03.02.19 · differential-geometry / manifolds

Jacobi fields, conjugate points, and the Morse Index Theorem

shipped3 tiersLean: none

Anchor (Master): Milnor Morse Theory Part III §§13--15; Morse 1934 The Calculus of Variations in the Large; Bott 1956 Ann. Math. 60; Cheeger-Ebin Comparison Theorems in Riemannian Geometry Ch. 1

Intuition Beginner

Drop two beads onto a globe at the equator and send them straight north along nearby lines of longitude. At first they march apart almost as if the surface were flat. But the lines of longitude bend toward each other, and the two beads meet again at the north pole. The pole is where the spreading reverses: paths that started off heading apart are brought back together by the curvature of the sphere.

A point where neighbouring straightest-possible paths refocus is called a conjugate point. On a flat plane, straight lines that start apart only ever get farther apart, so there are no conjugate points at all. On a sphere, every shortest path runs into one. The amount of refocusing is a fingerprint of how much, and which way, the space is curved.

There is a second story hiding here. Past the pole, a longitude line is no longer the shortest route between its endpoints; you could have done better by going the other way around. Conjugate points are exactly where a straightest path stops being a shortest path. Counting them measures how badly a long geodesic fails to minimise distance.

Visual Beginner

Alt text: On the sphere, two geodesics start at the same equatorial point, spread apart as they climb toward the pole, then curve back and reconverge exactly at the north pole, which is labelled a conjugate point. On the flat plane underneath, two geodesics that start apart stay parallel and never meet, so the plane has no conjugate points. The contrast shows that conjugate points are produced by positive curvature, which acts like a focusing lens on families of geodesics.

Worked example Beginner

Take the unit sphere and the geodesic that runs along a meridian from the equator up over the north pole and down to the equator on the far side. Mark three checkpoints: the starting point on the equator, the north pole, and the ending point on the equator.

Watch a family of nearby meridians. Leaving the start, they fan out. By the time they reach the north pole, a quarter of the way plus a quarter of the way — a half-circle in all — they have all crowded back to a single point. So the north pole is conjugate to the start. The first checkpoint and the pole are a conjugate pair, and the spacing between them is a quarter of the full great circle, an arc of length $π /2$ times the radius, which on the unit sphere is just $π$ .

What this tells us: on the unit sphere the conjugate distance is exactly $π$ , the half-circumference. A meridian shorter than $π$ has no conjugate point in its interior and is the genuine shortest path. A meridian longer than $π$ has passed a conjugate point, and you can find a shorter route. Counting the interior conjugate points — here, one — measures exactly how far past "shortest" the path has gone.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $(M, g)$ is a Riemannian manifold with Levi-Civita connection $\nabla$ and Riemann curvature tensor $R$ , where $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ . Sign convention: with this definition the sectional curvature of the unit sphere is $+ 1$ and the Jacobi equation reads $V^{''} + R (V, \overset{γ}{˙}) \overset{γ}{˙} = 0$ ; this matches the convention of unit 03.02.05 and of Milnor ^{[Milnor Part III §13]}. Let $γ : [0, 1] \to M$ be a geodesic, so $\nabla_{\overset{γ}{˙}} \overset{γ}{˙} = 0$ , and write $V^{'} := \nabla_{\overset{γ}{˙}} V$ for covariant differentiation of a vector field $V$ along $γ$ .

The objects of the theory live on the path space $Ω = Ω (M; p, q)$ of piecewise-smooth curves $ω : [0, 1] \to M$ with $ω (0) = p$ and $ω (1) = q$ . The energy of such a path is $$ E(\omega) = \tfrac12 \int_0^1 |\dot\omega(t)|^2 , dt . $$ A tangent vector to $Ω$ at $ω$ is a piecewise-smooth vector field $W$ along $ω$ with $W (0) = W (1) = 0$ ; these arise as velocity fields $\partial_{s} ∣_{s = 0} α (s, \cdot)$ of variations $α$ of $ω$ with fixed endpoints. The first variation of energy is $$ \tfrac{d}{ds}\Big|{s=0} E(\alpha(s,\cdot)) = -\int_0^1 \langle W, \nabla{\dot\omega}\dot\omega \rangle , dt - \sum_i \langle W(t_i),, \Delta_{t_i}\dot\omega \rangle, $$ where $Δ_{t_{i}} \overset{ω}{˙}$ is the jump in $\overset{ω}{˙}$ at a corner $t_{i}$ . The right-hand side vanishes for every admissible $W$ exactly when $ω$ is a (smooth, corner-free) geodesic. Thus geodesics are precisely the critical points of $E$ on $Ω$ , the variational fact that makes Morse theory applicable.

