The Cut Locus and Closed Geodesics
Throughout, \(M\) is a smooth, connected, complete Riemannian manifold without boundary. Unless explicitly stated otherwise, geodesics used to measure cut times have unit speed. Minimizing curves are unique only up to increasing reparametrization; uniqueness below means uniqueness of their constant-speed parametrizations.
The cut locus marks where radial geodesics stop minimizing. There are two possible mechanisms: the exponential map can become singular, or a second minimizing geodesic can reach the same endpoint. The distinction matters even in flat examples.
A geodesic ceases to minimize after its first conjugate point, but it can cease to minimize earlier. For the flat cylinder \(\mathbb S^1\times\mathbb R\), the obstruction is angular winding, not height: a lift with angular displacement greater than \(\pi\) has a shorter lift to the same endpoint. Vertical geodesics remain minimizing for all time.
Definition 1. Let \((M,g)\) be a complete, connected differentiable manifold. Let \(p \in M\), and \(v \in T_p M\). The cut time of \((p,v)\) is defined as
where \(\gamma_v\) is the maximal geodesic starting at \(p\), with initial velocity \(v\).
For \(v\ne0\), the definition allows any speed, with \(t_{\rm cut}(p,av)=t_{\rm cut}(p,v)/a\) for \(a>0\). For \(v=0\), the constant geodesic has cut time \(+\infty\). In proofs using length equal to elapsed time, we normalize \(|v|=1\). The minimizing times form an initial interval: a shorter path to an earlier point could be concatenated with the remaining segment to shorten any later segment.
A sufficiently short initial segment is minimizing by the normal-neighborhood theorem and Gauss's lemma, so the cut time is positive. Merely remaining inside an arbitrary ball does not imply minimality.
Definition 2. Let \((M,g)\) be a complete, connected differentiable manifold. Let \(p \in M\), and \(v \in T_p M\). Suppose \(t_{\rm cut}(p,v) < \infty\). \(\gamma_v (t_{\rm cut}(p,v)) \in M\) is called the cut point of \(p\) along the geodesic \(\gamma_v\). The cut locus of \(p \in M\) is the set of all points \(q \in M\) such that \(q\) is the cut point of \(p\) along some geodesic, denoted \(\operatorname{Cut}(p)\).
The cut locus depends on the global geometry. In the following models, explicit geodesics make it possible to identify the cut points directly.
Example 3. In Euclidean and hyperbolic space, every geodesic segment minimizes, so the cut locus is empty. On a positive-dimensional compact manifold, every unit-speed geodesic has finite cut time, since a minimizing segment has length at most the diameter; hence every point has nonempty cut locus. The dimension restriction excludes a one-point manifold.
On the unit round sphere \(\mathbb S^n\), \(n\ge1\), minimizing great-circle arcs have length at most \(\pi\), and the cut locus of \(p\) is \(\{-p\}\). For \(n\ge2\), this is also its first conjugate point. For the circle \(n=1\), there are no conjugate points: an endpoint-zero tangential Jacobi field is identically zero.
On the flat cylinder, lift \(p=(e^{i\theta_0},z_0)\) and \(q=(e^{i\theta},z)\) to the Euclidean plane. Geodesic lengths are
The least value is unique unless the angular difference is \(\pi\) modulo \(2\pi\). Thus \(\operatorname{Cut}(p)=\{(e^{i(\theta_0+\pi)},z):z\in\mathbb R\}\). Flatness makes every Jacobi equation \(J''=0\) in a parallel frame, so there are no conjugate points.
The next theorem identifies what can happen at the first loss of minimality. Before that time, the geodesic is unique and has no conjugate point; at the cut time, singularity or nonuniqueness must occur.
Theorem 4. Let \((M,g)\) be a complete, connected differentiable manifold. Let \(p \in M\), and \(v \in T_p M\). Suppose \(c = t_{\rm cut}(p,v) \in (0,\infty]\). Then,
If \(0 < b < c\), then \(\gamma_v|_{[0,b]}\) has no conjugate points. Furthermore, \(\gamma_v\) is the unique minimizing curve between its endpoints.
