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On continuous piecewise smooth perpendicular fields with zero endpoints, the Riemannian index form is nonnegative exactly when no point strictly inside the geodesic segment is conjugate to its initial point. It is positive on every nonzero field exactly when no conjugate point occurs up to and including the final endpoint. At a first conjugate endpoint its nullspace consists of the endpoint-vanishing Jacobi fields. These statements distinguish nonnegative quadratic forms from strictly positive ones and do not assume global minimization.

In unit-curvature Real projective space of dimension at least two, Riemannian distance is $d([v],[u])=\arccos |⟨v,u⟩|$ for unit lifts. A unit-speed geodesic has lift $\cos tv+\sin tw$, with $w$ unit and perpendicular to $v$, and therefore ceases to minimize after $\pi /2$. Its normal Jacobi fields with zero initial value are $\sin t$ times parallel vector fields along a curve, giving first conjugate point at time $\pi$. The two times differ in every direction.

A Riemannian cut point along a geodesic in a complete Riemannian manifold is either its first conjugate point or the endpoint of another minimizing geodesic from the initial point. To prove this, approach the cut time from above and choose minimizing geodesics to the later endpoints using the Hopf-Rinow theorem. A subsequence of their initial unit vectors converges. If its limit is the original initial vector and the cut endpoint is not conjugate, the inverse function theorem for the Riemannian exponential map identifies the two geodesics locally and contradicts the shorter length of the chosen minimizers. A conjugate point earlier than the cut time is excluded by the negative Riemannian index form direction obtained by extending an endpoint-vanishing Jacobi field by zero and perturbing its endpoint corner.

For $\gamma (t)={\exp }_p(tV)$, every Jacobi field with $J(0)=0$ is uniquely of the formDifferentiate the geodesic variation ${\exp }_p(t(V+sB))$ in $s$ to obtain the formula. Uniqueness of the Jacobi equation with prescribed initial value and derivative shows that all such fields are obtained. A nontrivial field vanishing at $t=L>0$ therefore corresponds to a nonzero vector in the kernel of $(d{\exp }_p{)}_{LV}$, giving the conjugate point criterion.

Let a nonzero Jacobi field $J$ vanish at times zero and $a$. On a longer geodesic segment ending at $b>a$, extend it by zero to a continuous piecewise smooth field $Z$. Then $I(Z,Z)=0$. Choose an endpoint-vanishing field $P$ with $P(a)=-D_{t}J(a)$; integration by parts gives $I(Z,P)=-|D_{t}J(a){|}^2<0$. Thus $I(Z+\epsilon P,Z+\epsilon P)<0$ for small positive $\epsilon$. Smooth approximation preserves negativity. The second variation of geodesic energy supplies a shorter path, so a minimizing geodesic cannot extend beyond its first conjugate point.

// Target: differential-geometry.bigb

The Morse index of a fixed-endpoint geodesic equals the number of conjugate points strictly between its endpoints, counted with multiplicity. Its nullity equals the multiplicity of the terminal conjugate point. A conjugate multiplicity is the dimension of the space of Jacobi fields vanishing at the initial point and at the point in question.

Along a nonconstant affinely parametrized geodesic, the multiplicity of a conjugate point at positive time $t_0$ is the dimension of the space of Jacobi fields vanishing at both endpoints. The tangential component of such a Jacobi field is affine and vanishes at both endpoints, hence is zero. Uniqueness for the Jacobi field equation embeds this space into the $(n-1)$-dimensional normal space by $J\mapsto D_{t}J(0)$. Thus its dimension is at most $n-1$.

For fixed-endpoint variations, the second variation iswith $p=f_{vv}$, $r=f_{uv}$, $s=f_{uu}$ evaluated along $(x,\bar{u},{\bar{u}}^{\prime })$. The Jacobi equation, written without differentiating $r$ separately, isWhen $r$ is differentiable this is $-(p{z}^{\prime }{)}^{\prime }+(s-r^{\prime })z=0$.

Positive definiteness means $q(\phi )>0$ for every nonzero admissible variation. Uniform or coercive positivity of a second variation is the stronger assertion that $q(\phi )\ge c\int {\phi }^{\prime 2}$ for some fixed $c>0$. On a fixed-endpoint interval this is equivalent to a positive lower bound in the $W^{1,2}$ norm.

