Second Variation and Minimizing Geodesics
Let \((M,g)\) be a smooth Riemannian manifold, with Levi–Civita connection \(\nabla\). See the connection notes for the notation.
Every energy minimizing curve is a geodesic; as we have noted, the first variation of the energy functional must vanish for every proper variation of such curve. From calculus, the second variation of the energy functional must be nonnegative. The second derivative measures which variations can lower the energy. Its curvature term is the source of the relation between conjugate points and the failure of a geodesic to minimize.
Proposition 1 (Second variation). Let \(\gamma:[a,b]\to M\), \(a<b\), be a geodesic of constant speed \(c>0\), and \(\Gamma(s,t)\) a smooth variation. Set \(T=\gamma'\), \(V=\partial_s\Gamma|_{s=0}\), \(A=D_s\partial_s\Gamma|_{s=0}\), and \(V^\perp=V-c^{-2}\langle V,T\rangle T\). For \(E(s)=\frac12\int_a^b|\partial_t\Gamma|^2dt\) and \(L(s)=\int_a^b|\partial_t\Gamma|dt\),
These signs use \(R(X,Y)Z=\nabla_Y\nabla_XZ-\nabla_X\nabla_YZ+\nabla_{[X,Y]}Z\).
Proof. Compactness of \([a,b]\) and \(|\partial_t\Gamma(0,t)|=c>0\) give a uniform small strip with \(|\partial_t\Gamma|\geq c/2\). Thus the length integrand, as well as the energy integrand, is smooth there and may be differentiated under the integral. On the variation write \(S=\partial_s\Gamma\), \(T_s=\partial_t\Gamma\). Torsion-freeness gives \(D_sT_s=D_tS\), hence
The commutator formula gives \(D_sD_tS=D_tD_sS-R(S,T_s)S=D_tD_sS+R(T_s,S)S\). At zero, skew-adjointness gives \(\langle R(T,V)V,T\rangle=-\langle R(T,V)T,V\rangle\). Integration by parts in the first term, using \(D_tT=0\), yields the energy formula.
For length, the chain rule for the square root gives
At zero the first numerator integrates to the just computed energy expression. Since \(T/c\) is parallel,
Finally the curvature term is unchanged when either occurrence of \(V\) is replaced by \(V^\perp\), because \(R(T,T)=0\) and \(\langle R(T,V)T,T\rangle=0\). This proves the length formula with the same endpoint acceleration term. ◻
Remark 2. For fixed endpoints, the second variation of length depends only on the normal component of the variational field. For moving endpoints, the endpoint acceleration term in the formula must also be retained. Tangential reparametrizations with fixed endpoints do not change length.
Remark 3. In a proper variation the endpoint curves are constant, so their velocity and covariant acceleration both vanish. Thus the boundary terms in the second-variation formulas vanish. The condition \(V(a)=V(b)=0\) alone, without fixed endpoint curves, would not imply zero endpoint acceleration.
The interior terms in the second variation form a symmetric bilinear expression in vector fields along the geodesic. We give this expression a name so that its positivity and null space can be studied directly.
Definition 4. For continuous piecewise smooth vector fields along a geodesic, the index form is
The curvature symmetries make \(X\mapsto R(T,X)T\) self-adjoint, so \(I\) is a symmetric bilinear form. Equivalently, by skew-adjointness,
A field is called proper when it vanishes at both endpoints and normal when it is orthogonal to \(T\).
For a fixed-endpoint normal variation of a unit-speed geodesic, the boundary terms disappear. The second variation of length is then given by the index form.
Corollary 5. Let \(\gamma:[a,b]\to M\) be a nonconstant geodesic of constant speed \(c=|\dot\gamma|>0\), and let \(\Gamma\) be a proper variation with variational field \(V\). Then
In particular, for a unit-speed geodesic and a normal variational field, \(L''(0)=I(V,V)\). If \(\gamma\) minimizes length, then \(I(V,V)\ge0\) for every continuous piecewise smooth field \(V\) vanishing at the endpoints.
