The Isotopy Lemma and Injectivity Radius
Throughout, \(M\) is a connected complete smooth Riemannian manifold without boundary. All minimizing geodesics used in distance arguments have unit speed. Compactness is stated where it is needed. The smooth-function results, Lemma 10 and Proposition 11, need only their stated compact-strip hypotheses.
Morse theory suggests following level sets by a flow, but the distance function is not smooth at the cut locus. To retain this approach, we use the directions of all minimizing geodesics in place of a single gradient. We apply this idea to the distance function \(d_p\) (defined in Proposition 8). However, note that we cannot directly apply the theory, because—as we have seen in the proof of Proposition 8—\(d_p\) is not a smooth function on all of \(M\). Furthermore, outside of \(\operatorname{Cut}(p) \cup \{p\}\) and in neighbourhoods where \(d_p\) is smooth, \(|\nabla d_p| = 1\), meaning there are no critical points to examine. Therefore, to apply the insights of Morse theory to the distance function, we need to adapt a few definitions.
Definition 1 (Critical point of \(d_p\)). A point \(q \neq p\) is called a critical point of \(d_p\) if, for all \(X_q \in T_q M\), there exists a minimizing geodesic \(\gamma\) from \(q=\gamma(0)\) to \(p\) such that \(\langle \gamma'(0), X_q \rangle \geq 0\).
We regard \(p\) as a trivial critical point by convention; Definition 1 concerns \(q\ne p\). Let \(\mathcal U_q\) be the compact set of initial unit directions of minimizing geodesics from \(q\) to \(p\). Noncriticality means that some outward vector \(V\) satisfies \(\langle V,u\rangle<0\) for every \(u\in\mathcal U_q\); its negative is an inward vector with all inner products positive. Equivalently, \(q\) is critical if and only if \(0\) belongs to the convex hull of \(\mathcal U_q\), by finite-dimensional separation. A singleton direction set is not critical, so every nontrivial critical point lies in the cut locus.
Example 2. Consider the distance function \(d_p\) on the following manifolds.
On \(\mathbb S^2\), the only critical point of \(p\) is \(-p\).
On the cylinder \(\mathbb S^1 \times \mathbb R\), the only critical point of \((x, y)\) is \((-x, y)\).
On the flat torus \(\mathbb S^1 \times \mathbb S^1\), the critical points of \(p\) are the two barycenters of the sides and the corner point.
On a compact manifold \(M\), if \(q\) is a farthest point from \(p\), then \(q\) is a critical point of \(p\).
If \(q \in \operatorname{Cut}(p)\) is not conjugate to \(p\), and \(d(p, q) = d(p, \operatorname{Cut}(p))\), then \(q\) is a critical point of \(p\).
These examples use the standard product metrics, with the base point itself omitted. On the sphere the antipode has all inward directions and is critical. On the cylinder, the two opposite angular choices can have convex hull containing zero only when their common vertical component is zero; this gives \((-x,y)\). On the square flat torus, represented by \([-\frac12,\frac12]^2\) centered at \(p\), the distinct critical points are the two side-midpoint classes and the corner class: zero lies in the convex hull of the shortest lift vectors precisely at those points. The farthest-point assertion follows from the inward field argument below. The last assertion follows from the nearest-cut-point theorem: in the absence of conjugacy its two minimizing terminal directions are opposite.
The isotopy lemma uses this notion of critical point to identify an annular region with a product. Its proof constructs a vector field along which distance changes strictly.
Lemma 3 (Isotopy lemma). If \(0<r_1<r_2<\infty\) and the closed annulus \(A=\{r_1\le d_p\le r_2\}\) contains no critical point of \(d_p\), then
with distance equal to the second coordinate. If the entire exterior \(\{d_p\ge r_1\}\) has no critical points, the corresponding conclusion is \(d_p^{-1}(r_1)\times[r_1,\infty)\); there is no level at \(+\infty\).
Proof. The minimizing-direction sets are upper semicontinuous. Indeed, if \(q_i\to q\ne p\) and \(u_i\in\mathcal U_{q_i}\), a convergent subsequence in a local orthonormal trivialization has limit \(u\). Smooth geodesic dependence and \(d(p,q_i)\to d(p,q)\) show that the limiting segment reaches \(p\) with length \(d(p,q)\); hence \(u\in\mathcal U_q\).