Definition (index form). At a geodesic $γ$ , the second variation or Hessian of $E$ , written $E_{**}$ , is the symmetric bilinear form on the tangent space $T_{γ} Ω$ given for vector fields $V, W$ along $γ$ vanishing at the endpoints by $$ I(V, W) ;=; E_{\ast\ast}(V, W) ;=; \int_0^1 \Big( \langle V', W' \rangle - \langle R(V, \dot\gamma)\dot\gamma,, W \rangle \Big), dt . $$ This is the index form of $γ$ . Symmetry $I (V, W) = I (W, V)$ follows from the pair symmetry $⟨ R (V, \overset{γ}{˙}) \overset{γ}{˙}, W ⟩ = ⟨ R (W, \overset{γ}{˙}) \overset{γ}{˙}, V ⟩$ of the curvature tensor 03.02.05. Integrating the first term by parts on each smooth subinterval, $$ I(V, W) = -\int_0^1 \big\langle V'' + R(V,\dot\gamma)\dot\gamma,; W\big\rangle, dt - \sum_i \big\langle \Delta_{t_i} V',; W(t_i)\big\rangle, $$ where $Δ_{t_{i}} V^{'}$ is the jump in $V^{'}$ at a corner. The differential operator appearing here is the Jacobi operator $J (V) = V^{''} + R (V, \overset{γ}{˙}) \overset{γ}{˙}$ .

Definition (Jacobi field). A smooth vector field $J$ along $γ$ is a Jacobi field if it satisfies the Jacobi equation $$ J'' + R(J, \dot\gamma)\dot\gamma = 0 . $$ Equivalently, $J = \partial_{s} ∣_{s = 0} Γ (s, \cdot)$ is the variation field of a one-parameter family $Γ (s, \cdot)$ of geodesics. Jacobi fields are the infinitesimal geodesics through $γ$ : the Jacobi equation is the linearisation of the geodesic equation. Because it is a second-order linear ODE for a vector field along $γ$ , the space of Jacobi fields has dimension $2 n$ where $n = dim M$ , a Jacobi field being determined by the initial data $J (0), J^{'} (0) \in T_{γ (0)} M$ .

Definition (conjugate point). The point $γ (c)$ is conjugate to $γ (0)$ along $γ$ if there is a nonzero Jacobi field $J$ with $J (0) = J (c) = 0$ . The multiplicity of the conjugate point is the dimension of the vector space of such Jacobi fields; it is at most $n - 1$ , since the tangential field $t \overset{γ}{˙} (t)$ is the only Jacobi field proportional to $\overset{γ}{˙}$ and it does not vanish twice.

The tangential and normal parts of a Jacobi field evolve independently. The tangential Jacobi fields are exactly $a \overset{γ}{˙} + b t \overset{γ}{˙}$ (two dimensions, never vanishing at two points unless zero), so conjugate points are entirely a phenomenon of the normal Jacobi fields $J ⊥ \overset{γ}{˙}$ . This is why the multiplicity bound is $n - 1$ rather than $n$ .

Counterexamples to common slips

Conjugate point versus self-intersection. A geodesic can cross itself without either crossing point being conjugate (a figure-eight geodesic on a flat torus), and a conjugate point need not be a self-intersection (the antipode on the sphere is conjugate even though the meridian reaches it without crossing). Conjugacy is about an infinitesimal family refocusing, not about the single curve meeting itself.
"Index = number of conjugate points" without multiplicity. On a round sphere $S^{n}$ the first conjugate point along a great circle has multiplicity $n - 1$ , not $1$ . Dropping the multiplicity undercounts the index by a factor that grows with dimension.
Cut point versus conjugate point. The first conjugate point is where the geodesic stops being a local minimum among nearby paths; the cut point is where it stops being the global minimiser. The cut point comes no later than the first conjugate point but can come strictly earlier (e.g. on a flat cylinder, where there are no conjugate points at all but cut points exist).

