Jacobi Fields and Conjugate Points

Let \((M,g)\) be a smooth Riemannian manifold, with Levi–Civita connection \(\nabla\). See the connection notes for the notation.

How does curvature affect neighbouring geodesics? The relation between geometry and topology for surfaces in \(\mathbb R^3\) suggests asking for analogous consequences in higher dimensions. Jacobi fields measure the first-order separation of a family of geodesics, turning this question into a linear differential equation along one geodesic. Roughly speaking, positive curvature causes nearby geodesics to converge, while negative curvature causes them to spread out. To understand the topological consequences from this fact, we need a quantitative way to measure the effect of curvature on a family of geodesics.

We begin by differentiating the geodesic equation with respect to the variation parameter. Interchanging covariant derivatives introduces curvature, giving the Jacobi equation.

Proposition 1. Let \(\Gamma(s,t)\) be a smooth variation for which every \(t\)-curve is an affinely parametrized geodesic, and set \(\gamma(t)=\Gamma(0,t)\), \(J(t)=\partial_s\Gamma(0,t)\). Then

\[D_t^2J+R(\gamma',J)\gamma'=0.\tag{1}\]

This is the Jacobi equation in the convention \(R(X,Y)Z=\nabla_Y\nabla_XZ-\nabla_X\nabla_YZ+\nabla_{[X,Y]}Z\).

Proof. Write \(S=\partial_s\Gamma\) and \(T=\partial_t\Gamma\), vector fields along the map, and let \(D_s,D_t\) denote the pullback covariant derivatives. In coordinates,

\[D_tT=\left(\partial_t^2x^k+\Gamma^k_{ij}\partial_tx^i\partial_tx^j\right)\partial_k=0\]

by the geodesic equation. Torsion-freeness gives \(D_sT=D_tS\), and the curvature commutator in our convention is \(D_tD_sV-D_sD_tV=R(S,T)V\). Therefore

\[0=D_sD_tT=D_tD_sT-R(S,T)T=D_t^2S+R(T,S)T.\]

Restriction to \(s=0\) proves (1). The pullback formulation is valid even when the variation map is not an immersion, so no independence of \(S,T\) is assumed. ◻

Remark 2. Primes on fields denote covariant differentiation along the geodesic. If another text uses \(R_{\mathrm{other}}=-R\), its formula is \(J''+R_{\mathrm{other}}(J,\gamma')\gamma'=0\). These are the same equation. With our convention a unit sphere has \(R(T,J)T=J\) for \(J\perp T\), giving \(J''+J=0\).

By Proposition 1, the Jacobi equation is linear and second order. Prescribing a field and its covariant derivative at one point therefore determines a solution along the given geodesic.

Proposition 3 (Existence and Uniqueness of Jacobi Fields). Let \((M,g)\) be a differentiable manifold. Let \(I = [a,b] \subset \mathbb R\), \(\gamma : I \to M\) be a geodesic, \(a \in I\), and \(p = \gamma(a)\). For every pair of vectors \(v,w \in T_p M\), there exists a unique Jacobi field \(J\) along \(\gamma\) such that \(J(a) = v\) and \(\nabla_{\partial/\partial t} J(a) = w\).

Proof. Parallel transport an orthonormal basis along \(\gamma\) to obtain \(E_1,\ldots,E_d\) on the entire interval. Write \(J=j^iE_i\) and put

\[A^i{}_j(t)=\langle R(\gamma'(t),E_j(t))\gamma'(t),E_i(t)\rangle.\]

The coefficients are smooth and \(A\) is symmetric by the curvature symmetries. Because the frame is parallel, (1) becomes \(j''+A(t)j=0\). With \(z=(j,j')\), this is the first-order linear system

\[z'=\begin{pmatrix}0&I\\-A(t)&0\end{pmatrix}z,\qquad z(a)=(v^i,w^i).\]

Linear ODE existence and uniqueness give a local solution. On every compact subinterval the coefficient matrix is bounded, and Grönwall's inequality bounds \(|z(t)|\) by its initial norm times an exponential. Thus no finite-time blowup occurs and continuation covers \([a,b]\). This proves existence and uniqueness for both specified initial vectors; the equations are coupled, not generally decoupled. ◻

Remark 4. The Jacobi equation is a linear second-order ODE along the geodesic. Prescribing \(J(a)\) and \(\nabla_{\dot\gamma}J(a)\) determines a unique field throughout the geodesic interval. Conjugate points do not obstruct this initial-value problem. They obstruct the different problem of prescribing the two endpoint values \(J(a)\) and \(J(b)\).