At a noncritical \(q\), choose an inward vector \(X_q\) with \(\langle X_q,u\rangle>0\) for all \(u\in\mathcal U_q\). Compactness of \(\mathcal U_q\) gives a strict margin. Extend \(X_q\) smoothly in a chart; the margin persists nearby by the preceding subsequence argument. A locally finite partition of unity combines these fields into a smooth field \(X\) with the same strict inequality for every minimizing direction. Normalize it to unit length. On a compact annulus the margin has a uniform positive lower bound \(c\), using compactness of the set of all minimizing directions over that annulus.
Distance need not be differentiable at a cut point, so we justify the monotonicity without choosing a smooth family of minimizers. Given \(u\in\mathcal U_q\), choose a point \(z\) a short distance along that minimizer toward \(p\). The smooth local function \(h(y)=d(p,z)+d(z,y)\) lies above \(d_p(y)\), agrees at \(q\), and has \(\nabla h(q)=-u\). Along an integral curve \(\sigma\) of \(X\), this bounds the upper right derivative of \(d_p(\sigma(t))\) by \(-\langle X,u\rangle\le-c\). Since distance is Lipschitz, its composition with this smooth curve is absolutely continuous, differentiable almost everywhere, and therefore
whenever the intervening trajectory lies in the compact annulus. This is the required strict decrease.
Hopf–Rinow makes the annulus compact. The flow exists while it remains there, and the displayed rate forces it to reach the lower boundary in finite time. Reversing the flow reaches the upper boundary. Each trajectory crosses each distance level exactly once. If \(\phi_t\) is the inward flow, define \(H(x,r)\), for \(d_p(x)=r_1\), by following the reversed flow until its distance is \(r\). The hitting time depends continuously on \((x,r)\): continuous flow dependence gives convergence of distance functions, and the uniform strictly monotone rate bounds the change of hitting time by the change in the attained level divided by \(c\). The same argument for the inverse, which follows an arbitrary point to level \(r_1\), proves that \(H\) is a homeomorphism.
For an infinite exterior choose one locally finite field on the whole noncritical region. Every finite annulus has its own positive rate bound and compactness argument, so its trajectories reach each prescribed finite level. Unit speed prevents finite-time escape on a complete manifold. These compatible hitting maps give the product with \([r_1,\infty)\). \(\square\)
Thus topology cannot change across a compact interval of distance levels containing no critical points. The converse is not asserted: encountering a critical point need not force a change in topology, and a critical point need not be a farthest point.
Lemma 4. Let \((M,g)\) be a compact differentiable manifold. Let \(p \in M\), and \(q\) be the farthest point from \(p\) in terms of the usual distance function. Then, \(q\) is a critical point of \(p\). Furthermore, if \(d(p,q) = \operatorname{diam}(M)\), then for any \(X_p \in T_p M\), there exists a minimal geodesic \(\gamma\) from \(p=\gamma(0)\) to \(q\) such that \(\langle \gamma'(0), X_p \rangle \geq 0\).
Proof. If a farthest point \(q\ne p\) were noncritical, the reversed inward field just constructed would strictly increase distance from \(p\) for a short time, contradicting maximality. If \(d(p,q)\) is the diameter, \(p\) is also a farthest point from \(q\). Apply the same argument to \(d_q\) at \(p\), and unpack Definition 1 to obtain the stated inequality for every \(X_p\). \(\square\)
Lemma 5. Let \((M,g)\) be a compact differentiable manifold of dimension \(n\). Let \(p \in M\). If \(d_p\) has only one nontrivial critical point \(q \neq p\), then \(M\) is homeomorphic to the \(n\)-sphere.
Proof. Every farthest point is critical by Lemma 4. Thus \(q\) is the unique farthest point, with \(D=d(p,q)>0\). Choose a small normal radius \(r_0>0\) at \(p\). On \(M\setminus\{p,q\}\), construct one inward field as in Lemma 3. Its flow gives compatible product coordinates for all levels in \([r_0,D)\): every finite subannulus is compact and noncritical. Attach this product to the normal ball at \(p\). Since \(\partial B_{r_0}(p)\cong\mathbb S^{n-1}\), it follows that
The radial coordinate on \([r_0,D)\) can be stretched to \([r_0,\infty)\). Finally, a compact Hausdorff space obtained by adding one point to \(\mathbb R^n\) is its one-point compactification: complements of neighborhoods of the added point are compact, and every compact subset has a neighborhood of the added point disjoint from it. Hence \(M\cong\mathbb S^n\). This also handles \(n=1\); the hypothesis cannot occur for a connected zero-dimensional manifold. \(\square\)
Before using the isotopy lemma in global comparison arguments, we describe the region where exponential coordinates are injective. Its size is measured by the injectivity radius.