The strict Legendre implication uses the uniform interpretation of positivity. If $q\ge c\int {\phi }^{\prime 2}$, localized rapidly oscillating variations show $p(x)\ge c$: otherwise support them in an interval where $p-c<0$, so the derivative term of $q-c\int {\phi }^{\prime 2}$ is negative of order frequency squared, overwhelming its mixed and zero-order terms. Continuity extends the bound to the endpoints, giving $p>0$ on $[a,b]$. Under mere positive definiteness this strict conclusion is false: $f=u^2$ at $\bar{u}=0$ gives $q=2\int {\phi }^2>0$ but $p=0$. In that weaker convention only the ordinary Legendre necessary condition $p\ge 0$ follows. This distinction makes the implication precise.

Under the stated regular condition $p>0$, a conjugate point $c\in (a,b]$ to $a$ is an endpoint admitting a nonzero Jacobi solution with $z(a)=z(c)=0$. The disconjugacy form of the Jacobi positivity criterion says that no such point in $(a,b]$ exists if and only if there is a Jacobi solution strictly positive on all of $[a,b]$.

For the required identity, let that positive solution be $z$ and write $\phi =zh$. Set $A=pz{z}^{\prime }+r{z}^2$. The Jacobi equation givesExpanding the second-variation integrand therefore yieldsFor compactly supported $\phi$, the boundary term vanishes, provingThis is the Jacobi identity for the second variation. The divergence form is also appropriate under the original $C^1$ regularity; with $p>0$, the Euler-Lagrange equation gives the usual further regularity along the extremal.

For every $0<\lambda <1$, positivity in the preceding part gives $\int u^{\prime 2}\ge {\lambda }^2\int u^2$ for each fixed admissible $u$. Let $\lambda \uparrow 1$ to obtainThe sharp constant is attained by $u(x)=\sin x$, consistently with the first conjugate point occurring at the endpoint for $\lambda =1$.

For a fixed-endpoint variation $\phi \in W_0^{1,\infty }(a,b)$, the second variation iswhere all coefficients are evaluated at $(x,\bar{u}(x),{\bar{u}}^{\prime }(x))$. The notation ${\delta }^{2}I(\bar{u})>0$ means this is strictly positive for every nonzero admissible variation. This is pointwise positive definiteness of the quadratic form, not an assertion of a prescribed uniform lower bound.

PutAfter integration by parts, the form is $q(\phi )={\int }_a^b(P{\phi }^{\prime 2}+Q{\phi }^2)dx$. Its Jacobi equation isIn the regular case $P>0$ on $[a,b]$, a conjugate point $c\in (a,b]$ to $a$ is a point for which a nonzero solution satisfies $w(a)=w(c)=0$. Equivalently it is a zero of the normalized solution with $w(a)=0$, $P(a)w^{\prime }(a)=1$.

Under this regular Strengthened Legendre condition, the Jacobi positivity criterion isThe endpoint $b$ must be excluded as a conjugate point too. To prove necessity, suppose a Jacobi solution vanishes at $a$ and $c$. Define $\phi =w$ on $[a,c]$ and zero on $[c,b]$. It is a nonzero Lipschitz admissible variation, anda contradiction. For completeness, the converse follows from the Jacobi identity for the second variation: when the normalized $w$ has no further zero,The boundary contribution $[P(w^{\prime }/w){\phi }^2{]}_a^b$ vanishes, including the limit at $a$ since $w$ has a simple zero and $\phi =O(x-a)$. Equality forces $\phi =Cw$, and $\phi (b)=0$ with $w(b)\ne 0$ forces $C=0$.

Independently of regularity, positivity necessarily implies the ordinary Legendre necessary condition $P\ge 0$. If $P<0$ on an interval, localized rapidly oscillating test functions have a negative derivative contribution of order frequency squared, dominating the bounded zero-order term. This contradicts positivity.

The assumptions stated here do not themselves impose $P>0$. That strengthened condition is a regularity hypothesis for the familiar Jacobi criterion, and is not necessary for strict positivity: the integrand $f=u^2$ gives $q(\phi )=2\int {\phi }^2>0$ with $P=0$. For a criterion covering the full possibly degenerate class, use Regularized Jacobi criterion for a degenerate second variation. The following conditions are necessary and sufficient:

These conditions are necessary because $q>0$ gives $q_{\eta }(\phi )=q(\phi )+\eta \int {\phi }^{\prime 2}>0$, so the regular criterion just proved applies; a weak Jacobi solution tested against itself would give $q=0$. They are sufficient as well: the regular criterion gives $q_{\eta }>0$ for every $\eta >0$, hence $q\ge 0$ by letting $\eta \downarrow 0$. If $q(\phi )=0$, nonnegativity of $q(\phi +t\psi )$ for every real $t$ forces its mixed bilinear term to vanish for every test $\psi$. This is precisely the weak Jacobi equation. The third condition then forces $\phi =0$. Thus the criterion treats degeneracy without silently assuming the strengthened condition.