If \(c < \infty\), then \(\gamma_v|_{[0,c]}\) is minimizing, and one or both of the following conditions are true:
\(\gamma_v(c)\) is conjugate to \(p\) along \(\gamma_v\).
There are at least two minimizing geodesics from \(p\) to \(\gamma_v (c)\).
Here \(|v|=1\). In the first assertion, there is no point conjugate to \(p=\gamma_v(0)\) in \(0<t\le b\). Uniqueness concerns the unit-speed minimizing geodesic with the prescribed endpoints.
Proof. Write \(\gamma=\gamma_v\). If \(b<c\), choose \(b'\in(b,c)\). The segment to \(b'\) minimizes. An interior conjugate point would give a negative index direction on this longer segment, contradicting the second variation necessary condition. If a second unit-speed minimizer \(\xi\) reached \(\gamma(b)\), its terminal velocity would differ from \(\gamma'(b)\), since equal terminal data determine the same geodesic backwards. Concatenating \(\xi\) with \(\gamma|_{[b,b']}\) gives a curve of length \(b'\) with a corner. In normal coordinates at that corner, replacing two short pieces by their chord shortens length: the old length is \(2\varepsilon\), while the new length is \(\varepsilon|u+w|+O(\varepsilon^2)<2\varepsilon\) for distinct incoming and outgoing unit velocities \(u,w\). This contradicts minimality.
Suppose \(c<\infty\). Continuity gives \(d(p,\gamma(c))=\lim_{t\uparrow c}t=c\), so the segment through \(c\) minimizes. Take \(b_i\downarrow c\) with \(b_i>c\), and choose unit-speed minimizers \(\sigma_i:[0,a_i]\to M\) from \(p\) to \(\gamma(b_i)\). Hopf–Rinow supplies them. Their lengths satisfy \(a_i=d(p,\gamma(b_i))<b_i\) and \(a_i\to c\). Compactness of the unit sphere at \(p\) gives a subsequence \(w_i=\sigma_i'(0)\to w\). Hence \(\exp_p(cw)=\gamma(c)\), and the length-\(c\) segment with initial direction \(w\) is minimizing.
If \(cv\) is singular for \(\exp_p\), the endpoint is conjugate to \(p\). Otherwise \(cv\) is a regular point of \(\exp_p\), with an injective neighborhood \(U\). If \(w=v\), both distinct vectors \(a_iw_i\) and \(b_iv\) eventually lie in \(U\), yet their exponential images agree. They are distinct because their norms are \(a_i<b_i\). This is impossible. Thus \(w\ne v\) and there are two distinct minimizing geodesics. \(\square\)
We now sharpen the description by locating the first occurrence of either obstruction along the geodesic.
Proposition 5. Let \(\gamma\) be a geodesic on \(M\). Suppose \(\gamma(t_0)\) is the cut point of \(p = \gamma(0)\) along \(\gamma\). Then, at least one of the following statements is true:
\(\gamma(t_0)\) is the first conjugate point to \(p\) along \(\gamma\).
\(\gamma(t_0)\) is the first point along \(\gamma\) such that there exists another geodesic \(\sigma \neq \gamma\), connecting \(p\) to \(\gamma(t_0)\), with length \(L(\sigma) = L(\gamma|_{[0,t_0]}) = t_0\).
The geodesic is unit speed, and “another” means a distinct geodesic with its same unit-speed parametrization.
Proof. Theorem 4 already supplies one of the two obstructions at \(t_0\). Neither can occur earlier. An earlier conjugate point would lie strictly inside the minimizing segment \([0,t_0]\), and its Jacobi field would yield a shortening variation. An earlier second minimizing geodesic could be concatenated with the remaining segment to form a corner, which can be shortened as in the preceding proof. A second geodesic of length \(t<t_0\) to \(\gamma(t)\) is automatically minimizing because \(d(p,\gamma(t))=t\). Thus the obstruction at \(t_0\) is its first occurrence. \(\square\)
Proposition 5 carries two immediate consequences.
