03.02.35 · differential-geometry / manifolds

Synge's theorem and the second variation of arc length

shipped3 tiersLean: none

Anchor (Master): Synge 1936 Quart. J. Math. 7; Weinstein 1968 Ann. Math. 87; Cheeger-Ebin Comparison Theorems in Riemannian Geometry Ch. 1; do Carmo Riemannian Geometry Ch. 9

Intuition Beginner

Imagine a smooth closed world shaped like a slightly squashed ball, and draw a loop on it that you cannot shrink to a point without leaving the surface. Walk that loop carrying an arrow, always keeping the arrow pointing the same way relative to the path. On a curved world the arrow comes back rotated, but in just the right even-dimensional setting some direction comes back pointing exactly as it started.

That fixed direction is a free escape hatch. Push the whole loop a tiny step in that direction and, because the world curves toward you, the new loop is shorter. You can keep shortening. But you started with the shortest loop in its family, so there was nothing shorter to find. The only way out of the contradiction is that no such stubborn loop existed at all.

This is the heart of Synge's theorem: on a tightly curved even-dimensional world, every loop can be shrunk to a point. Positive curvature plus even dimension plus two-sidedness forces the space to be simply connected. The same bending that focuses nearby straight paths together also lets you slide any shortest loop into a shorter one, so a shortest stubborn loop cannot survive.

Visual Beginner

Alt text: A shortest closed loop is drawn on a curved, rounded surface. An arrow is carried around the loop by parallel transport and returns to its starting point pointing the same way, marking a preserved direction. A second, slightly smaller loop sits just inside the first along that preserved direction, and a label notes that the inner loop is shorter. The picture shows that on a positively curved even-dimensional space a supposedly shortest loop always has a shorter neighbour, so no shortest stubborn loop can exist and the space has no stubborn loops at all.

Worked example Beginner

Take the round sphere, the surface of a ball, in two dimensions. It curves the same amount everywhere and that amount is positive. Its dimension, two, is even. And it is two-sided: it has a consistent notion of left and right, an inside and an outside.

Now try to find a loop on the sphere that you cannot shrink. Pick any loop at all: a small circle, a great circle around the equator, a wandering scribble. Every one of them can be slid and tightened down to a single point, like a rubber band sliding off a melon. There is no stubborn loop anywhere on the sphere.

Compare this with a flat doughnut surface, the torus. The torus is even-dimensional and two-sided, but it is flat, not positively curved. On the torus the loop that goes around the hole genuinely cannot be shrunk. The missing ingredient is curvature: take the curvature away and the conclusion fails.

What this tells us: the three ingredients work together. The sphere has all three (positive curvature, even dimension, two-sidedness) and has no stubborn loops; the torus drops curvature and keeps a stubborn loop. Synge's theorem says you need all three at once.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $(M, g)$ is a complete Riemannian manifold with Levi-Civita connection $\nabla$ and curvature tensor $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ . Sign convention: with this definition the sectional curvature of the unit sphere is $+ 1$ and the index form carries the curvature term with a minus sign, matching unit 03.02.19. For a vector field $V$ along a curve $γ$ write $V^{'} := \nabla_{\overset{γ}{˙}} V$ . A unit-speed closed geodesic of length $L$ is a geodesic $γ : [0, L] \to M$ with $γ (0) = γ (L)$ and $\overset{γ}{˙} (0) = \overset{γ}{˙} (L)$ , equivalently a smooth geodesic loop with no corner at the basepoint.

The variational object is the arc-length (equivalently, the energy) of a smooth one-parameter variation $α : (- ε, ε) \times [0, L] \to M$ of $γ = α (0, \cdot)$ through closed curves, with variation field $V (t) = \partial_{s} ∣_{s = 0} α (s, t)$ . For a closed variation the field is periodic: $V (0) = V (L)$ and $V^{'} (0) = V^{'} (L)$ , the loop-space replacement for the Dirichlet endpoint conditions of the open-path index theory in 03.02.19.