Lemma 5. Let \((M,g)\) be a differentiable manifold of dimension \(d\). Let \(\gamma\) be a geodesic in \(M\). Let \(\mathscr J (\gamma)\) denote the collection of all Jacobi fields along \(\gamma\), and \(\mathscr X (\gamma)\) denote the collection of smooth vector fields along \(\gamma\). Then, \(\mathscr J(\gamma)\) is a \(2d\)-dimensional linear subspace of \(\mathscr X(\gamma)\).

Proof. Linearity of the subspace follows from the linearity of the Jacobi equation. The map \(\Phi : \mathscr J(\gamma) \to T_p M \oplus T_p M : J \mapsto (J(a), \nabla_{\partial/\partial t} J(a))\), for \(p = \gamma(a)\), is linear and bijective by Proposition 3. This concludes the argument. ◻

We can characterize Jacobi fields along a geodesic \(\gamma\) as the variation fields of a variation along geodesics.

Proposition 6. Every Jacobi field \(J\) along a geodesic \(\gamma:[a,b]\to M\), \(a<b\), is realized by some smooth variation through affinely parametrized geodesics:

\[\Gamma(0,t)=\gamma(t),\qquad J(t)=\partial_s\Gamma(0,t).\tag{2}\]

Conversely, every such variation field is Jacobi. Different Jacobi fields generally require different variations.

Proof. The converse was proved in Proposition 1. To realize prescribed \(J\), choose a smooth curve \(z(s)\) with \(z(0)=\gamma(a)\), \(z'(0)=J(a)\). Parallel transport \(\gamma'(a)\) and \(J'(a)\) along \(z\), obtaining \(V(s),W(s)\). Set

\[\Gamma(s,t)=\exp_{z(s)}\bigl((t-a)(V(s)+sW(s))\bigr).\tag{3}\]

There is one common small interval in \(s\) on which this is defined for every \(t\in[a,b]\): the original geodesic extends a little beyond its compact interval, and the open domain and smooth dependence of the geodesic flow contain a product neighborhood of its compact trajectory. Thus finitely many local flow neighborhoods supply a uniform permitted change in initial data. Completeness of \(M\) is unnecessary.

Clearly \(\Gamma(0,t)=\gamma(t)\). Its field \(F=\partial_s\Gamma|_{s=0}\) is Jacobi by Proposition 1, and

\[F(a)=z'(0)=J(a),\qquad F'(a)=D_s\partial_t\Gamma(0,a)=D_s(V+sW)|_{s=0}=W(0)=J'(a).\]

Here torsion-freeness exchanges \(D_t\partial_s\Gamma\) and \(D_s\partial_t\Gamma\); parallelness kills the derivatives of \(V,W\). Initial-value uniqueness gives \(F=J\) on the whole interval. ◻

Proposition 7. If \(\varphi:(M,g)\to(\widetilde M,\widetilde g)\) is a local isometry, \(\widetilde\gamma=\varphi\circ\gamma\) on the same interval \(I\), and \(\widetilde J(t)=d\varphi_{\gamma(t)}J(t)\), then \(J\) is Jacobi if and only if \(\widetilde J\) is Jacobi.