Key theorem with proof Intermediate+

Theorem (Morse Index Theorem). Let $γ : [0, 1] \to M$ be a geodesic from $p = γ (0)$ to $q = γ (1)$ with no conjugate point at the endpoint $q$ . Then the index of the index form $E_{**}$ — the dimension of a maximal subspace of $T_{γ} Ω$ on which $E_{**}$ is negative definite — is finite and equals the number of points $γ (c)$ , $0 < c < 1$ , conjugate to $p$ along $γ$ , each counted with its multiplicity. The nullity of $E_{**}$ equals the multiplicity of $q$ as a conjugate point (zero, under the stated hypothesis).

Proof. Write $λ (c)$ for the index of the index form $I_{c}$ of the restricted geodesic $γ ∣_{[0, c]}$ , taken on vector fields vanishing at $0$ and $c$ . The claim is $λ (1) = \sum_{0 < c < 1} μ (c)$ , where $μ (c)$ is the multiplicity of $γ (c)$ as a conjugate point.

Step 1: reduction to a finite-dimensional form. Choose $0 = t_{0} < t_{1} < \dots < t_{k} = 1$ so fine that each $γ ∣_{[t_{i - 1}, t_{i}]}$ has no interior conjugate point and is hence the unique minimal geodesic between its endpoints — possible because conjugate points are isolated (Step 4). Let $Ω (t_{0}, \dots, t_{k}) \subset Ω$ be the subspace of broken geodesics: paths that are geodesic on each $[t_{i - 1}, t_{i}]$ and pass through fixed breakpoints. This is a finite-dimensional manifold, coordinatised by the breakpoint positions $γ (t_{1}), \dots, γ (t_{k - 1})$ , of dimension $(k - 1) n$ . Restricting $E$ to it gives a smooth function whose critical points are the smooth geodesics. Milnor's finite-dimensional approximation ^{[Milnor Part III §16]} shows that on the tangent space to this submanifold, $E_{**}$ has the same index and the same nullity as it does on the full $T_{γ} Ω$ : the broken-geodesic directions carry all the negativity, and the complementary directions (fields that are Jacobi on each subinterval with matching values) contribute a positive-definite form. So it suffices to compute the index of the Hessian of $E$ on the finite-dimensional manifold $Ω (t_{0}, \dots, t_{k})$ .

Step 2: the index is monotone and right-continuous in $c$ . For $0 < c \leq 1$ the index $λ (c)$ of $I_{c}$ is a monotone non-decreasing function of $c$ : enlarging the domain enlarges the space of admissible negative directions (extend a field by zero). It is also **continuous from the left** and jumps only at conjugate values. Concretely, if $V$ makes $I_{c}$ negative on $[0, c]$ , the same $V$ extended by zero makes $I_{c^{'}}$ negative for $c^{'} > c$ , so $λ$ cannot drop; and a compactness argument on the unit sphere of a negative subspace shows $λ (c - ε) = λ (c)$ for small $ε$ . Thus $λ$ is a step function, constant except for upward jumps.

Step 3: the jump at a conjugate value equals the multiplicity. Fix $c$ . By the symmetric-bilinear-form structure, the nullity of $I_{c}$ is precisely the number of independent Jacobi fields vanishing at both $0$ and $c$ , i.e. the multiplicity $μ (c)$ — a field $J$ lies in the null space of $I_{c}$ iff $J (J) = 0$ with $J (0) = J (c) = 0$ , by the integrated-by-parts formula. The crux is that as $c$ increases through a conjugate value, the index jumps by exactly this nullity:

λ (c + ε) = λ (c) + μ (c) for all small ε > 0,

while $λ (c + ε) = λ (c)$ at non-conjugate $c$ . This is the heart of the theorem. To see it, let $J_{1}, \dots, J_{μ (c)}$ span the Jacobi fields vanishing at $0$ and $c$ . Each $J_{a}$ , extended by zero past $c$ , is a continuous broken field that is not $C^{1}$ at $c$ (its left derivative $J_{a}^{'} (c^{-}) \neq = 0$ , else $J_{a} \equiv 0$ by ODE uniqueness). The second variation in the direction of a perturbation $J_{a} + δ W_{a}$ , with $W_{a}$ chosen so that $⟨ Δ_{c} (J_{a} + δ W_{a})^{'}, \cdot ⟩$ is corrected, becomes strictly negative just past $c$ : the jump term $- ⟨ Δ_{c} J_{a}^{'}, J_{a} (c)⟩$ in the integrated index form is what supplies a new negative direction for each $J_{a}$ . A careful accounting of these $μ (c)$ new directions, and the verification that no others appear, yields the jump formula. (This is a Sturm-type oscillation argument; Milnor presents it via the broken-geodesic Hessian, where the jump is read off the change of inertia of an explicit symmetric matrix ^{[fasttrack-texts the index theorem (Part III analogue)]}.)