Definition (second variation of arc length). Let $γ$ be a unit-speed geodesic and $V$ a piecewise-smooth periodic field along $γ$ that is everywhere orthogonal to $\overset{γ}{˙}$ . The second variation of arc length of a variation with field $V$ is $$ L''(0) ;=; I(V,V) ;=; \int_0^L \Big( |V'|^2 - \langle R(V,\dot\gamma)\dot\gamma,, V\rangle \Big), dt , $$ the index form of $γ$ now taken on periodic orthogonal fields, following do Carmo ^{[do Carmo Ch. 9]} and the energy-functional treatment of the second variation ^{[fasttrack-texts second variation of energy (Part III analogue)]}. (For a unit-speed geodesic and an orthogonal variation field the second variations of length and of energy agree; the tangential part of $V$ contributes only reparametrisation and is discarded.) The single curvature term is $⟨ R (V, \overset{γ}{˙}) \overset{γ}{˙}, V ⟩ = K (V, \overset{γ}{˙}) ∣ V ∣^{2}$ when $V ⊥ \overset{γ}{˙}$ is a unit field, with $K$ the sectional curvature of the plane $span (V, \overset{γ}{˙})$ .

Definition (free-homotopy class and minimising closed geodesic). A free-homotopy class of loops is a conjugacy class in $π_{1} (M)$ ; loops in the same class are deformable into one another without a fixed basepoint. By Hopf-Rinow 03.02.32, on a compact $M$ each noncontractible free-homotopy class contains a shortest closed geodesic — a smooth closed geodesic of minimal length in its class — obtained by the direct method, since the infimum of length over a class is attained and an attaining loop is a smooth closed geodesic.

Definition (parallel transport around a loop / holonomy). Parallel transport once around $γ$ defines a linear isometry $P_{γ} : T_{γ (0)} M \to T_{γ (0)} M$ , the holonomy of the loop. It fixes $\overset{γ}{˙} (0)$ (the velocity is parallel along a geodesic) and restricts to a special-orthogonal transformation $P_{γ}^{⊥} \in S O (\overset{γ}{˙} (0)^{⊥})$ of the orthogonal complement when $M$ is oriented and $γ$ preserves a chosen orientation.

Counterexamples to common slips

Orientability is not optional. Real projective space $RP^{2 n}$ carries a metric of constant curvature $+ 1$ and is even-dimensional, yet $π_{1} (RP^{2 n}) = Z /2 \neq = 0$ . It is non-orientable, which is exactly the hypothesis Synge's even-dimensional theorem needs. Dropping orientability voids the conclusion.
Even dimension is not optional. Odd-dimensional positively curved spaces need not be simply connected: lens spaces $S^{2 n + 1} / Z_{m}$ are positively curved with fundamental group $Z / m$ . The even-dimensional argument hinges on a parallel transformation of an even-dimensional orthogonal space having $+ 1$ as an eigenvalue; this fails in odd codimension.
Positive curvature is not optional. The flat torus $T^{2 n}$ is even-dimensional and orientable but flat; it has $π_{1} = Z^{2 n}$ . With non-positive curvature the index form is positive on every loop and the shortening step never appears.

Key theorem with proof Intermediate+

Theorem (Synge, 1936). Let $(M, g)$ be a compact, orientable, even-dimensional Riemannian manifold with strictly positive sectional curvature. Then $M$ is simply connected ^{[Synge 1936]}.

Proof. Argue by contradiction. If $π_{1} (M) \neq = 0$ , some free-homotopy class of loops is noncontractible. Since $M$ is compact, Hopf-Rinow 03.02.32 and the direct method of the calculus of variations provide a shortest loop $γ$ in that class; an arc-length minimiser within a free-homotopy class is a smooth closed geodesic, of length $L > 0$ . Parametrise $γ$ by arc length on $[0, L]$ .