Proof. Substitute the fixed endpoints into Proposition 1. A length-minimizing geodesic, with constant speed on this fixed parameter interval, also minimizes energy by the length-energy inequality. Every endpoint-vanishing field is realized by the proper variation \(\exp_{\gamma(t)}(sV(t))\); hence the energy has a local minimum at \(s=0\), giving \(I(V,V)=E''(0)\ge0\). For piecewise smooth fields, apply the calculation on each smooth subinterval. The boundary terms cancel at internal partition points because \(\dot\gamma\) and the transversal acceleration are continuous there. ◻
For the stated piecewise-smooth class, each piece extends smoothly to its closed subinterval; continuity at a junction for all \(s\) makes the transversal endpoint curve and its \(s\)-acceleration agree from both sides. These are the boundary terms that cancel. No equality of the one-sided derivatives \(D_tV\) is required.
Lemma 6. If \(\gamma\) is a geodesic and \(V\) is a proper normal (piecewise) smooth vector field along \(\gamma\), then \(I(V,W)=0\) for every proper normal (piecewise) smooth vector field \(W\) along \(\gamma\) if and only if \(V\) is a Jacobi field.
Proof. Let \([a, b]\) be the domain of the geodesic \(\gamma\). Recall that a piecewise smooth vector field along \(\gamma\) is continuous everywhere on \([a,b]\) and smooth on each subinterval of a suitable partition of \([a,b]\). Furthermore, a vector field \(X\) is proper if it vanishes at the endpoints, meaning \(X(a) = X(b) = 0\), and normal if it is everywhere orthogonal to the velocity vector \(\dot{\gamma}\).
Let \(a = t_0 < t_1 < \dots < t_k = b\) be a common partition of \([a,b]\) such that both \(V\) and \(W\) are smooth on each subinterval \([t_{i-1}, t_i]\). We will denote the covariant derivative \(\nabla_{\dot{\gamma}} X\) by \(X'\) and \(\nabla_{\dot{\gamma}}^2 X\) by \(X''\).
The index form \(I(V, W)\) for proper piecewise smooth vector fields \(V\) and \(W\) along \(\gamma\) is defined as
Using integration by parts on each subinterval \([t_{i-1}, t_i]\), we can rewrite the first term in the integral:
Summing this expression over all \(k\) subintervals yields
Because \(W\) is globally continuous, \(W(t_i^-) = W(t_i^+) = W(t_i)\) for all \(i\). We regroup the boundary sum at the partition points as
Since \(W\) is a proper vector field, it vanishes at the endpoints (i.e., \(W(a) = W(b) = 0\)), so the evaluations at \(a\) and \(b\) vanish. Let \(\Delta V'(t_i) = V'(t_i^+) - V'(t_i^-)\) denote the jump in the covariant derivative of \(V\) at the partition point \(t_i\). Substituting this result back into the index form gives
(\(\Leftarrow\)) Assume \(V\) is a Jacobi field. By definition, a Jacobi field is smooth everywhere on \([a, b]\), which implies that its covariant derivative is continuous across all partition points. Thus, \(\Delta V'(t_i) = 0\) for all \(i = 1, \dots, k-1\). Furthermore, \(V\) identically satisfies the Jacobi equation \(V'' + R(\dot{\gamma},V)\dot{\gamma} = 0\) for all \(t \in [a, b]\). Substituting these facts into (4), both the integral and the boundary sum vanish entirely. Hence, \(I(V, W) = 0\) for every proper normal piecewise smooth vector field \(W\).
(\(\Rightarrow\)) Conversely, assume that \(I(V, W) = 0\) for every proper normal piecewise smooth vector field \(W\) along \(\gamma\). We wish to show that \(V\) is a Jacobi field.
First, we show that \(V\) satisfies the Jacobi equation on the interior of each subinterval \((t_{i-1}, t_i)\). Let \(U = V'' + R(\dot{\gamma},V)\dot{\gamma}\). We claim that \(U\) is normal to \(\dot{\gamma}\). Since \(V\) is a normal vector field (\(\langle V, \dot{\gamma} \rangle = 0\)) and \(\gamma\) is a geodesic (\(\nabla_{\dot{\gamma}}\dot{\gamma} = 0\)), differentiating the inner product yields \(\langle V', \dot{\gamma} \rangle = 0\) and subsequently \(\langle V'', \dot{\gamma} \rangle = 0\). By the skew-symmetry properties of the Riemann curvature tensor, \(\langle R(\dot{\gamma},V)\dot{\gamma}, \dot{\gamma} \rangle = 0\). Therefore, \(\langle U, \dot{\gamma} \rangle = 0\), meaning \(U\) is indeed everywhere normal to \(\dot{\gamma}\).