Step 4: conjugate points are isolated and finite in number. The Jacobi fields vanishing at $0$ form an $n$ -dimensional space $J_{0}$ ; evaluation $J \mapsto J (c)$ is a linear map $J_{0} \to T_{γ (c)} M$ whose kernel has dimension $μ (c)$ . The conjugate values are the zeros of the analytic function $c \mapsto det (matrix of J \mapsto J (c))$ in a parallel frame, hence isolated; on the compact interval $[0, 1]$ there are finitely many. Combined with Steps 2–3, $λ$ starts at $λ (0^{+}) = 0$ (a short geodesic has positive-definite index form, since $\int ∣ V^{'} ∣^{2}$ dominates the curvature term for small length by the Poincaré inequality) and accumulates, jump by jump, the sum of the multiplicities of the interior conjugate values: $$ \lambda(1) = \sum_{0 < c < 1} \mu(c) . $$ Finally, the nullity of $I_{1} = E_{**}$ is $μ (1)$ , the multiplicity of the right endpoint, which is zero by hypothesis. $■$

This identifies the geodesic $γ$ , viewed as a critical point of $E$ on the path space, as a Morse critical point of index $\sum μ (c)$ whenever it has no conjugate endpoint. It is the variational analogue of the Morse index of a function at a nondegenerate critical point 03.02.30, with the curvature-driven Jacobi operator playing the role of the Hessian.

Bridge. The index theorem builds toward the topology of the path space: once each geodesic is a Morse critical point with computable index, $Ω (M; p, q)$ acquires the homotopy type of a CW complex with one cell per geodesic, the engine of Bott periodicity. The conjugate-point count appears again in 03.02.06, where the closed-form Jacobi fields in constant curvature pin down the injectivity radius and the diameter bound, and in 13.02.02, where the second variation of the action that begins the construction is the same object that distinguishes minimising from non-minimising geodesics. The curvature tensor $R$ of 03.02.05 enters as the coefficient of the Jacobi operator, so the entire focusing phenomenon is read off the sign of the sectional curvature.

Exercises Intermediate+

Exercise (medium, short-answer). State precisely why a geodesic that has passed an interior conjugate point cannot be length-minimising, using the index theorem.

Hint

A minimiser must have non-negative second variation of length/energy. What does a positive index say about the second variation?

Answer

Rubric. If $γ$ has an interior conjugate point, the index theorem gives index $\geq 1$ , so there is a variation field $W$ with $E_{**} (W, W) < 0$ . The corresponding variation strictly decreases energy (and, after reparametrisation, length) to second order, so a nearby path is strictly shorter. A length-minimiser must have positive-semidefinite second variation, hence index $0$ , hence no interior conjugate point. Full credit names the negative direction and the minimiser-implies-nonnegative-Hessian step.

Exercise (hard, short-answer). Prove the easy half of Bonnet–Myers infinitesimally: if the sectional curvatures of $(M,g)$ are all $\ge K > 0$, show that any geodesic of length $> \pi/\sqrt K$ has an interior conjugate point. (Use the index form as a comparison.)

Hint

Take a unit normal parallel field $E$ and the test field $V (t) = sin (π t / L) E (t)$ on a geodesic of length $L > π / K$ . Compute $I (V, V)$ and show it is negative; conclude index $\geq 1$ , hence a conjugate point.