Consider the holonomy $P := P_{γ}^{⊥}$ , parallel transport once around $γ$ restricted to the orthogonal complement $W := \overset{γ}{˙} (0)^{⊥} \subset T_{γ (0)} M$ . Since $\nabla g = 0$ , $P$ is orthogonal; since $M$ is oriented and parallel transport preserves orientation, and since $\overset{γ}{˙} (0)$ is fixed by $P_{γ}$ , the restriction $P$ preserves the induced orientation of $W$ , hence $det P = + 1$ , i.e. $P \in S O (W)$ . The space $W$ has dimension $dim M - 1$ , which is odd.

A special-orthogonal transformation of an odd-dimensional real inner-product space always has $+ 1$ as an eigenvalue. Indeed the non-real eigenvalues of an orthogonal map occur in conjugate pairs $e^{\pm i θ}$ , the real eigenvalues are $\pm 1$ , and $det P = + 1$ ; in odd dimension the product of all eigenvalues being $+ 1$ forces an odd number of $+ 1$ 's, so at least one eigenvector $e \in W$ has $P e = e$ . Let $E (t)$ be the parallel field along $γ$ with $E (0) = e$ ; then $E (L) = P e = e = E (0)$ , so $E$ is a periodic, unit-length, parallel field orthogonal to $\overset{γ}{˙}$ .

Use $V = E$ as the variation field. Because $E$ is parallel, $E^{'} = \nabla_{\overset{γ}{˙}} E = 0$ , so the second variation of arc length is $$ I(E,E) = \int_0^L \Big( |E'|^2 - \langle R(E,\dot\gamma)\dot\gamma, E\rangle \Big),dt = -\int_0^L K(E,\dot\gamma),dt < 0 , $$ since $∣ E ∣ = 1$ and the sectional curvature $K (E, \overset{γ}{˙}) > 0$ everywhere by hypothesis. Because $E$ is periodic, it integrates to an honest variation through closed loops (translate $γ$ along the flow of a field extending $E$ ), and the variation has no first-order length change — $γ$ is a geodesic, so $L^{'} (0) = 0$ — while $L^{''} (0) = I (E, E) < 0$ . Hence nearby loops in the same free-homotopy class are strictly shorter than $γ$ , contradicting the minimality of $γ$ .

The contradiction shows no noncontractible free-homotopy class exists, so $π_{1} (M) = 0$ and $M$ is simply connected. $■$

Bridge. Synge's argument builds toward the entire program of curvature-controls-topology, and it is exactly the closed-geodesic counterpart of the open-path index theory of 03.02.19: there the index form on Dirichlet fields detected conjugate points, here the same index form on periodic fields detects a length-decreasing direction, and the bridge is the second variation $I (V, V) = \int ∣ V^{'} ∣^{2} - ⟨ R (V, \overset{γ}{˙}) \overset{γ}{˙}, V ⟩ d t$ read with new boundary conditions. The foundational reason the proof closes is that a shortest loop has positive-semidefinite second variation, so a single negative direction is fatal — this is exactly the minimiser-implies-nonnegative-Hessian principle that drove the conjugate-point obstruction. Putting these together, Synge generalises Bonnet-Myers from the diameter to the fundamental group: where Bonnet-Myers 03.02.06 uses positive curvature to bound size, Synge uses the same index form to kill loops, and both are the central insight that a lower curvature bound is a topological constraint, not merely a metric one. The minimisation step appears again in the odd-dimensional orientability theorem and in Weinstein's fixed-point theorem below, which run the identical parallel-field-plus-shortening machine.

Exercises Intermediate+

Exercise (medium, short-answer). Explain why the variation field in Synge's proof must be periodic rather than vanishing at endpoints, and why this distinguishes the closed-geodesic setting from the conjugate-point setting of [03.02.19].

Hint

A closed-loop variation must keep the loop closed; what does that demand of $V$ at $t = 0$ and $t = L$ ?