Choose a smooth, non-negative bump function \(f: [a, b] \to \mathbb{R}\) such that \(f(t) > 0\) for \(t \in (t_{i-1}, t_i)\) and \(f(t) = 0\) for \(t \notin (t_{i-1}, t_i)\). Define the vector field \(W(t) = f(t) U(t)\) along \(\gamma\). By construction, \(W\) is proper, normal, and piecewise smooth. Since \(W(t_j) = 0\) for all partition points \(t_j\), the boundary sum in (4) vanishes. Substituting this \(W\) into (4) gives
Since \(f(t) > 0\) on \((t_{i-1}, t_i)\) and \(\|U(t)\|^2 \geq 0\), we must have \(U(t) = 0\) strictly on \((t_{i-1}, t_i)\). This holds for every subinterval, proving that \(V'' + R(\dot{\gamma},V)\dot{\gamma} = 0\) wherever \(V\) is smooth.
Since \(U = 0\) uniformly on all subintervals, the integral term in (4) drops out for any \(W\), reducing the index form to
We now show that \(V'\) is globally continuous, i.e., the jumps \(\Delta V'(t_i)\) are all zero. Since the one-sided limits of \(V'\) at \(t_j\) are both normal to \(\dot{\gamma}(t_j)\), their difference \(\Delta V'(t_j)\) is also normal to \(\dot{\gamma}(t_j)\). For any fixed \(j \in \{1, \dots, k-1\}\), we construct a proper, normal, piecewise smooth vector field \(W\) such that \(W(t_j) = \Delta V'(t_j)\) and \(W(t_i) = 0\) for all \(i \neq j\). To do this, let \(E_j(t)\) be the parallel transport of the vector \(\Delta V'(t_j)\) along \(\gamma\). Since parallel transport preserves inner products and \(\dot{\gamma}\) is parallel along \(\gamma\), \(E_j(t)\) is orthogonal to \(\dot{\gamma}(t)\) for all \(t \in [a, b]\). Multiply this by a piecewise smooth scalar bump function \(g: [a, b] \to \mathbb{R}\) that equals \(1\) at \(t_j\) and has a sufficiently small support avoiding all other partition points and endpoints. Then \(W(t) = g(t) E_j(t)\) is a proper, normal, and piecewise smooth vector field.
Substituting this \(W\) into the reduced index form yields
Consequently, \(\Delta V'(t_j) = 0\) for all \(j\), proving that \(V'\) is globally continuous on \([a, b]\).
At this stage, we know that \(V\) is a \(C^1\) vector field on \([a, b]\) that satisfies the linear second-order ordinary differential equation \(V'' + R(\dot{\gamma},V)\dot{\gamma} = 0\) on each subinterval. By the standard uniqueness and regularity theory for linear ODEs, a solution to this equation that is \(C^1\) must in fact be smooth everywhere. Therefore, \(V\) extends to a globally smooth vector field on the entire interval \([a, b]\) satisfying the Jacobi equation. This makes \(V\) a Jacobi field, completing the proof. ◻
Lemma 7 (The Index Lemma). Let \(\gamma:[0,a]\to M\) be a nonconstant geodesic with no conjugate point to \(\gamma(0)\) along \(\gamma\), on the interval \((0,a]\). Let \(J\) be a Jacobi field along \(\gamma\) with \(\langle J, \gamma' \rangle = 0\), and \(V\) be a (piecewise) differentiable vector field along \(\gamma\) with \(\langle V, \gamma' \rangle = 0\). Suppose further that \(J(0)=V(0)=0\) and \(J(t_0)=V(t_0)\) for \(t_0 \in (0,a]\). Then,
with equality if and only if \(V=J\) on \([0,t_0]\).