Lemma 6. Let \(M\) be a differentiable manifold, and \(q, p \in M\). If \(q \notin \operatorname{Cut}(p)\), then there exists a unique minimizing geodesic joining \(p\) to \(q\).
Proof. Completeness and connectedness give at least one minimizing geodesic by Hopf–Rinow. If \(p=q\), the only minimizing curve is constant. Otherwise, two distinct minimizers would make either one nonminimizing immediately after \(q\), by the corner-shortening argument. Its cut time would therefore be exactly \(d(p,q)\), putting \(q\) in \(\operatorname{Cut}(p)\), a contradiction. \(\square\)
Lemma 7. If \(q \in \operatorname{Cut}(p)\), then \(p \in \operatorname{Cut}(q)\).
Proof. Reverse a minimizing segment \(\gamma:[0,\ell]\to M\) from \(p\) to its cut point \(q\). If there are two minimizers, reversing both gives two minimizers from \(q\) to \(p\), and the corner argument prevents either from minimizing beyond \(p\). Otherwise Theorem 4 gives a nonzero Jacobi field \(J\) along \(\gamma\) vanishing at both endpoints. The field \(\bar J(t)=J(\ell-t)\) is a Jacobi field along \(\bar\gamma(t)=\gamma(\ell-t)\), also zero at both endpoints: the Jacobi equation is unchanged because both appearances of the tangent change sign. Thus \(p\) is conjugate to \(q\) along the reversed minimizing segment. Any extension beyond \(p\) has an interior conjugate point and cannot minimize. In either case the reversed cut time is exactly \(\ell\). \(\square\)
In what follows, for \(p, q \in M\), let
be the distance function from \(p\) to \(q\), where \(p\) is fixed.
Proposition 8. The function \(d_p\) is smooth on \(M \setminus (\operatorname{Cut}(p) \cup \{p\})\). Furthermore, for each \(q \in M \setminus (\operatorname{Cut}(p) \cup \{p\})\), if \(\gamma^q\) is the unique minimizing geodesic from \(p\) to \(q\), then the gradient of \(d_p\) at \(q\) is given by
Proof. Let \(q\ne p\) lie outside the cut locus, let \(r=d(p,q)\), and let \(u=r\gamma'(0)\) be the initial vector of its unique unit-speed minimizer. This endpoint occurs before the cut time; otherwise it would itself be a cut point. Thus \(d(\exp_p)_u\) is nonsingular. Choose a local inverse near \(q\).
We must check that nearby minimizing vectors use this same inverse branch. If \(q_i\to q\) and \(u_i\) are any minimizing initial vectors with \(\exp_pu_i=q_i\), then \(|u_i|=d(p,q_i)\to r\). Every subsequential limit \(u_*\) satisfies \(|u_*|=r\) and \(\exp_pu_*=q\), so it is the vector of a minimizer to \(q\); uniqueness gives \(u_*=u\). Consequently all such \(u_i\) lie in the chosen inverse neighborhood for large \(i\). This proves that, on a neighborhood of \(q\), the minimizer is unique and its vector is regular. Theorem 4 then excludes cut points in that neighborhood. It also proves, rather than assumes, that the domain outside \(p\) and its cut locus is open.