Answer

Rubric. A variation through closed loops needs $α (s, 0) = α (s, L)$ and matching velocities, so the field satisfies $V (0) = V (L)$ and $V^{'} (0) = V^{'} (L)$ — periodic boundary conditions. The Dirichlet conditions $V (0) = V (L) = 0$ of the Morse index theory would force the loop's basepoint fixed and forbid the closed-loop deformation. Synge needs a periodic field, which is why he produces an eigenvector of the holonomy $P$ (giving $E (L) = E (0)$ ) rather than a Jacobi field vanishing at conjugate points. Full credit names both the periodicity requirement and the holonomy source of the periodic field.

Exercise (hard, short-answer). Prove the odd-dimensional Synge theorem: a compact positively curved manifold of odd dimension is orientable. (Run the same shortening argument on an assumed non-orientable example and derive a contradiction.)

Hint

If $M$ is non-orientable, there is an orientation-reversing loop $γ$ ; take a shortest such closed geodesic. Now the normal holonomy $P$ has $det P = - 1$ on an even-dimensional normal space. Find a periodic parallel field again.

Answer

Rubric. Suppose $M^{2 k + 1}$ is positively curved but non-orientable. Then some loop reverses orientation; take a shortest closed geodesic $γ$ in a free-homotopy class of orientation-reversing loops. Its normal holonomy $P$ acts on $W = \overset{γ}{˙} (0)^{⊥}$ of dimension $2 k$ (even); because $γ$ reverses orientation while fixing $\overset{γ}{˙}$ , $det P = - 1$ on $W$ . An orientation-reversing orthogonal map of an even-dimensional space has $+ 1$ as an eigenvalue (pairs $e^{\pm i θ}$ have product $1$ ; $det = - 1$ forces an odd number of $- 1$ 's, hence an even number — at least zero — but the parity count with even total dimension and an odd number of $- 1$ 's leaves a $+ 1$ ). The corresponding parallel field $E$ is periodic, and $I (E, E) = - \int K < 0$ shortens $γ$ , contradicting minimality. So no orientation-reversing loop exists and $M$ is orientable. Full credit handles the determinant sign on the even-dimensional normal space and produces the periodic field.

Exercise (hard, short-answer). Deduce from Synge's theorem that a compact orientable even-dimensional positively curved manifold has fundamental group equal to the identity, and contrast this with the weaker conclusion Bonnet-Myers alone gives for $\pi_1$.

Hint

Bonnet-Myers 03.02.06 bounds the diameter and so makes the universal cover compact, giving finite $π_{1}$ . Synge sharpens "finite" to "equal to the identity" in the even-orientable case.

Answer

Rubric. Bonnet-Myers gives a diameter bound $\leq π / K_{0}$ for curvature $\geq K_{0} > 0$ , so the universal cover is compact, hence $π_{1} (M)$ is finite — but it does not say which finite group. Synge's even-dimensional orientable theorem upgrades this to $π_{1} (M) = 0$ outright. The contrast is sharp: lens spaces and $RP^{2 n}$ realise nonzero finite $π_{1}$ allowed by Bonnet-Myers but excluded by Synge once even dimension and orientability are imposed. Full credit states the Bonnet-Myers "finite" conclusion and the Synge " $π_{1} = 0$ " sharpening, and names a space that distinguishes them.

Lean formalization Intermediate+

Mathlib has the curvature tensor, sectional curvature, parallel transport, and geodesics, but no second-variation calculus on loop space and no closed-geodesic minimisation, so Synge's theorem is not formalisable end-to-end (lean_status: none). The load-bearing linear-algebra step — a special-orthogonal map of an odd-dimensional space fixing a vector — is expressible against current Mathlib; the rest is statement-level pseudo-Lean.