Proof. On a neighborhood where \(\varphi\) is an isometry onto its image, pulling back the target Levi–Civita connection preserves metric compatibility and torsion-freeness. Uniqueness of that connection gives

\[d\varphi(\nabla_XY)=\widetilde\nabla_{d\varphi X}(d\varphi Y),\qquad d\varphi(R(X,Y)Z)=\widetilde R(d\varphi X,d\varphi Y)d\varphi Z.\tag{4}\]

These identities are used locally or along the curve; no global pushforward field on the target is presumed. Applying the first twice and the second once gives

\[d\varphi\bigl(J''+R(\gamma',J)\gamma'\bigr)=\widetilde J''+\widetilde R(\widetilde\gamma',\widetilde J)\widetilde\gamma'.\]

Because each \(d\varphi\) is invertible, either side vanishes if and only if the other does. Geometrically the forward implication also follows by composing a realizing geodesic variation with \(\varphi\); the covariant identity supplies both directions without any global inverse. ◻

Uniqueness identifies the Jacobi field obtained by varying the initial velocity of a geodesic. This relates Jacobi fields to the differential of the exponential map.

Corollary 8. Let \(M\) be a differentiable manifold, and \(\gamma : [0,a] \to M\) be a geodesic. Then, a Jacobi field along \(\gamma\) with \(J(0)=0\) is given by \(J(t) = (\dif\, \exp_p)_{t \gamma'(0)} (tJ'(0))\).

Here \(p=\gamma(0)\), and tangent spaces of the vector space \(T_pM\) are canonically identified with \(T_pM\). To prove the formula, vary the initial velocity: \(\Gamma(s,t)=\exp_p(t(\gamma'(0)+sJ'(0)))\). Differentiating in \(s\) yields the displayed field, whose initial data are \(0,J'(0)\). Initial-value uniqueness identifies it with \(J\).

The next identity describes the tangential component of a Jacobi field. It is useful for separating changes of parametrization from transverse separation of geodesics.

Lemma 9. Let \(J\) be a Jacobi field along the geodesic \(\gamma:[0,a] \to M\). Then,

\[ \langle J(t), \gamma'(t) \rangle = \langle J'(0), \gamma'(0) \rangle t + \langle J(0), \gamma'(0) \rangle ~,~ \quad t \in [0,a] ~.\tag{5}\]

Proof. Since the geodesic velocity is parallel,

\[\frac{d^2}{dt^2}\langle J,\gamma'\rangle=\langle J'',\gamma'\rangle=-\langle R(\gamma',J)\gamma',\gamma'\rangle=0.\]

The last equality is skew-adjointness in the two inner-product slots. Thus this scalar function is affine, and its initial value and derivative give (5). ◻

Theorem 10. Let \(\gamma(t)=\exp_p(tv)\), and let \(J(t)=(d\exp_p)_{tv}(tw)\), with \(|w|=1\). Then

\[|J(t)|^2=t^2-\frac13\langle R(v,w)v,w\rangle t^4+O(t^5).\tag{6}\]

In normal coordinates built from an orthonormal basis at \(p\),

\[g_{ij}(x)=\delta_{ij}-\frac13R_{ik\ell j}(p)x^kx^\ell+O(|x|^3),\qquad R_{abcd}=\langle R(\partial_c,\partial_d)\partial_b,\partial_a\rangle.\tag{7}\]

Stating the index convention is essential: the corresponding formula in a convention with opposite curvature sign has correspondingly different indices or sign.

Proof. Put \(q(t)=|J(t)|^2\). The initial data are \(J(0)=0,J'(0)=w\). The Jacobi equation gives \(J''(0)=0\); differentiating it gives

\[J'''(0)=-R(v,w)v,\]

since all terms involving \(J(0)\) or \(D_t\gamma'\) vanish. Metric compatibility and repeated product differentiation now yield

\[q(0)=q'(0)=0,\quad q''(0)=2,\quad q'''(0)=0,\quad q^{(4)}(0)=2\bigl(\langle J^{(4)},J\rangle+4\langle J''',J'\rangle+3|J''|^2\bigr)_{t=0}=-8\langle R(v,w)v,w\rangle.\]

Taylor's theorem, with \(4!=24\), proves (6). For arbitrary \(w\), its first term is \(t^2|w|^2\).

