s$, so each $H(\cdot, s)$ is a closed loop, freely homotopic to $\gamma$ via the family $H(\cdot, \tau s)$, $\tau \in [0, 1]$. Hence $H(\cdot, s) \in [\alpha]$ for all $s \in (-\varepsilon, \varepsilon)$. The variation field is $\partial_s H(t, 0) = W(t)$ by definition of the exponential map at $s = 0$. [/step] [step:Compute the second variation of length and derive the contradiction] Apply Part 2 of the [Second Variation Formula](/theorems/2729) to the variation $H$ above. The variation field $Y := W$ is everywhere normal to $\dot\gamma$ (Step 3), so $Y_n = Y = W$ and $Y'_n = \nabla_t W = 0$ (Step 3 again). The boundary term \begin{align*} g\!\left(\frac{\nabla Y_n}{ds}(t, 0),\, \dot\gamma(t)\right)\Big|_0^T = 0 \end{align*} vanishes because the variation $H(t, s) = \exp_{\gamma(t)}(s W(t))$ matches at the closed-curve endpoints $t \in \{0, T\}$ for every $s$ (so $\frac{\nabla}{ds} \partial_s H$ at $t = 0$ and $t = T$ are equal as elements of $T_p M$, and the closed condition makes the boundary contribution telescope to zero — concretely, in the closed-curve setup we identify $\gamma(0) = \gamma(T)$ and the contribution of the boundary term is $g(\nabla_s \partial_s H(T, 0), \dot\gamma(T)) - g(\nabla_s \partial_s H(0, 0), \dot\gamma(0)) = 0$ by $W(0) = W(T)$ and $\dot\gamma(0) = \dot\gamma(T)$). Hence the second variation of length is \begin{align*} \frac{d^2}{ds^2} \ell(H(\cdot, s))\Big|_{s = 0} = \int_0^T \big[ |Y'_n|_g^2 - R(Y_n, \dot\gamma, Y_n, \dot\gamma) \big] d\mathcal{L}^1(t) = -\int_0^T R(W, \dot\gamma, W, \dot\gamma) \, d\mathcal{L}^1(t). \end{align*} By the definition of sectional curvature applied to the orthonormal pair $(W(t), \dot\gamma(t))$ — recall $|W(t)|_g = |\dot\gamma(t)|_g = 1$ and $g(W, \dot\gamma) = 0$: \begin{align*} K(\sigma_t) = \frac{R(W, \dot\gamma, W, \dot\gamma)}{|W|_g^2 |\dot\gamma|_g^2 - g(W, \dot\gamma)^2} = R(W, \dot\gamma, W, \dot\gamma), \end{align*} where $\sigma_t = \mathrm{span}(W(t), \dot\gamma(t)) \subseteq T_{\gamma(t)} M$. By hypothesis $K(\sigma_t) > 0$ for every $t$, and by continuity and compactness of $[0, T]$, \begin{align*} \int_0^T R(W, \dot\gamma, W, \dot\gamma) \, d\mathcal{L}^1(t) = \int_0^T K(\sigma_t)\, d\mathcal{L}^1(t) > 0. \end{align*} Therefore \begin{align*} \frac{d^2}{ds^2} \ell(H(\cdot, s))\Big|_{s = 0} < 0. \end{align*} The first variation of length at $s = 0$ vanishes because $\gamma$ is a closed geodesic — by the [First Variation Formula](/theorems/2728) and $\nabla_t \dot\gamma = 0$, the integral term vanishes; the boundary term vanishes because the variation is closed (the contribution at $t = 0$ and $t = T$ cancels). So the function $s \mapsto \ell(H(\cdot, s))$ has a strict local maximum at $s = 0$ on $(-\varepsilon, \varepsilon)$ (zero first derivative, negative second derivative). Therefore there exists $s_0 \in (-\varepsilon, \varepsilon) \setminus \{0\}$ with \begin{align*} \ell(H(\cdot, s_0)) < \ell(\gamma). \end{align*} But $H(\cdot, s_0) \in [\alpha]$ (Step 3), contradicting the length-minimising property of $\gamma$ within $[\alpha]$. The contradiction shows our assumption $\pi_1(M, p_0) \ne 0$ is false: $M$ is simply connected. [guided] We have invested significant effort to construct the parallel normal field $W$. The payoff comes now: we plug $W$ into the second variation of length and show this second variation is strictly negative. Combined with the fact that the first variation vanishes (because $\gamma$ is a geodesic and the variation is closed), this means $\gamma$ is a strict local *maximum* of length within the family $H(\cdot, s)$ — but $\gamma$ was supposed to be a length *minimiser* in its homotopy class. That is the contradiction. We invoke Part 2 of the [Second Variation Formula](/theorems/2729). The formula's hypotheses require a smooth proper variation of a unit-speed geodesic with a designated normal component of the variation field; we verify these. The variation $H$ is smooth (Step 3); the curve $\gamma$ is a unit-speed geodesic (Step 1); and the variation field $\partial_s H(\cdot, 0) = W$ is everywhere normal to $\dot\gamma$ by the orthogonality computation in Step 3. Hence $Y := W$, $Y_n = Y = W$, and $Y'_n = \nabla_t W = 0$ (parallelism, Step 3). Why does the boundary term in the second variation formula vanish? Recall the formula has a contribution \begin{align*} g\!