Answer

Rubric. Reparametrise to $[0, L]$ with unit speed. With $V = sin (π t / L) E$ , $E$ parallel unit normal, $I (V, V) = \int_{0}^{L} ((π / L)^{2} cos^{2} (π t / L) - ⟨ R (V, \overset{γ}{˙}) \overset{γ}{˙}, V ⟩) d t$ . The curvature term is $sin^{2} (π t / L) sec (E, \overset{γ}{˙}) \geq K sin^{2} (π t / L)$ . So $I (V, V) \leq \int_{0}^{L} ((π / L)^{2} cos^{2} - K sin^{2}) = \frac{L}{2} ((π / L)^{2} - K) < 0$ when $L > π / K$ . A negative direction forces index $\geq 1$ , so by the index theorem there is an interior conjugate point. Full credit computes $I (V, V)$ and uses the curvature lower bound correctly. This is the analytic engine of the Bonnet–Myers diameter bound, applied globally in 03.02.06.

Exercise (hard, short-answer). Explain why, on a manifold of non-positive sectional curvature, the index form is positive definite on every geodesic, and deduce that there are no conjugate points (the infinitesimal Cartan–Hadamard statement).

Hint

Look at the sign of each term in $I (V, V) = \int ∣ V^{'} ∣^{2} - ⟨ R (V, \overset{γ}{˙}) \overset{γ}{˙}, V ⟩ d t$ when sectional curvature is $\leq 0$ .

Answer

Rubric. When all sectional curvatures are $\leq 0$ , $⟨ R (V, \overset{γ}{˙}) \overset{γ}{˙}, V ⟩ = ∣ V ∣^{2} ∣ \overset{γ}{˙} ∣^{2} sec (V, \overset{γ}{˙}) - (proj) \leq 0$ for the normal part, so $- ⟨ R (V, \overset{γ}{˙}) \overset{γ}{˙}, V ⟩ \geq 0$ and $I (V, V) \geq \int ∣ V^{'} ∣^{2} d t > 0$ for any nonzero $V$ vanishing at the endpoints (else $V^{'} \equiv 0$ and $V$ constant $= 0$ ). Positive-definite index form means index $0$ and nullity $0$ on every geodesic, so by the index theorem no interior conjugate points exist. This is the local statement behind Cartan–Hadamard's global diffeomorphism $exp_{p} : T_{p} M \to M$ .

Lean formalization Intermediate+

Mathlib has the Levi-Civita connection, the curvature tensor, and the exponential map, but no path-space variational layer, so the central results here are not yet formalisable end-to-end. The Jacobi equation itself is expressible as a second-order linear ODE for a section along a curve; the following is the statement-level shape one would target, written in Lean-compatible pseudo-Lean. It does not compile against current Mathlib (lean_status: none).

-- Statement target (NOT compiling against current Mathlib):
-- The Jacobi equation for a vector field J along a geodesic γ.
variable {M : Type*} [RiemannianManifold M] (γ : ℝ → M) (hγ : IsGeodesic γ)

def IsJacobiField (J : ∀ t, TangentSpace M (γ t)) : Prop :=
  ∀ t, covariantDeriv γ (covariantDeriv γ J) t
        + riemann (J t) (deriv γ t) (deriv γ t) = 0

-- A point γ c is conjugate to γ 0 along γ:
def IsConjugate (c : ℝ) : Prop :=
  ∃ J, IsJacobiField γ J ∧ J ≠ 0 ∧ J 0 = 0 ∧ J c = 0

-- Morse Index Theorem (target): index of the index form on Ω(p,q)
-- equals the sum of conjugate-point multiplicities in (0,1).
-- theorem morse_index : indexForm γ |>.index
--   = ∑ c in conjugatePoints γ, multiplicity γ c

the Mathlib gap analysis above enumerates the missing primitives: the energy functional on a Sobolev/piecewise-smooth path space, the index form as its Hessian, the $2 n$ -dimensional Jacobi solution space, conjugate-point multiplicity, and the index-theorem identity itself.

Advanced results Master

The index form as a Hessian on a Hilbert manifold. Completing $Ω (M; p, q)$ to the $H^{1}$ Sobolev manifold $Ω^{1}$ of paths with square-integrable derivative turns $E$ into a smooth function satisfying the Palais–Smale condition, and the index form $I = E_{**}$ becomes the genuine Hessian of $E$ at a geodesic ^{[Milnor Part III §16]}. The Morse Index Theorem then reads: a geodesic is a nondegenerate critical point iff its endpoints are non-conjugate, and its Morse index in the sense of 03.02.30 is the sum of interior conjugate multiplicities. This is the bridge by which the function-theoretic Morse theory of Part I and the variational geodesic theory of Part III become one subject. The finite-dimensional broken-geodesic model $Ω (t_{0}, \dots, t_{k})$ used in the proof is the deformation retract that makes the infinite-dimensional space tractable.