Proof. Restrict to \([0,t_0]\). Choose an orthonormal basis \(e_i\) of \(\gamma'(0)^\perp\), and let \(J_i\) have initial data \(J_i(0)=0,J_i'(0)=e_i\). They remain normal because \(\langle J_i,\gamma'\rangle\) is affine with zero initial data. At every \(t\in(0,t_0]\) they form a basis of the normal space: a vanishing linear combination there would be a Jacobi field vanishing at zero and \(t\), contrary to nonconjugacy.
Thus \(V=f^iJ_i\) for \(t>0\). The coefficients extend piecewise smoothly to zero. To see this in a parallel frame, write the Jacobi matrix as \(A(t)=tB(t)\), where \(B\) is smooth and \(B(0)=I\), and write \(V(t)=t\widetilde V(t)\) by the one-variable integral formula \(V(t)/t=\int_0^1V'(ut)du\). Then \(f=B^{-1}\widetilde V\) near zero. At every other point \(A\) is invertible; the coefficients are continuous across all partition points.
The Wronskian satisfies
and its initial value is zero. Put \(X=f^iJ_i'\), \(Y=(f^i)'J_i\), so \(V'=X+Y\). Jacobi's equation and the zero Wronskians give
Consequently, on every smooth piece,
Integrate and sum: \(V,X\) are continuous at the partition points, and the term at zero is zero. The Jacobi field with the same endpoints is \(J=f^i(t_0)J_i\), so integration by parts gives \(I(J,J)=\langle J'(t_0),J(t_0)\rangle=\langle X(t_0),V(t_0)\rangle\). Hence
Equality forces every derivative \((f^i)'\) to vanish on every open piece. Continuity makes these constants agree across junctions, and the endpoint value gives \(V=J\). Conversely \(V=J\) gives equality. ◻
A positive lower bound for Ricci curvature can force a long geodesic to admit a shortening variation. The next proposition applies this argument outside a compact set; completeness then converts a bound on distance into compactness.
Proposition 8. Let \(M\) be a complete, connected Riemannian \(n\)-manifold with \(n\geq2\) and let \(k > 0\). Suppose that there is a compact subset \(D \subset M\) such that for all \(p \notin D\), \(\mathop{\mathrm{\mathrm{Ric}}}(U,U) \ge (n-1)k\) for all \(U \in T_pM\) such that \(\|U\| = 1\). Then, \(M\) is compact.
Proof of Proposition 8. Suppose, for the sake of contradiction, that \(M\) is not compact. Because \(M\) is a complete Riemannian manifold, the Hopf-Rinow theorem dictates that \(M\) must be unbounded.
Fix any \(x_0\in M\), and choose \(R>0\) with \(D\subset B_R(x_0)\); this also covers \(D=\varnothing\). Choose \(L>\pi/\sqrt{k}\). Unboundedness provides a point at distance greater than \(R+L\); following a minimizing geodesic to it for time \(R+L\) gives a point \(q\) at exactly that distance.
By the Hopf-Rinow theorem, there exists a globally length-minimizing, unit-speed geodesic \(\gamma: [0, R+L] \to M\) connecting \(\gamma(0) = x_0\) to \(\gamma(R+L) = q\). For any \(t \in [R, R+L]\), since \(\gamma\) is globally length-minimizing, the distance from \(x_0\) to \(\gamma(t)\) is exactly \(t \ge R\). Consequently, \(\gamma(t) \notin B_R(x_0)\), which implies \(\gamma(t) \notin D\). By the given hypothesis, the Ricci curvature evaluated along the velocity vector of this subsegment strictly satisfies
We analyze the second variation of length (the index form) on the subsegment \(\sigma = \gamma|_{[R, R+L]}\). Let \(E_1(t), \dots, E_{n-1}(t)\) be parallel orthonormal vector fields along \(\sigma\) that are everywhere orthogonal to \(\gamma'(t)\). Define the proper variation vector fields \(V_i(t)\) along \(\sigma\) by:
Because \(V_i(R) = V_i(R+L) = 0\), these correspond to valid proper variations holding the endpoints of \(\sigma\) fixed. The sum of the index forms for these variations is exactly
Since the \(E_i\) fields are parallel, their covariant derivatives vanish, giving \(\nabla_{\gamma'} V_i = \frac{\pi}{L} \cos\left(\frac{\pi (t-R)}{L}\right) E_i(t)\). Furthermore, taking the trace of the Riemann curvature tensor over the orthogonal frame evaluates exactly to the Ricci curvature. Hence, it follows that
Applying the lower bound \(\mathop{\mathrm{\mathrm{Ric}}}(\gamma', \gamma') \ge (n-1)k\), we have
Integrating the squared trigonometric functions over the half-period \(L\) yields \(\int_R^{R+L} \cos^2\left(\frac{\pi (t-R)}{L}\right) \dif t = \int_R^{R+L} \sin^2\left(\frac{\pi (t-R)}{L}\right) \dif t = \frac{L}{2}\). Thus, the inequality becomes:
Because we deliberately chose \(L > \frac{\pi}{\sqrt{k}}\), we have \(\frac{\pi^2}{L^2} - k < 0\), which strictly forces the sum of the index forms to be negative. Therefore, \(I(V_j, V_j) < 0\) for at least one \(j \in \{1, \dots, n-1\}\).