-- The eigenvalue-1 lemma (closest to compiling against current Mathlib):
-- An element of SO(2k+1) fixes a nonzero vector.
example {n : ℕ} (P : Matrix (Fin (2*n+1)) (Fin (2*n+1)) ℝ)
    (hP : P ∈ Matrix.specialOrthogonalGroup (Fin (2*n+1)) ℝ) :
    ∃ v ≠ 0, P.mulVec v = v := by
  sorry  -- det(P - 1) = 0 in odd dimension with det P = 1

-- Statement target (NOT compiling against current Mathlib):
-- Synge's theorem.
variable {M : Type*} [RiemannianManifold M]

-- theorem synge
--     (hcompact : CompactSpace M) (horient : Orientable M)
--     (heven : Even (finrank ℝ (TangentSpace M)))
--     (hpos : ∀ p (σ : Plane (TangentSpace M p)), 0 < sectionalCurvature σ) :
--     IsSimplyConnected M

the Mathlib gap analysis enumerates the missing primitives: the second variation of arc length on loop space with periodic boundary conditions, the shortest-closed-geodesic-in-a-class minimisation built on Hopf-Rinow, and the holonomy-eigenvalue-to-periodic-field passage.

Advanced results Master

Weinstein's fixed-point theorem. The same parallel-transport mechanism proves a statement about isometries rather than loops. Let $M$ be a compact orientable even-dimensional manifold of positive sectional curvature and let $f : M \to M$ be an orientation-preserving isometry. Then $f$ has a fixed point ^{[Weinstein 1968]}. The proof minimises the displacement function $p \mapsto d (p, f (p))$ ; if the minimum $ℓ$ were positive, the minimising point $p$ and a shortest geodesic $γ$ from $p$ to $f (p)$ produce, via $f$ acting on $\overset{γ}{˙}$ and parallel transport, a special-orthogonal map of an odd-dimensional space, hence a fixed direction along which $d (\cdot, f (\cdot))$ decreases to second order — contradicting minimality. Synge's theorem is the special case $f =$ a deck transformation of the universal cover: a deck transformation other than the identity would be a fixed-point-free orientation-preserving isometry, which Weinstein forbids, so the cover is one-sheeted and $π_{1} = 0$ . The two theorems are one argument, with the closed geodesic of Synge replaced by the displacement geodesic of Weinstein.

Synge's lemma and the second variation in higher generality. The computation $I (E, E) = - \int K < 0$ for a parallel field is the simplest instance of Synge's lemma: if along a geodesic one can find $k$ linearly independent periodic (or boundary-vanishing) fields on which the index form is negative, the geodesic has index $\geq k$ , and on a positively curved space parallel fields supply such directions whenever holonomy permits. Iterating with several eigenvectors of $P$ controls the index of the closed geodesic, the periodic analogue of the conjugate-point count of 03.02.19, and feeds the Morse theory of the free loop space $Λ M$ that underlies closed-geodesic existence results (Lyusternik-Fet) and the Gromoll-Meyer theorem on infinitely many closed geodesics.

The Frankel theorem. A cousin of Synge with the same second-variation engine: in a compact positively curved manifold, any two compact totally geodesic submanifolds whose dimensions add to at least $dim M$ must intersect ^{[Cheeger-Ebin Ch. 1]}. The proof minimises distance between the submanifolds and uses positive curvature to make a connecting geodesic non-minimising unless its length is zero. Frankel's theorem and Synge's theorem are the intersection-theoretic and the loop-theoretic faces of the principle that positive curvature focuses geodesics so strongly that supposedly-extremal configurations always admit improvement.

Sharpness and the failure boundary. Each hypothesis is sharp. $RP^{2 n}$ (positive curvature, even-dimensional, non-orientable, $π_{1} = Z /2$ ) shows orientability is essential; lens spaces $S^{2 n + 1} / Z_{m}$ (positive curvature, odd-dimensional, $π_{1} = Z / m$ ) show even dimension is essential; flat tori show positive curvature is essential. The Bonnet-Myers theorem 03.02.06 survives all three weakenings — it only needs positive curvature to bound the diameter and make $π_{1}$ finite — so Synge is precisely the surplus that even dimension plus orientability buy on top of the diameter bound.