The Morse index for the path space and the homotopy type. Once every geodesic is a Morse critical point with index equal to its conjugate-point sum, the full machinery of Part I transfers: $Ω (M; p, q)$ (for non-conjugate $p, q$ ) has the homotopy type of a CW complex with one cell of dimension $λ$ for each geodesic from $p$ to $q$ of index $λ$ . Applied to a symmetric space, with the set of minimal geodesics forming a manifold, this is the engine of Bott's proof of periodicity for the stable orthogonal and unitary groups ^{[Bott 1956]}. The index of a geodesic — computed by counting conjugate points — is precisely the cell dimension.

Index form versus the path-space Laplacian; the Sturm comparison. On normal fields, the index form is the quadratic form of the self-adjoint Sturm–Liouville operator $- D^{2} / d t^{2} - R (\cdot, \overset{γ}{˙}) \overset{γ}{˙}$ with Dirichlet conditions. Its eigenvalues are real and bounded below, and the number of negative eigenvalues is the index. The Sturm oscillation theorem then is the index theorem in the one-dimensional (curve-by-curve) case, and the Morse Index Theorem is its $n$ -dimensional vector generalisation. Bott's 1956 reformulation reads the index of an iterated closed geodesic off the Sturm intersection number, which is how the periodicity calculation iterates ^{[Bott 1956]}.

Curvature comparison and the Rauch theorem. Comparing the index form of $γ$ in $M$ against the index form of a model geodesic in a constant-curvature space yields the Rauch comparison theorem: an upper curvature bound delays conjugate points (pushes them farther along the geodesic, so geodesics minimise longer), and a lower bound advances them. The Bonnet–Myers and Cartan–Hadamard statements are the two extreme corollaries, derived globally in 03.02.06 and 03.02.05; here they appear infinitesimally as definiteness statements for $I$ , worked out in the exercises.

Synthesis. The Morse Index Theorem stitches together at least four threads. It is the variational image of the Morse-function index of 03.02.30, with the energy functional, the index form, and the Jacobi operator playing the roles of the Morse function, its Hessian, and the inertia data. It consumes the curvature tensor of 03.02.05 as the coefficient of the Jacobi equation, making conjugate-point structure a direct readout of sectional curvature. It supplies the general theory whose constant-curvature special solutions are used in 03.02.06 to classify space forms and bound diameters. It is the linearised refinement of the geodesic and second-variation calculus of 13.02.02, turning "geodesic" into "critical path." And through the focusing of geodesic congruences it is the Riemannian prototype of the conjugate-point and Raychaudhuri analysis underlying Lorentzian causality in 03.02.17. The common invariant across all five is the index — a count of how the curvature refocuses an infinitesimal family of geodesics — which serves equally as a Morse index, a cell dimension, and an obstruction to minimisation.

Full proof set Master

Proposition (the index form has finite index). The index of $E_{**}$ on $T_{γ} Ω$ is finite.

Proof. By Step 1 of the index theorem, the index equals that of the Hessian of $E$ restricted to the finite-dimensional broken-geodesic manifold $Ω (t_{0}, \dots, t_{k})$ , whose dimension is $(k - 1) n$ . A symmetric bilinear form on a finite-dimensional space has finite index, bounded by the dimension. Hence the index is finite and, in fact, $\leq (k - 1) n$ . $■$

Proposition (nullity equals endpoint conjugate multiplicity). The nullity of $E_{**}$ equals the multiplicity of $q = γ (1)$ as a conjugate point of $p$ .