To obtain (7) without assuming the expansion we seek, use the same field in normal coordinates. There \(\Gamma(s,t)\) has coordinate vector \(t(v+sw)\), so \(J(t)=tw^i\partial_i|_{tv}\) and

\[t^2g_{ij}(tv)w^iw^j=t^2|w|^2-\frac13\langle R(v,w)v,w\rangle t^4+O(t^5).\]

Dividing by \(t^2\), comparing the ordinary Taylor coefficients of the smooth metric at zero, and polarizing in \(w\) determines the constant, linear and quadratic coefficients. Since

\[\langle R(v,w)v,w\rangle=R_{ik\ell j}w^iv^kv^\ell w^j,\]

the quadratic term is the one in (7). Smooth multivariable Taylor expansion makes the remaining term uniformly \(O(|x|^3)\) for directions in the compact unit sphere. This explains why a one-ray calculation determines the full metric expansion. ◻

1. Conjugate Points

The exponential map gives normal coordinates near \(p\) because its differential at the origin is the identity. Indeed, for \(w\in T_pM\), the curve \(t\mapsto\exp_p(tw)\) has initial velocity \(w\), so \((\dif\exp_p)_0w=w\). Thus \(0\) is a regular point, and the inverse function theorem gives a neighbourhood of \(0\) on which \(\exp_p\) is a diffeomorphism. Farther out in its domain the differential may become singular. Jacobi fields characterize those singularities.

Definition 11. Let \(M\) be a differentiable manifold, \(\gamma: I \to M\) be a geodesic, \(a,b \in I,\ a<b\), and \(p = \gamma(a)\), \(q = \gamma(b)\). \(p\) and \(q\) are said to be conjugate along \(\gamma\) if there exists a Jacobi field vanishing at \(t=a\) and \(t=b\), but is not identically zero. The order or multiplicity of conjugacy is the dimension of the space of Jacobi fields vanishing at \(p\) and \(q\).

Remark 12. Fields with \(J(a)=0\) form a \(d\)-dimensional space, parametrized by \(J'(a)\). For a nonconstant geodesic, Lemma 9 says \(\langle J,\gamma'\rangle\) is affine; if it vanishes at two distinct times, it is identically zero. Thus a field vanishing at both endpoints satisfies \(J'(a)\perp\gamma'(a)\), leaving at most \(d-1\) independent initial derivatives. The tangential component might be identically zero; it is only a nonzero affine component that has at most one zero.

The bound is sharp on the unit \(d\)-sphere: along a unit-speed great circle, each parallel normal field \(E_i\) gives \(J_i(t)=\sin(t-a)E_i(t)\), which vanishes at \(a,a+\pi\), yielding \(d-1\) independent fields. A constant geodesic has \(J''=0\), so a field vanishing at two distinct times is zero and there are no conjugate times.

The definition of conjugate points allow us to characterize them as critical points of \(\exp_p\).

Proposition 13. Let \(M\) be a differentiable manifold, \(p \in M\), \(v \in U \subset T_p M\). Let \(\gamma = \gamma_v : [0,a] \to M\) be the geodesic segment \(\gamma(t) = \exp_p (\frac{t}{a}v)\). Let \(q = \gamma(a) = \exp_p (v)\). Then, \(v\) is a critical point of \(\exp_p\) if and only if \(q\) is conjugate to \(p\) along \(\gamma\). Furthermore, the order of conjugacy of \(q\) to \(p\) is given by the dimension of the kernel of the linear map \((\dif \exp_p)_v\).

The interval parameter here satisfies \(a>0\), and \(v\) belongs to the domain of \(\exp_p\). Before rescaling to \(a=1\), the kernel correspondence is explicitly

\[w\longmapsto J_w(t)=(d\exp_p)_{(t/a)v}\bigl((t/a)w\bigr),\qquad J_w'(0)=w/a,\quad J_w(a)=(d\exp_p)_v w.\]

Thus \(w\ne0\) implies a nonzero field by its initial derivative; this supplies the nontriviality in both directions of the proof. The identification \(T_v(T_pM)\cong T_pM\) is canonical for a vector space and requires no chart choice.