Synthesis. Synge's theorem is the central insight that a curvature sign plus a parity-and-orientation hypothesis is a topological theorem about $π_{1}$ , and it is dual to the conjugate-point obstruction of 03.02.19: there the index form on Dirichlet fields obstructs minimisation between fixed endpoints, here the same form on periodic fields obstructs minimisation within a free-homotopy class, and the bridge is the second variation $I (V, V)$ read with the two boundary conditions. The foundational reason all four theorems of this unit — Synge even, Synge odd, Weinstein, Frankel — collapse into one is the shortening lemma: a configuration minimising length (a loop, a displacement, an inter-submanifold distance) cannot survive a parallel field on which $I < 0$ , and positive curvature manufactures that field whenever a determinant-and-parity count yields a $+ 1$ eigenvalue of holonomy. Putting these together, Synge generalises Bonnet-Myers from "diameter bounded, $π_{1}$ finite" to " $π_{1} = 0$ ," and this is exactly the step from a metric constraint to a homotopy constraint; the same second-variation machinery appears again in the Morse theory of the loop space and in the comparison geometry of 03.02.06, so the index form is the single invariant carrying curvature into topology across the entire cluster.

Full proof set Master

Proposition (the eigenvalue- $+ 1$ lemma). A special-orthogonal transformation $P$ of a real inner-product space $W$ of odd dimension has $+ 1$ as an eigenvalue.

Proof. Consider the characteristic polynomial $χ_{P} (λ) = det (P - λ I)$ , of odd degree $dim W$ . Its non-real roots come in conjugate pairs (real coefficients), so the number of real roots counted with multiplicity is odd, hence at least one real root exists. Real eigenvalues of an orthogonal map are $\pm 1$ . Group the eigenvalues over $C$ : each conjugate pair $e^{\pm i θ}$ multiplies to $1$ , so $det P = (+ 1)^{a} (- 1)^{b}$ where $a, b$ are the multiplicities of $+ 1, - 1$ . As $det P = + 1$ , $b$ is even. Since $a + b = dim W$ is odd, $a$ is odd, in particular $a \geq 1$ . Thus $+ 1$ is an eigenvalue. $■$

Proposition (parallel eigenfield is periodic). Let $γ : [0, L] \to M$ be a closed geodesic with normal holonomy $P \in S O (\overset{γ}{˙} (0)^{⊥})$ , and let $e$ be a unit $+ 1$ -eigenvector of $P$ . The parallel field $E$ along $γ$ with $E (0) = e$ is a periodic, unit-length, parallel field orthogonal to $\overset{γ}{˙}$ .

Proof. Parallel transport is a linear isometry, so $E (t)$ has unit length for all $t$ and is orthogonal to $\overset{γ}{˙}$ throughout (orthogonality to the parallel field $\overset{γ}{˙}$ is preserved). By definition of holonomy, $E (L) = P e = e = E (0)$ . Parallelism gives $E^{'} (0) = E^{'} (L) = 0$ , so the periodic conditions $E (0) = E (L)$ , $E^{'} (0) = E^{'} (L)$ both hold. $■$

Proposition (negative second variation). For the field $E$ of the previous proposition on a positively curved manifold, $I (E, E) < 0$ .