There \(d_p(z)=|\exp_p^{-1}z|\), which is smooth since the inverse vector is nonzero. Given \(X\in T_qM\), choose a smooth endpoint curve \(\sigma(s)\) with \(\sigma(0)=q,\sigma'(0)=X\). The variation \(c(s,t)=\exp_p(t\exp_p^{-1}\sigma(s)/r)\), \(0\le t\le r\), consists of the minimizing radial segments. First variation gives
Thus \(\nabla d_p(q)=\gamma'(r)\) and \(|\nabla d_p|=1\). \(\square\)
Remark 9. The arguments above shows that, if we appeal to the second variation formula, we can calculate the Hessian of \(d_p\) on \(M \setminus (\operatorname{Cut}(p) \cup \{p\})\). Furthermore, the arguments (almost) directly show that if there are two minimizing geodesics from \(p\) to \(q\), then \(d_p\) cannot be differentiable at \(q\). An argument of this kind will appear in the next subsection.
Here is the nondifferentiability argument. For each minimizing geodesic to \(q\), choose an interior point \(x\) sufficiently close to \(q\) that \(d_x\) is smooth near \(q\). The smooth function \(f_x(z)=d(p,x)+d(x,z)\) satisfies \(d_p\le f_x\) and equality at \(q\). If \(d_p\) were differentiable there, \(f_x-d_p\) would have derivative zero at its minimum, forcing \(\nabla d_p(q)\) to equal that geodesic's terminal unit tangent. Distinct minimizing geodesics have distinct terminal tangents by ODE uniqueness, a contradiction.
For the Hessian away from the cut locus, let \(X\perp\gamma'(r)\), and let \(J\) be the unique Jacobi field with \(J(0)=0,J(r)=X\). Differentiating the terminal radial tangent gives \(\operatorname{Hess}d_p(X,X)=\langle J'(r),J(r)\rangle=I(J,J)\), the last equality following by integration by parts and the Jacobi equation. The radial and mixed terms vanish because \(|\nabla d_p|=1\). These identities underlie Hessian comparison.
1. Application: Existence of Closed Geodesics
Theorem 10. Let \(M\) be complete differentiable manifold, and let \(p \in M\). Suppose there exists a point \(q \in \operatorname{Cut}(p)\) such that the shortest distance from \(p\) to \(\operatorname{Cut}(p)\) is realized. Then,
Either there exists a minimizing geodesic \(\gamma\) from \(p\) to \(q\), along which \(q\) is conjugate to \(p\);
Or there exists exactly two minimizing geodesics from \(p\) to \(q\), denoted \(\gamma\) and \(\sigma\). Furthermore, \(\gamma'(\ell) = -\sigma'(\ell)\), where \(\ell = d(p,q)\).
Proof. Put \(\ell=d(p,q)=d(p,\operatorname{Cut}(p))>0\). If any minimizing geodesic from \(p\) to \(q\) is conjugate at its endpoint, the first alternative holds. Otherwise all these minimizing vectors are regular points of \(\exp_p\), and Theorem 4 supplies at least two distinct minimizers \(\gamma,\sigma\).
Suppose their terminal unit vectors \(a=\gamma'(\ell)\), \(b=\sigma'(\ell)\) are not opposite. The vector \(v=-(a+b)\) then has \(\langle v,a\rangle=\langle v,b\rangle=-(1+\langle a,b\rangle)<0\). Choose a curve \(\tau(s)\) through \(q\) with velocity \(v\). The two disjoint local inverse branches of \(\exp_p\) near \(\ell\gamma'(0)\) and \(\ell\sigma'(0)\) produce distinct geodesic variations ending at \(\tau(s)\). By first variation both lengths have negative derivative at \(0\), so both are less than \(\ell\) for small \(s>0\).
Every geodesic from \(p\) of length less than \(\ell\) is minimizing: otherwise its first cut point would occur at a smaller length than the minimum distance to the cut locus. It follows that both varied segments minimize to \(\tau(s)\). Their lengths must agree with \(d(p,\tau(s))<\ell\); having two distinct minimizers makes \(\tau(s)\) a cut point, again a contradiction. Thus any two distinct minimizers have opposite terminal tangents.