\left(\frac{\nabla Y_n}{ds}(t, 0),\, \dot\gamma(t)\right)\Big|_0^T = 0. \end{align*} For an open curve with fixed endpoints we would set this term to zero by hypothesis on the variation. Here $\gamma$ is closed, and the boundary contribution is $g(\nabla_s \partial_s H(T, 0), \dot\gamma(T)) - g(\nabla_s \partial_s H(0, 0), \dot\gamma(0))$. By the closed condition $W(0) = W(T)$ and $\dot\gamma(0) = \dot\gamma(T)$ we have identical contributions at the two endpoints, so the boundary term telescopes to zero. With the boundary term gone and $Y'_n = 0$, the second variation formula reduces to \begin{align*} \frac{d^2}{ds^2} \ell(H(\cdot, s))\Big|_{s = 0} = \int_0^T \big[ |Y'_n|_g^2 - R(Y_n, \dot\gamma, Y_n, \dot\gamma) \big] d\mathcal{L}^1(t) = -\int_0^T R(W, \dot\gamma, W, \dot\gamma) \, d\mathcal{L}^1(t). \end{align*} This is the heart of the Synge mechanism: parallelism of $W$ killed the gradient term $|Y'_n|^2$, leaving only the curvature term — and the sign in front of $R$ is *negative*. Positive curvature now becomes our friend: the more positive the curvature, the more negative the second variation. We make the curvature explicit. The integrand $R(W, \dot\gamma, W, \dot\gamma)$ is, by the very definition of sectional curvature applied to the orthonormal pair $(W(t), \dot\gamma(t))$, \begin{align*} K(\sigma_t) = \frac{R(W, \dot\gamma, W, \dot\gamma)}{|W|_g^2 |\dot\gamma|_g^2 - g(W, \dot\gamma)^2} = R(W, \dot\gamma, W, \dot\gamma), \end{align*} where $\sigma_t = \mathrm{span}(W(t), \dot\gamma(t)) \subseteq T_{\gamma(t)} M$. The denominator is $1$ because $|W(t)|_g = |\dot\gamma(t)|_g = 1$ and $g(W, \dot\gamma) = 0$ (Step 3) — this is exactly why we built $W$ to be a *unit* vector orthogonal to $\dot\gamma$, and parallel transport preserves both norms and the orthogonality. By hypothesis $K > 0$ on every $2$-plane in $TM$, so in particular $K(\sigma_t) > 0$ for every $t \in [0, T]$. Moreover $t \mapsto K(\sigma_t)$ is continuous on the compact interval $[0, T]$, so it attains a positive minimum, and \begin{align*} \int_0^T R(W, \dot\gamma, W, \dot\gamma) \, d\mathcal{L}^1(t) = \int_0^T K(\sigma_t)\, d\mathcal{L}^1(t) > 0. \end{align*} Substituting back, \begin{align*} \frac{d^2}{ds^2} \ell(H(\cdot, s))\Big|_{s = 0} < 0. \end{align*} Now we extract the contradiction. We need the first variation at $s = 0$ to vanish, otherwise a strictly negative second derivative does not by itself force $\ell(H(\cdot, s_0)) < \ell(\gamma)$. By the [First Variation Formula](/theorems/2728), the first variation along a unit-speed curve with variation field $Y$ is \begin{align*} \frac{d}{ds} \ell(H(\cdot, s))\Big|_{s = 0} = -\int_0^T g(\nabla_t \dot\gamma, Y) \, d\mathcal{L}^1(t) + g(Y, \dot\gamma)\big|_0^T. \end{align*} The integral term vanishes because $\gamma$ is a geodesic, $\nabla_t \dot\gamma = 0$. The boundary term vanishes because the variation is closed: $Y(0) = W(0) = W(T) = Y(T)$ and $\dot\gamma(0) = \dot\gamma(T)$, so $g(Y, \dot\gamma)|_0^T = 0$. So $s \mapsto \ell(H(\cdot, s))$ has zero first derivative and strictly negative second derivative at $s = 0$ on $(-\varepsilon, \varepsilon)$. By the second-derivative test, $s = 0$ is a strict local maximum, so there exists $s_0 \in (-\varepsilon, \varepsilon) \setminus \{0\}$ with \begin{align*} \ell(H(\cdot, s_0)) < \ell(\gamma). \end{align*} But $H(\cdot, s_0)$ is a closed loop (because $W(0) = W(T)$, so $H(0, s_0) = H(T, s_0)$) and freely homotopic to $\gamma$ via the family $H(\cdot, \tau s_0)$, $\tau \in [0, 1]$. Hence $H(\cdot, s_0) \in [\alpha]$ — yet it is strictly shorter than the supposed length-minimiser $\gamma$ in $[\alpha]$. This contradicts the construction of $\gamma$ in Step 1 as a length-minimiser in $[\alpha]$. The contradiction shows that our assumption $\pi_1(M, p_0) \ne 0$ is false. Since the basepoint $p_0$ was arbitrary and $M$ is path-connected, $\pi_1(M) = 0$, so $M$ is simply connected. A few words on why each hypothesis was used. **Compactness** of $M$ entered when we invoked [Minimal Geodesic in a Non-Trivial Homotopy Class](/theorems/2731) to produce the length-minimising $\gamma$; without compactness the infimum need not be attained. **Even dimension** was the parity input that made $\dim N = \dim M - 1$ odd, which is what powered the eigenvalue lemma in Step 2 to give us the fixed unit vector $w$. **Orientability** ensured that $P_\gamma \in \mathrm{SO}(T_p M)$ (rather than just $\mathrm{O}$) by lifting parallel transport to a path inside the identity component — without it, $P_\gamma|_N$ could be orientation-reversing on $N$, and the eigenvalue $+1$ would not be guaranteed (the Möbius strip illustrates this). **Positive sectional curvature** was the only ingredient that produced the strict negativity of the second variation — for $K \ge 0$ we would only get $\ell''(0) \le 0$, which is consistent with $\gamma$ being a (non-strict) minimiser. [/guided] [/step]