1. Application: Estimating Injectivity Radius
Definition 6. The injectivity radius at \(p\) is
For complete manifolds, the cut-locus theorem gives \(\iota(p)=\inf_{|v|=1}t_{\rm cut}(p,v)=d(p,\operatorname{Cut}(p))\), with the empty-set distance \(+\infty\). Proposition 9 explains why “injective” gives the same definition in this setting.
Example 7. The unit round sphere has \(\iota=\pi\): open radial balls of that radius map diffeomorphically onto the complement of the antipode. For the flat torus \(\mathbb R^2/\mathbb Z^2\), \(\iota=\frac12\). More generally, for a full lattice \(\Lambda\), the value is half the shortest length of a nonzero lattice vector: distinct tangent vectors have the same exponential image exactly when their difference lies in \(\Lambda\). Thus normalization of the torus metric matters. Euclidean space and the hyperbolic upper half-plane with metric \((dx^2+dy^2)/y^2\) have infinite injectivity radius by their global exponential coordinates.
Remark 8. Every point has positive injectivity radius by a normal-neighborhood argument. A compact manifold has a uniform positive lower bound, using the inverse function theorem near its compact zero section (or the cut-time continuity proved in the previous note). Completeness and noncompactness alone do not determine the global infimum: Euclidean space has infinite injectivity radius and a standard flat cylinder has positive finite injectivity radius. It can also be zero, for example on a complete hyperbolic cusp quotient, where the nontrivial horizontal loops have lengths tending to zero toward the cusp. Lifting these loops to the hyperbolic plane shows that the shortest deck-displacement lengths, and hence the injectivity radii, tend to zero.
It is instructive to see the following equivalent formulation of the injectivity radius.
Proposition 9. Let \(M\) be a differentiable manifold, and \(p \in M\). Let \(r > 0\) be such that \(\exp_p\) is defined, and injective on the open ball \(B_r (0)\) in \(T_p M\). Then, \(\exp_p|_{B_r(0)}\) is a diffeomorphism of \(B_r(0)\) onto an open subset of \(M\).
Proof. We use the standing completeness assumption. Suppose \(d(\exp_p)_v\) were singular at some \(v\in B_r(0)\). It is nonsingular at \(0\), so \(v\ne0\). Choose \(a>1\) with \(a|v|<r\). The segment \(\gamma(t)=\exp_p(tv)\), \(0\le t\le a\), has an interior conjugate point at \(t=1\), and therefore is not minimizing to \(q=\gamma(a)\). By Hopf–Rinow there is a minimizing initial vector \(w\) to \(q\), with \(|w|=d(p,q)<a|v|<r\). Then \(w\ne av\), but \(\exp_pw=\exp_p(av)\), contradicting injectivity. Thus the differential is nonsingular throughout the ball. The inverse function theorem makes its image open and gives smooth local inverses; injectivity makes them agree on overlaps, producing a smooth global inverse. \(\square\)
The following fact is proven in (Milnor 1963).
Lemma 10 (Regular interval theorem). Let \(f:M\to\mathbb R\) be smooth and \(a<b\). If \(f^{-1}([a,b])\) is compact and contains no critical point, then \(M^a=\{f\le a\}\) is a strong deformation retract of \(M^b\). Explicitly, there are continuous maps \(h_s:M^b\to M^b\) with \(h_0=\mathrm{id}\), \(h_s|_{M^a}=\mathrm{id}\), and \(h_1(M^b)\subset M^a\). The two sublevels are also diffeomorphic as manifolds with boundary. No nondegeneracy assumption outside the strip is needed.
In the proof below, \(x_1=a\) and \(x_2=b\). The homotopy takes values in \(M^b\), not in \(M^a\) at every intermediate time.
Proof. First, we equip \(M\) with a Riemannian metric \(g = \langle \cdot, \cdot \rangle\), which is always possible using a partition of unity. This allows us to define the gradient vector field \(\nabla f\) on \(M\), uniquely characterized by the relation \(\langle \nabla f, Y \rangle = d f(Y)\) for any smooth vector field \(Y \in \mathscr{X}(M)\).
Let \(K = f^{-1}([x_1, x_2])\). By hypothesis, \(K\) is compact and contains no critical points of \(f\). Thus, the gradient \(\nabla f\) is non-zero everywhere on \(K\), meaning \(\lVert \nabla f \rVert^2 > 0\) for all \(x \in K\).