Proof. $V$ lies in the null space of $E_{**}$ iff $I (V, W) = 0$ for all admissible $W$ . By the integrated index form, $I (V, W) = - \int ⟨ J (V), W ⟩ d t - \sum_{i} ⟨ Δ_{t_{i}} V^{'}, W (t_{i})⟩$ . Vanishing for all $W$ forces $J (V) = 0$ on each smooth piece and $Δ_{t_{i}} V^{'} = 0$ at each corner, so $V$ is a smooth Jacobi field; and $V (0) = V (1) = 0$ by admissibility. Thus the null space is exactly the space of Jacobi fields vanishing at both endpoints, whose dimension is $μ (1)$ by definition. $■$

Proposition (Jacobi fields are variation fields of geodesics). Every Jacobi field $J$ along $γ$ arises as $\partial_{s} ∣_{s = 0} Γ (s, \cdot)$ for a smooth family $Γ$ of geodesics with $Γ (0, \cdot) = γ$ .

Proof. Given $J$ , set $Γ (s, t) = exp_{σ (s)} (t (\overset{γ}{˙} (0) + s W (0)))$ where $σ$ is a curve with $σ (0) = p$ , $\overset{σ}{˙} (0) = J (0)$ , and $W$ is the parallel field with $W (0) = J^{'} (0)$ , all transported appropriately. Each $Γ (s, \cdot)$ is a geodesic. Its variation field $V (t) = \partial_{s} ∣_{0} Γ (s, t)$ satisfies the Jacobi equation — differentiate the geodesic equation $\nabla_{\partial_{t}} \partial_{t} Γ = 0$ in $s$ and commute derivatives using the curvature tensor — with $V (0) = J (0)$ and $V^{'} (0) = J^{'} (0)$ . By ODE uniqueness $V = J$ . $■$

Proposition (the jump formula, restated). As $c$ increases through a conjugate value, the index $λ (c)$ increases by exactly $μ (c)$ ; elsewhere it is locally constant. The proof is Step 3 of the index theorem; the upward jump is supplied by the corner-jump term $- ⟨ Δ_{c} J_{a}^{'}, J_{a} (c)⟩$ for each null Jacobi field $J_{a}$ , and the verification that no further negative directions appear comes from the left-continuity of $λ$ (Step 2). The complete accounting is the inertia-change computation for the broken-geodesic Hessian matrix ^{[fasttrack-texts the index theorem (Part III analogue)]}. $■$

The Rauch comparison theorem and the full Palais–Smale verification for $E$ on $Ω^{1}$ are stated above without proof — see do Carmo ^{[do Carmo Ch. 10]} for Rauch and Milnor ^{[Milnor Part III §16]} for the Hilbert-manifold setup.

Connections Master

Morse functions and the Morse index 03.02.30. This unit is the variational incarnation of that one: the energy functional replaces a generic Morse function, the index form replaces the Hessian, and the Morse Index Theorem computes the index of a geodesic exactly as the Morse lemma computes the index of a critical point. A geodesic with no conjugate endpoint is a nondegenerate critical point of $E$ in the sense defined there.
Geodesics and parallel transport 13.02.02. The Jacobi equation is the linearisation of the geodesic equation established there; covariant differentiation along $γ$ and the second variation of the action both reappear here as the operators $V^{'}$ and $E_{**}$ . The geodesic-as-extremal-of-energy fact is the starting point of the whole construction.
Sectional, Ricci, and scalar curvature 03.02.05. The curvature tensor $R$ entering the Jacobi equation is precisely the object built there; whether conjugate points appear, and how soon, is governed by the sectional curvature in the plane spanned by the Jacobi field and the geodesic. The Bonnet–Myers and Cartan–Hadamard dichotomy is the global shadow of the sign of this curvature.
Constant-curvature spaces and Killing–Hopf 03.02.06. That unit already uses closed-form Jacobi fields $J (t) = f (t) E (t)$ with $f$ a sine, cosine, or hyperbolic solution; this unit supplies the general theory those special solutions instantiate, and the Bonnet–Myers diameter bound it invokes is derived infinitesimally here in the exercises.
Lorentzian Hopf–Rinow and global hyperbolicity 03.02.17. Conjugate points and the focusing of geodesic congruences are the Riemannian prototype of the focusing theorems (Raychaudhuri) and the conjugate-point structure used in Lorentzian causality and singularity theorems.

Historical & philosophical context Master

Jacobi introduced the equation now bearing his name in the 1830s while studying the second variation in the calculus of variations and the question of when an extremal ceases to minimise; his Vorlesungen über Dynamik (lectures delivered 1842–43, published 1866) contains the conjugate-point criterion for geodesics on surfaces. The modern global theory is due to Marston Morse, whose The Calculus of Variations in the Large (American Mathematical Society Colloquium Publications 18, 1934) proved the index theorem for geodesics and recast the calculus of variations as the study of critical points of functionals on path spaces ^{[Morse 1934]}.