There are exactly two: a third would have terminal tangent opposite to the first, hence equal to the second, and uniqueness backwards from terminal data would identify those two geodesics. Their concatenation, with the second reversed, is smooth at \(q\). This gives a geodesic loop based at \(p\); smoothness at \(p\) requires the extra argument in Proposition 11. \(\square\)
Proposition 11. Let \(M\) be a complete connected Riemannian manifold of dimension \(n\ge2\), with \(0<L\le K\le H\). Write
Either \(\iota(M)\ge\pi/\sqrt H\), or there is a nonconstant smooth closed geodesic of length \(2\iota(M)\), and every nonconstant closed geodesic has at least this length. Equal-length shortest geodesics are allowed.
Proof. Since \(\operatorname{Ric}\ge(n-1)Lg\), Bonnet–Myers makes \(M\) compact. We justify the attainment assertion used in the source. The cut-time function \(c(p,v)\) on the unit tangent bundle is continuous. For upper semicontinuity, if \(t>c(p,v)\), then \(d(p,\exp_p(tv))<t\); this strict inequality persists for nearby \((p,v)\), so their cut times are less than \(t\). For lower semicontinuity, suppose \((p_i,v_i)\to(p,v)\) and \(c_i\to a<c(p,v)\). A uniform normal neighborhood near \(p\) makes \(a>0\). Theorem 4 says that each endpoint at \(c_i\) is conjugate or has a second minimizing initial direction. A subsequence of conjugate cases would make \(d(\exp_p)_{av}\) singular by continuity, impossible before the cut time. In the other case choose the second directions \(w_i\). Compactness in a local tangent trivialization gives a subsequential limit \(w\). The two limiting geodesics to \(\exp_p(av)\) minimize, so uniqueness before the cut time gives \(w=v\). But the map \((x,u)\mapsto(x,\exp_xu)\) is locally injective near \((p,av)\); it cannot identify the distinct vectors \((p_i,c_iv_i)\) and \((p_i,c_iw_i)\) converging there. This also is impossible. The same argument excludes any finite lower limit when \(c(p,v)=+\infty\).
The unit tangent bundle is compact, and \(0<c(p,v)\le\operatorname{diam}M\). Thus its minimum \(\ell=\iota(M)>0\) is attained. Choose a minimizing pair \(p,q\) at distance \(\ell\). If \(\ell\ge\pi/\sqrt H\), the first alternative holds. Assume \(\ell<\pi/\sqrt H\). No unit-speed segment of length \(\ell\) has conjugate endpoints. Indeed an endpoint-zero Jacobi field has zero tangential part and index form zero, whereas the curvature upper bound and the one-dimensional Dirichlet inequality give
for any nonzero endpoint-zero normal field. The scalar inequality follows by writing \(f(t)=\sin(\pi t/\ell)u(t)\), integrating the cross term, and obtaining \(\int(f'^2-(\pi/\ell)^2f^2)=\int\sin^2(\pi t/\ell)u'^2\ge0\), first for compactly supported smooth \(f\), then by density; apply it componentwise in a parallel frame.
Theorem 10 therefore gives exactly two minimizers \(\eta,\xi\) from \(p\) to \(q\), with opposite terminal velocities. Cut-locus symmetry and the global minimality of \(\ell\) show that \(p\) realizes the nearest cut point of \(q\) as well. Apply Theorem 10 to the reversed segments, again with no conjugacy: their terminal velocities are opposite, which says \(\eta'(0)=-\xi'(0)\). Concatenating \(\eta\) with the reverse of \(\xi\) is thus smooth at both joining points and gives a closed geodesic of length \(2\ell\).
Finally, a nonconstant closed geodesic of length \(A\) supplies two distinct half-arcs from its starting point to its midpoint. If \(A/2<\ell\), both half-arcs would be minimizing and would remain before their cut times, contradicting uniqueness in Theorem 4. Hence \(A\ge2\ell\), proving the claimed shortest-length property. \(\square\)
Reading: Carmo (1992), Riemannian geometry.