Proof. Since $E$ is parallel, $E^{'} = 0$ , so $I (E, E) = \int_{0}^{L} (0 - ⟨ R (E, \overset{γ}{˙}) \overset{γ}{˙}, E ⟩) d t$ . With $E, \overset{γ}{˙}$ orthonormal, $⟨ R (E, \overset{γ}{˙}) \overset{γ}{˙}, E ⟩ = K (E, \overset{γ}{˙}) > 0$ by the positive-curvature hypothesis. Hence the integrand is strictly negative and $I (E, E) = - \int_{0}^{L} K (E, \overset{γ}{˙}) d t < 0$ . Because $E$ is periodic it is the field of a genuine variation through closed loops, and $γ$ being a geodesic gives $L^{'} (0) = 0$ , so $γ$ is not a local minimum of length in its free-homotopy class. $■$

Proposition (Synge from the three lemmas). Synge's even-dimensional theorem follows.

Proof. If $π_{1} (M) \neq = 0$ , a shortest closed geodesic $γ$ in a noncontractible class exists by Hopf-Rinow 03.02.32 and the direct method. Its normal space $\overset{γ}{˙} (0)^{⊥}$ has odd dimension $dim M - 1$ , and orientability makes the normal holonomy lie in $S O$ . The first proposition yields a $+ 1$ -eigenvector; the second yields a periodic parallel field $E$ ; the third yields $I (E, E) < 0$ , contradicting minimality of $γ$ . So $π_{1} (M) = 0$ . $■$

Weinstein's fixed-point theorem and Frankel's theorem are stated above; their proofs run the identical eigenvalue- $+ 1$ -then-shorten scheme on the displacement geodesic and the inter-submanifold geodesic respectively — see Weinstein ^{[Weinstein 1968]} and Cheeger-Ebin ^{[Cheeger-Ebin Ch. 1]}.

Connections Master

Jacobi fields, conjugate points, and the Morse Index Theorem 03.02.19. Synge's theorem is the closed-geodesic dual of that unit's open-path index theory. The very same index form $I (V, V) = \int ∣ V^{'} ∣^{2} - ⟨ R (V, \overset{γ}{˙}) \overset{γ}{˙}, V ⟩ d t$ reappears, but evaluated on periodic fields rather than fields vanishing at the endpoints; where the Morse index counts conjugate points obstructing minimisation between fixed endpoints, Synge produces a single negative periodic direction obstructing minimisation within a homotopy class. Synge's lemma — index $\geq k$ from $k$ negative directions — is the periodic refinement of the index count established there.
Hopf-Rinow and completeness 03.02.32. The existence of a shortest closed geodesic in each noncontractible free-homotopy class rests on Hopf-Rinow: completeness plus compactness makes length-minimisation over a homotopy class attain its infimum at a smooth closed geodesic. Without this minimiser there is no object to shorten, and the whole argument is vacuous.
Constant-curvature spaces, Bonnet-Myers, and Killing-Hopf 03.02.06. Bonnet-Myers is the companion theorem: positive curvature bounds the diameter and forces $π_{1}$ to be finite, and Synge sharpens "finite" to " $π_{1} = 0$ " under the extra even-dimension-plus-orientability hypotheses. The sharpness examples $RP^{2 n}$ and the lens spaces are space forms classified there, and they mark exactly where Synge's surplus over Bonnet-Myers expires.
The Morse theory of the free loop space. Synge's lemma feeds the index theory of closed geodesics on $Λ M$ , the foundation of the Lyusternik-Fet existence theorem and the Gromoll-Meyer infinitely-many-closed-geodesics theorem; positive curvature controlling the periodic index is the entry point to that program.

Historical & philosophical context Master

John Lighton Synge proved the theorem in "On the connectivity of spaces of positive curvature" (Quarterly Journal of Mathematics, Oxford series, 7, 316–320, 1936), introducing the second variation of arc length for closed geodesics and the parallel-field argument that bears his name ^{[Synge 1936]}. The method was a decisive demonstration that the sign of sectional curvature, processed through the second variation, controls global topology — a theme Bonnet had opened for surfaces and Myers extended to the diameter bound. Synge's use of a parallel eigenfield of the loop holonomy made the role of dimension parity and orientability explicit for the first time.