Corollary 5 says that a minimizing segment must have nonnegative index on every endpoint-vanishing field. Thus the negative value just obtained directly contradicts minimality of \(\sigma\). Equivalently, realizing the normal field as a proper variation gives \(L'(0)=0,L''(0)<0\), and Taylor's theorem produces strictly shorter nearby curves. Hopf–Rinow then proves compactness. Connectedness and \(n\geq2\) matter: in dimension one the Ricci lower bound is vacuous, and an infinite disjoint union of positively curved compact components need not be compact. ◻
1. Application: Geodesics Minimize Up to Conjugate Points
The second variation formula can be used to establish an important fact: no geodesic is minimizing past its first conjugate point. A standard example is the \(n\)-sphere, where, if a geodesic starts from the North pole and travels in a given direction towards the South pole, it cannot be minimizing after the South pole, i.e. the conjugate point to the North pole along this given geodesic. The proof constructs a field from a Jacobi field that vanishes at the conjugate point, then perturbs it to make the index form negative.
Theorem 9. Let \(M\) be a differentiable manifold, and take \(p, q \in M\). If \(\gamma : [a,b] \to M\) is a geodesic from \(p\) to \(q\), and has an interior conjugate point, i.e. there exists \(d \in (a,b)\) such that \(\gamma(d)\) is conjugate to \(\gamma(a)\) along \(\gamma\), then there exist a proper normal vector field \(X\) along \(\gamma\) such that \(I(X,X) < 0\).
Proof. Rescale the parameter to \([0,1]\). Choose a nonzero Jacobi field \(J\) vanishing at zero and \(d\in(0,1)\); its tangential component is affine and vanishes at both times, so it is normal. Initial-value uniqueness gives \(J'(d)\ne0\). Parallel transport \(-J'(d)\) along \(\gamma\) and multiply it by a smooth bump equal to one at \(d\) and zero near both endpoints. This constructs a smooth normal field \(K\) with \(K(0)=K(1)=0\), \(K(d)=-J'(d)\). For \(\varepsilon>0\), define
Clearly, this is a proper, normal, and (piecewise) smooth vector field along \(\gamma\). A direct calculation of the index form of \(N_\varepsilon (t)\) shows that
Clearly, the linear term dominates for small enough \(\varepsilon > 0\), rendering the last line negative. Take \(X(t) = N_\varepsilon (t)\) to conclude. ◻
Proposition 10. Suppose \(\gamma:[a,b]\to M\), \(a<b\), is a nonconstant geodesic and no \(\gamma(t)\), \(a<t\leq b\), is conjugate to \(\gamma(a)\) along it. Then for some \(\varepsilon>0\), every piecewise smooth curve \(c\) with the same endpoints and \(d(c(t),\gamma(t))<\varepsilon\) for all \(t\) satisfies \(L(c)\geq L(\gamma)\). Also \(I(V,V)>0\) for every nonzero proper normal continuous piecewise smooth field \(V\). Equivalently \(I(V,V)=0\) precisely for the proper normal Jacobi fields, which under this hypothesis consist only of zero.
Proof. The index assertion follows at once from Lemma 7 with the comparison Jacobi field \(J=0\), after translating the parameter. Positivity of a second variation alone does not establish the asserted neighborhood minimum; we prove that separately.