Proof. Without loss, let \(a=1\). Suppose that \(v\) is a critical point of \(\exp_p\). Then, there exists a nonzero vector \(w \in T_v(T_p M)\) such that \((\dif \exp_p)_v (w) = 0\). Identifying \(T_v(T_p M)\) with \(T_p M\) via charts as per usual, \(w \in T_p M\). Define the variation of geodesic \(\gamma\) by \(\Gamma(s,t) = \exp_p (t(v + sw))\). Let \(J(t) = \partial_s \Gamma(s,t)|_{s=0}\) denote the variation field. Direct calculations reveal that \(J(1) = 0\), i.e. \(J\) is a nontrivial Jacobi field vanishing at \(t=0\) and \(t=1\), establishing the conclusion that \(q\) is conjugate to \(p\) along \(\gamma\).

For the converse, if \(q\) is conjugate to \(p\) along \(\gamma\), then there is some nontrivial Jacobi field \(J\) along \(\gamma\) such that \(J(0) = 0\) and \(J(1) = 0\). Proposition 6 tells us that \(J\) is the variation field of a variation of geodesic \(\gamma\), of the form (2) with \(w = \nabla_{\partial/\partial t} J(0) \in T_p M\) and \((\dif \exp_p)_v (w) = J(1) = 0\). Therefore, \(v\) is a critical point of \(\exp_p\).

For the second assertion, again, without loss, let \(a=1\). Note that the order of conjugacy of \(q\) is equal to the number of linearly independent Jacobi fields \(J_1,\cdots,J_k\) such that they are zero at \(t=0\) and \(t=1\). A straightforward calculation shows that \(J_1,\cdots,J_k\) are linearly independent if and only if \(J'_1 (0), \cdots, J'_k (0)\) are linearly independent in \(T_p M\). (To rigorously see this, note that any linear combination \(J = c_1 J_1 + \cdots + c_k J_k\) is a Jacobi field with \(J(0) = 0\). By the uniqueness of solutions to the Jacobi equation, \(J \equiv 0\) if and only if \(J'(0) = c_1 J'_1(0) + \cdots + c_k J'_k(0) = 0\). This yields the equivalence of linear independence.) By the preceding construction, it follows that the order of conjugacy of \(q\) is exactly equal to the dimension of the kernel of \((\dif \exp_p)_v\). ◻

Initial values always determine a unique Jacobi field. To prescribe endpoint values instead, consider the linear endpoint-evaluation map on the space of Jacobi fields. Its kernel consists exactly of fields vanishing at both endpoints; this is where conjugate points enter.

Proposition 14 (Conjugate Points as Obstructions in Constructing Jacobi Fields). Let \((M,g)\) be a differentiable manifold of dimension \(n\), and \(\gamma : [a,b] \to M\) be a geodesic (segment). The two-point boundary problem for Jacobi fields along \(\gamma\) is uniquely solvable for every pair of vectors \(v \in T_{\gamma(a)} M\) and \(w \in T_{\gamma(b)} M\) if and only if \(\gamma(a)\) and \(\gamma(b)\) are not conjugate along \(\gamma\).

Here \(a<b\). In the conjugate case the same equal-dimension linear algebra also shows that not every endpoint pair is attainable: a nonzero kernel forces a proper image. For attainable data, adding a kernel field gives all the nonunique solutions.

Proof. Let \(\mathcal J\) be the \(2n\)-dimensional space of Jacobi fields along \(\gamma\), identified with initial data \((J(a),\nabla_{\dot\gamma}J(a))\). The endpoint map

\[\mathscr T:\mathcal J\longrightarrow T_{\gamma(a)}M\times T_{\gamma(b)}M,\qquad J\longmapsto(J(a),J(b))\]

is linear between spaces of equal finite dimension. Its kernel is zero exactly when the endpoints are not conjugate. In that case it is an isomorphism, giving existence and uniqueness for every endpoint pair. If the endpoints are conjugate, a nonzero element of the kernel and the zero field have the same zero endpoint values, so uniqueness fails. ◻

Reading: Carmo (1992), Riemannian geometry.