Milnor's Morse Theory (Annals of Mathematics Studies 51, 1963), based on notes by Spivak and Wells, gave the streamlined treatment that became canonical: Part II compresses Riemannian geometry to its load-bearing core, and Part III develops the path space, the index form, Jacobi fields, conjugate points, and the index theorem in roughly thirty pages, culminating in Bott periodicity ^{[Milnor Part III §§13--15]}. Raoul Bott's 1956 paper "On the iteration of closed geodesics and the Sturm intersection theory" (Communications on Pure and Applied Mathematics 9, 171–206) connected the geodesic index to classical Sturm oscillation theory and supplied the iteration formula that drives the periodicity computation ^{[Bott 1956]}. The comparison-geometry refinements — Rauch's 1951 theorem and the Cheeger–Ebin synthesis — turned the index form into the central tool for relating curvature bounds to topology ^{[do Carmo Ch. 10]}.

Bibliography Master

@book{Milnor1963Morse,
  author    = {Milnor, John W.},
  title     = {Morse Theory},
  series    = {Annals of Mathematics Studies},
  number    = {51},
  publisher = {Princeton University Press},
  year      = {1963},
  note      = {Based on lecture notes by M. Spivak and R. Wells. Part III, §§12--15}
}

@book{Morse1934CVL,
  author    = {Morse, Marston},
  title     = {The Calculus of Variations in the Large},
  series    = {American Mathematical Society Colloquium Publications},
  volume    = {18},
  publisher = {American Mathematical Society},
  year      = {1934}
}

@article{Bott1956Iteration,
  author  = {Bott, Raoul},
  title   = {On the iteration of closed geodesics and the {S}turm intersection theory},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {9},
  pages   = {171--206},
  year    = {1956}
}

@article{Rauch1951,
  author  = {Rauch, Harry E.},
  title   = {A contribution to differential geometry in the large},
  journal = {Annals of Mathematics},
  volume  = {54},
  pages   = {38--55},
  year    = {1951}
}

@book{Jacobi1866Dynamik,
  author    = {Jacobi, Carl Gustav Jacob},
  title     = {Vorlesungen {\"u}ber Dynamik},
  publisher = {G. Reimer},
  year      = {1866},
  note      = {Lectures delivered 1842--43; ed. A. Clebsch}
}

@book{doCarmo1992,
  author    = {do Carmo, Manfredo P.},
  title     = {Riemannian Geometry},
  publisher = {Birkh{\"a}user},
  year      = {1992}
}

@book{CheegerEbin1975,
  author    = {Cheeger, Jeff and Ebin, David G.},
  title     = {Comparison Theorems in Riemannian Geometry},
  publisher = {North-Holland},
  year      = {1975}
}

Prerequisites

03.02.05
03.02.30
13.02.02

Tier anchors

beginner: Needham Visual Differential Geometry (conjugate points on the sphere); do Carmo Riemannian Geometry Ch. 5 informal
intermediate: Milnor Morse Theory Part III §§13--15; do Carmo Riemannian Geometry Chs. 5, 9, 10; Lee Introduction to Riemannian Manifolds Chs. 10--12
master: Milnor Morse Theory Part III §§13--15; Morse 1934 The Calculus of Variations in the Large; Bott 1956 Ann. Math. 60; Cheeger-Ebin Comparison Theorems in Riemannian Geometry Ch. 1

References

03-modern-geometry/Bott-LecturesOnMorseTheory.pdf · Path space, the energy functional, the index theorem (Part III analogue)
Milnor — Morse Theory (Annals of Mathematics Studies 51, 1963) · Part III §§12--15 (the path space, the index theorem)
Morse — The Calculus of Variations in the Large (1934) · AMS Colloquium Publications 18, Ch. V
Bott — On the iteration of closed geodesics and the Sturm intersection theory (1956) · Comm. Pure Appl. Math. 9, pp. 171--206
do Carmo — Riemannian Geometry (1992) · Chs. 5, 9, 10 (Jacobi fields, the second variation, the index theorem)

Estimated time

beginner: 16m
intermediate: 42m
master: 80m