Put \(p=\gamma(a)\), \(v_0(t)=(t-a)\gamma'(a)\), and \(F(v)=\exp_p(v)\). By the Jacobi characterization and \(dF_0=\mathrm{id}\), \(dF\) is nonsingular on the compact segment \(v_0([a,b])\). Thus it remains nonsingular on an open neighborhood \(U\) of that segment. The parameter-preserving map
is a local diffeomorphism near the graph \((t,v_0(t))\), whose image is the graph \((t,\gamma(t))\). It is injective on a sufficiently small neighborhood of this compact graph. Indeed if arbitrarily close points in its domain had the same image, their common first coordinates would converge, and their vector coordinates would both converge to the corresponding \(v_0(t)\), contradicting local injectivity there. Its image therefore contains a uniform neighborhood \(d(y,\gamma(t))<\varepsilon\), by compactness.
Every curve in the statement has a unique continuous piecewise smooth lift \(v(t)\) through this chosen neighborhood, with \(F(v(t))=c(t)\), \(v(a)=0\), and \(v(b)=(b-a)\gamma'(a)\). This construction does not presume that \(F\) is globally injective or that \(\gamma\) has no self-intersections. Equip \(U\) with \(h=F^*g\), a positive metric because \(dF\) is nonsingular. The Gauss lemma says the radial function \(r(v)=|v|\) has \(|dr|_h=1\) away from zero. Thus along the lift, \(|(r\circ v)'|\leq|v'|_h\) almost everywhere; at zero the same inequality follows by the norm's local Lipschitz property and \(h_0=g_p\). Integrating gives
This proves local minimization in the stated uniform topology. It need not be a global minimum: conjugate points and the cut locus are different obstructions. ◻
The following identities justify the integrations by parts in the proof of Proposition 10.
Lemma 11. Let \(M\) be a differentiable manifold, \(p, q \in M\), and \(\gamma : [a,b] \to M\) be a geodesic from \(p\) to \(q\). Let \(V\) be a normal (piecewise) smooth vector field along \(\gamma\), \(\{a_0,\cdots,a_k\}\) be an admissible partition for \(V\), and
Then, the following identity holds where \(V\) is smooth:
Here the representation \(V=\sum_\alpha v^\alpha J_\alpha\) is part of the hypothesis, and the \(J_\alpha\) are Jacobi fields with zero pairwise Wronskians. In the preceding application all vanish at the initial time, which guarantees that condition by Lemma 12. Without it the displayed identity need not hold for an arbitrary family of Jacobi fields.
Proof. Note that on each interval \((a_i,a_{i+1})\) for \(0 \leq i \leq k-1\), \(\nabla_{\dif/\!\dif t} V = X + Y\) for all \(t \in (a_i,a_{i+1})\). It follows from definition that
Furthermore,
where the last equality follows from inspection and definition. It follows that
Apply Lemma 12 to see that \(\langle \nabla_{\dif/\!\dif t} J_\alpha, J_\beta \rangle = \langle J_\alpha, \nabla_{\dif/\!\dif t} J_\beta \rangle\) along all of \(\gamma\), since their difference vanishes at \(t=0\) (because \(J_\alpha(0)=0\)). As such,
To conclude, combining everything, we have
and rearranging everything gives the desired conclusion. ◻
Lemma 12. Let \(\gamma:[a,b]\to M\) be a geodesic, and \(J_1\), \(J_2\) are Jacobi fields along \(\gamma\). Then, \(\langle \nabla_{\dif/\!\dif t} J_1, J_2 \rangle - \langle J_1, \nabla_{\dif/\!\dif t} J_2 \rangle\) is a constant function along \(\gamma\).
Proof. This follows from direct calculations: let \(\varphi(t) = \langle \nabla_{\dif/\!\dif t} J_1 (t), J_2 (t) \rangle - \langle J_1 (t), \nabla_{\dif/\!\dif t} J_2 (t) \rangle\). It follows that
where the second equality follows from applying the Jacobi equation, and the third equality follows from symmetries of the Riemann curvature tensor. ◻
Reading: Carmo (1992), Riemannian geometry.