Since \(K\) is compact and the set of critical points of \(f\) is closed, we can find a relatively compact open neighborhood \(U\) of \(K\) such that its closure \(\overline{U}\) also contains no critical points. On \(U\), we can define a normalized smooth vector field that points in the direction of steepest descent:
Next, we choose a smooth bump function \(\rho : M \to [0,1]\) such that \(\rho \equiv 1\) on \(K\) and \(\operatorname{supp}(\rho) \subset U\). We then define a global vector field \(X\) on \(M\) by extending \(\rho V\) by zero:
Because \(\operatorname{supp}(\rho) \subset U\), \(X\) patches smoothly to zero and is a globally well-defined smooth vector field on \(M\). Furthermore, since \(X\) has compact support (contained entirely in \(\operatorname{supp}(\rho) \subset \overline{U}\)), it is a complete vector field. This means it generates a globally defined 1-parameter group of diffeomorphisms \(\phi : \mathbb{R} \times M \to M\). Let \(\phi_t(x) = \phi(t, x)\) denote the flow of \(X\) at time \(t\).
We now compute how the function \(f\) evaluates along the integral curves of \(X\). By the chain rule,
If \(x \notin U\), \(X(x) = 0\), so the derivative is zero. If \(x \in U\), we substitute \(X\) to get:
Since \(\rho \geq 0\) everywhere, \(f(\phi_t(x))\) is non-increasing for all \(t \geq 0\). In particular, whenever the trajectory \(\phi_t(x)\) is inside \(K\), we have \(\rho(\phi_t(x)) = 1\), yielding:
This means that as long as the flow remains inside \(K\), the value of \(f\) decreases at a uniform rate of exactly \(1\) unit per unit time.
To construct the deformation retract, we want points in \(M^{x_2}\) to flow “downward” until they hit the boundary of \(M^{x_1}\), and we want points already in \(M^{x_1}\) to remain stationary. Define a displacement time function \(\tau : M^{x_2} \to \mathbb{R}\) by:
Since \(f\) is continuous, \(\tau\) is continuous. Note that \(\tau(x) = 0\) if and only if \(f(x) \leq x_1\), i.e. \(x \in M^{x_1}\).
We now define the required homotopy \(h_s : M^{x_2} \to M^{x_2}\) for \(s \in [0,1]\) as:
We can verify that this homotopy satisfies the defining properties of a strong deformation retract:
At \(s = 0\), \(h_0(x) = \phi_0(x) = x\), so \(h_0\) is the identity map on \(M^{x_2}\).
For any \(x \in M^{x_1}\), \(f(x) \leq x_1\), meaning \(\tau(x) = 0\). Consequently, \(h_s(x) = \phi_0(x) = x\) for all \(s \in [0,1]\). This proves that \(M^{x_1}\) is pointwise fixed by the homotopy, meaning \(h_s|_{M^{x_1}} = \operatorname{id}_{M^{x_1}}\).
At \(s = 1\), let \(h = h_1\). We need to show that \(h(M^{x_2}) \subset M^{x_1}\). If \(x \in M^{x_1}\), then \(h(x) = x \in M^{x_1}\). If \(x \in M^{x_2} \setminus M^{x_1}\), then \(x_1 < f(x) \leq x_2\), so \(x\) initially belongs to \(K\). Since \(f\) is non-increasing and \(\frac{d}{\dif t} f(\phi_t(x)) = -\rho \geq -1\), the function value \(f(\phi_t(x))\) can decrease by at most \(t\). Thus, for all \(t \in [0, \tau(x)]\), we have \(f(\phi_t(x)) \geq f(x) - \tau(x) = x_1\). This ensures that the trajectory does not leave \(K\). Hence \(\rho(\phi_t(x)) = 1\) for the entire duration, and (1) strictly holds. At time \(t = \tau(x) = f(x) - x_1\), the function evaluates to:
\[f(h(x)) = f(\phi_{\tau(x)}(x)) = f(x) - \tau(x) = f(x) - (f(x) - x_1) = x_1 ~.\]Thus, \(h(x) \in f^{-1}(x_1) \subset M^{x_1}\).