Alan Weinstein's "A fixed point theorem for positively curved manifolds" (Annals of Mathematics 87, 29–33, 1968) recast Synge's mechanism as a statement about isometries, proving that an orientation-preserving isometry of a compact even-dimensional positively curved manifold has a fixed point, with Synge's theorem recovered by applying it to deck transformations ^{[Weinstein 1968]}. The unifying second-variation viewpoint, and the companion Frankel intersection theorem, were systematised in Cheeger and Ebin's Comparison Theorems in Riemannian Geometry (North-Holland, 1975), which presents Synge, Weinstein, and Frankel as one family driven by the index form ^{[Cheeger-Ebin Ch. 1]}. The standard textbook treatment is do Carmo's Riemannian Geometry (Birkhäuser, 1992), Chapter 9 ^{[do Carmo Ch. 9]}.

Bibliography Master

@article{Synge1936Connectivity,
  author  = {Synge, John Lighton},
  title   = {On the connectivity of spaces of positive curvature},
  journal = {The Quarterly Journal of Mathematics (Oxford Series)},
  volume  = {7},
  pages   = {316--320},
  year    = {1936}
}

@article{Weinstein1968FixedPoint,
  author  = {Weinstein, Alan},
  title   = {A fixed point theorem for positively curved manifolds},
  journal = {Annals of Mathematics},
  volume  = {87},
  pages   = {29--33},
  year    = {1968}
}

@book{CheegerEbin1975,
  author    = {Cheeger, Jeff and Ebin, David G.},
  title     = {Comparison Theorems in Riemannian Geometry},
  publisher = {North-Holland},
  year      = {1975},
  note      = {Ch. 1 §6: the second variation, Synge, Weinstein, Frankel}
}

@book{doCarmo1992,
  author    = {do Carmo, Manfredo P.},
  title     = {Riemannian Geometry},
  publisher = {Birkh{\"a}user},
  year      = {1992},
  note      = {Ch. 9: the second variation of energy and Synge's theorem}
}

@article{Frankel1961,
  author  = {Frankel, Theodore},
  title   = {Manifolds with positive curvature},
  journal = {Pacific Journal of Mathematics},
  volume  = {11},
  pages   = {165--174},
  year    = {1961}
}

@article{GromollMeyer1969,
  author  = {Gromoll, Detlef and Meyer, Wolfgang},
  title   = {Periodic geodesics on compact {R}iemannian manifolds},
  journal = {Journal of Differential Geometry},
  volume  = {3},
  pages   = {493--510},
  year    = {1969}
}

Prerequisites

03.02.19
03.02.32

Tier anchors

beginner: do Carmo Riemannian Geometry Ch. 9 (Synge informal); Needham Visual Differential Geometry (parallel transport on a curved loop)
intermediate: do Carmo Riemannian Geometry Ch. 9 §3; Cheeger-Ebin Comparison Theorems Ch. 1 §6; Lee Introduction to Riemannian Manifolds Ch. 12
master: Synge 1936 Quart. J. Math. 7; Weinstein 1968 Ann. Math. 87; Cheeger-Ebin Comparison Theorems in Riemannian Geometry Ch. 1; do Carmo Riemannian Geometry Ch. 9

References

03-modern-geometry/Bott-LecturesOnMorseTheory.pdf · Second variation of energy, index form on closed geodesics (Part III analogue)
Synge — On the connectivity of spaces of positive curvature (1936) · Quarterly Journal of Mathematics (Oxford) 7, pp. 316--320
Weinstein — A fixed point theorem for positively curved manifolds (1968) · Annals of Mathematics 87, pp. 29--33
do Carmo — Riemannian Geometry (1992) · Ch. 9 §3 (the second variation; Synge's theorem; Weinstein's theorem)
Cheeger-Ebin — Comparison Theorems in Riemannian Geometry (1975) · Ch. 1 §6 (the second variation formula and Synge's theorem)

Estimated time

beginner: 15m
intermediate: 40m
master: 78m