Therefore, \(h = h_1 : M^{x_2} \to M^{x_1}\) is a continuous map that restricts to the identity on \(M^{x_1}\), and \(h_s\) is the required continuous homotopy starting at the identity. This proves that \(M^{x_1}\) is a deformation retract of \(M^{x_2}\). ◻
The same complete flow also gives the promised diffeomorphism. Its derivative satisfies \(-1\le(d/dt)f(\phi_t(x))\le0\) everywhere, and equals \(-1\) in the strip. Consequently \(\phi_{b-a}\) maps \(M^b\) into \(M^a\). Conversely, if \(y\in M^a\), the reversed flow increases \(f\) by at most \(b-a\), so \(\phi_{-(b-a)}(y)\in M^b\). These inverse global flow maps restrict to a diffeomorphism of the sublevels. The fixed-sublevel deformation above is a different map and need only be continuous.
Proposition 11. Let \(M\) be a differentiable manifold of \(\dim M = n\). Let \(f : M \to \mathbb R\) be a smooth function whose critical points are all nondegenerate. Given \(p, q \in M\) and a differentiable curve \(\gamma : [0,1] \to M\) such that \(\gamma(0) = p\) and \(\gamma(1) = q\), let \(a = \max (f(p), f(q))\), and set \(M^a = \{x \in M : f(x) \leq a\}\), \(b = \max_{t \in [0,1]} (f \circ \gamma)(t)\). Suppose \(f^{-1}([a,b])\) is compact, and does not contain any critical points of index zero or one. Then, for all \(\delta > 0\), \(\gamma\) is homotopic to a curve \(\widetilde \gamma\) such that \(\widetilde \gamma([0,1]) \subset M^{a+\delta}\), where the endpoints are fixed.
The same conclusion holds for any \(a\ge\max(f(p),f(q))\). The homotopy need only be continuous; its resulting path may be smoothed while staying below \(a+\delta\), by first working below a slightly smaller level.
Proof. If \(b\le a+\delta\), use the original path. Otherwise choose a regular value \(a'\in(a,a+\delta)\). The compact strip contains only finitely many critical points: their set is closed and nondegenerate critical points are isolated. Work in small disjoint Morse neighborhoods of these points and in compact regular strips between their finitely many critical values. Several points at the same value can be handled simultaneously; no uncontrolled perturbation of \(f\) is required.
The regular-strip deformation pushes a path down between successive critical levels while fixing endpoints below \(a'\). At an index-\(\lambda\) critical point, \(\lambda\ge2\), use the handle description in the preceding note. Its co-core \(\{0\}\times\mathbb D^{n-\lambda}\) has codimension at least two. A path can be perturbed, relative to its endpoints and outside the handle, to avoid this co-core: relative transversality makes the inverse image have dimension \(1-\lambda<0\), hence empty. In the local coordinates this says \(u=x_-\ne0\). Under the negative adapted flow, \(u(t)=e^{2t}u(0)\) and \(v(t)=e^{-2t}v(0)\), so
strictly decreases and reaches the lower regular level in finite time. Hitting times are continuous on the compact perturbed path; regular flow outside the chart completes the downward deformation. The same local construction works at a critical point initially lying on the top level, by first perturbing the path in its compact coordinate neighborhood. No global compactness above \(b\) is needed.
Inducting downward over the finitely many levels gives a path below \(a'\), homotopic to the original relative to endpoints. Since \(a'<a+\delta\), this proves the result. The argument constructs a deformation of the path; it does not assert that deleting an entire noncompact co-core automatically produces a global deformation retraction. \(\square\)
Proposition 12 (Klingenberg). Let \(M\) be a simply connected, compact, differentiable manifold of \(\dim M = n \geq 3\), and \(n\) is odd. Furthermore, suppose \(\frac14 < K \leq 1\). Then, \(\iota (M) \triangleq \inf \{ \iota(p) : p \in M \} \geq \pi\).
Proof. Put \(\kappa=\min K>1/4\). The index-form Dirichlet estimate under \(K\le1\) excludes conjugate points on segments shorter than \(\pi\). If \(\iota(M)<\pi\), the shortest-loop alternative proved in the preceding note supplies a nonconstant closed geodesic \(\gamma\) of length \(\ell=2\iota(M)<2\pi\), based at \(p\).
Choose \(\varepsilon>0\) with \(5\varepsilon<\ell\) and \(3\varepsilon<2\pi-\max(\ell,\pi/\sqrt\kappa)\), and with \(\gamma(\ell-\varepsilon)\) nonconjugate to \(p\) along the indicated segment. Such a choice exists because conjugate instants are finite on a compact interval. The local inverse of \(\exp_p\) at \((\ell-\varepsilon)\gamma'(0)\) supplies nearby long geodesics. Sard's theorem lets us choose a nearby endpoint \(q\) that is a regular value of the whole map \(\exp_p\). Choose it so close that the corresponding geodesic \(\gamma_1\) has length between \(\ell-2\varepsilon\) and \(\ell\), while \(d(p,q)<2\varepsilon\), using the last short arc of the closed geodesic. Let \(\gamma_0\) be a minimizer to \(q\). Then the two geodesics are distinct and \(L_0=L(\gamma_0)<2\varepsilon\).
We make the finite-dimensional Morse reduction explicit. Simple connectivity gives a homotopy of paths from \(\gamma_0\) to \(\gamma_1\), fixing \(p,q\); choose it piecewise smooth with bounded energy \(E=\frac12\int_0^1|\dot c|^2\). Fix \(C\) above this bound and a uniform convex normal radius \(r>0\). Subdivide \([0,1]\) into \(N\) equal pieces, with \(\sqrt{2C/N}<r/2\). An energy-\(\le C\) path has each subarc of length at most \(\sqrt{2C/N}\), by Cauchy–Schwarz. Replace each subarc by its unique minimizing geodesic. This gives a continuous path in the finite-dimensional space of broken geodesics, described by their \(N-1\) vertices, and does not increase energy. It fixes \(\gamma_0,\gamma_1\) after they are given constant-speed parametrizations. The energy on vertices is
Its sublevel \(E_N\le C\) is compact inside the domain where all adjacent distances are \(<r\), because each adjacent distance is \(<r/2\) and \(M\) is compact.
First variation says that a critical broken geodesic has matching velocities at every vertex, hence is a smooth geodesic. Its Hessian is the index form on piecewise Jacobi fields determined by the vertex values. Every endpoint-zero field splits as such a field plus one vanishing at every vertex; integration by parts makes these pieces index-orthogonal. The latter piece has positive index form on each short segment, by the index lemma. Therefore the finite-dimensional Hessian has the same index and nullity as the full geodesic index form. Since \(q\) is a regular value of \(\exp_p\), nullity is zero: \(E_N\) is Morse throughout this compact sublevel.
A geodesic of length \(L>\pi/\sqrt\kappa\) has index at least \(n-1\ge2\). Indeed for parallel normal vectors \(V\), the fields \(Y(t)=\sin(\pi t/L)V(t)\) satisfy
and all their nonzero linear combinations satisfy the same inequality. Choose
Thus all critical points with energy \(>A^2/2\) have index at least two. Proposition 11 applied to the path in the finite-dimensional broken-geodesic space pushes it below \((A+\eta)^2/2\), for arbitrarily small \(\eta>0\), keeping its endpoints fixed. Choose \(\eta\) with \(A+\eta<2\pi-L_0\). Cauchy–Schwarz now gives a homotopy from \(\gamma_0\) to \(\gamma_1\) through paths of length \(<2\pi-L_0\).
For clarity, the last obstruction is the long-homotopy lemma: a nontrivial geodesic digon cannot be contracted through loops all shorter than \(2\pi\) when the conjugate radius is at least \(\pi\). Append the reverse of \(\gamma_0\) to the preceding homotopy. Its starting loop is the retraced minimizer, which contracts without increasing length; its final loop is the nontrivial digon \(\gamma_1*\bar\gamma_0\). All lengths are \(<2\pi\), a contradiction.
Here is the lifting reason for that lemma. Parametrize a proposed contraction proportionally to arclength and split every loop into two half-arcs from its basepoint to its midpoint. Each half has length \(<\pi\), hence lifts under the nonsingular exponential map on \(B_\pi(0)\): Gauss's lemma bounds the radial norm of a lift by its travelled length, keeping it in a compact subball, so local inverses continue the lift. The two lifted endpoints agree for the constant loop; agreement is open and closed along the homotopy, because the exponential map is locally injective and the lifts vary continuously. They therefore agree for the digon. One half of the digon lies entirely in its longer geodesic side and lifts radially, with length equal to its endpoint radius. The other lift has the same endpoints and length, so equality in Gauss's radial length inequality forces it to be the same radial segment. This would make the digon retraced, contradicting the distinctness of \(\gamma_0,\gamma_1\). This is the argument of Abresch–Meyer, Lemma 4.3 and Remark 4.4, pp.21–23.
The contradiction proves \(\iota(M)\ge\pi\). The argument uses \(n\ge3\); the stated odd-dimensional case is the one needed here. \(\square\)
Reading: Carmo (1992), Riemannian geometry.