Here $P=2$, $Q=-2{\lambda }^2$ and the Jacobi equation is $w^{\prime \prime }+{\lambda }^{2}w=0$. With $w(0)=0$ and $w^{\prime }(0)=1$, its solution is $w(x)=\sin (|\lambda |x)/|\lambda |$ for $\lambda \ne 0$, and $w(x)=x$ for $\lambda =0$. Its first positive zero is $\pi /|\lambda |$ when $\lambda \ne 0$.

If ${\lambda }^2<{\pi }^2/b^2$, no conjugate point occurs in $(0,b]$. The Jacobi positivity criterion givesSince $I_{\lambda }$ itself is this quadratic form divided by two, $I_{\lambda }(u)>0=I_{\lambda }(0)$ for every nonzero admissible $u$. Thus zero is the unique global minimizer. This argument establishes the result before using the sharp Poincaré inequality.

Write $T=\dot{\gamma }$ along an affinely parametrized geodesic and $D_t={\nabla }_T$ for the covariant derivative from the Levi-Civita connection. We use the Riemann curvature tensor convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, for which positive sectional curvature means $⟨R(X,Y)Y,X⟩>0$ on an orthonormal pair. A Jacobi field is a smooth vector field along $\gamma$ satisfyingSuch a field describes the first-order separation of nearby geodesics, as the geodesic variation construction in part (c) shows.

For $t_0>0$, $\gamma (t_0)$ is a conjugate point of $\gamma (0)$ along this geodesic if a nonzero Jacobi field satisfies $J(0)=J(t_0)=0$. The multiplicity of a conjugate point isIt is equivalently the dimension of the kernel of the differential of the Riemannian exponential map at $t_0\dot{\gamma }(0)$, by Jacobi fields vanishing at their initial point. Conjugacy concerns the specified geodesic and its two parameter values, even when its endpoints represent the same point of the Riemannian manifold.

Suppose $J$ is a Jacobi field vanishing at both endpoints of a nonconstant geodesic segment $[0,t_0]$. Its tangential component $h(t)=⟨J(t),T(t)⟩$ satisfiesbecause $D_{t}T=0$ and the Riemann curvature tensor has the usual skew-adjoint symmetry. The two zero endpoint values imply $h\equiv 0$. Both $J$ and $D_{t}J$ are therefore perpendicular to $T$.

The linear map from these Jacobi fields to $T(0{)}^{\perp }$ given by $J\mapsto D_{t}J(0)$ is injective: $J(0)=0$ together with $D_{t}J(0)=0$ forces $J\equiv 0$ by uniqueness for the linear ordinary differential equation defining a Jacobi field. Its target has dimension $n-1$, soThis proves the endpoint-vanishing normal Jacobi fields have dimension at most n-1 bound. For a constant geodesic the equation is $J^{\prime \prime }=0$, and two zero values instead force $J=0$, so there is no conjugate point.

Let $\tau >0$ be the first conjugate time and choose a nonzero Jacobi field $J$ with $J(0)=J(\tau )=0$. It is normal by part 1(b), and $D_{t}J(\tau )\ne 0$, since zero value and derivative there would force $J=0$. Fix any $T>\tau$ and extend $J$ by zero to a continuous piecewise smooth vector field $V$ on $[0,T]$.

For endpoint-vanishing fields, the Riemannian index form isIntegration by parts on $[0,\tau ]$ and the Jacobi field equation give $I(V,V)=0$ andChoose a smooth normal field $W$, zero at $0,T$, with $W(\tau )=-D_{t}J(\tau )$. Thenfor sufficiently small positive $\epsilon$. The corner in $V$ causes no difficulty: smooth endpoint-vanishing fields approximate $V+\epsilon W$ in the Sobolev space $H^1$, and the Riemannian index form is continuous in this norm on the compact segment. A smooth approximant therefore still has negative index form.

Realize this smooth field as a fixed-endpoint curve variation using the Riemannian exponential map. The allowed second variation of geodesic energy gives $E^{\prime }(0)=0$ and $E^{\prime \prime }(0)<0$, so some nearby curve has energy less than $T/2$, the energy of the unit-speed reference geodesic. The Cauchy-Schwarz inequality bounds its length byIt is shorter than the reference segment. Thus a geodesic cannot minimize beyond its first conjugate time. This argument deliberately uses $T>\tau$; it does not rule out minimization up to the conjugate point itself.

Let $t_0$ be the finite cut time and put $q=\gamma (t_0)$. Continuity of Riemannian distance shows that the segment to $q$ still minimizes and $d(p,q)=t_0$. Choose $t_j\downarrow t_0$ with $t_j>t_0$. By the Hopf-Rinow theorem, there is a minimizing geodesic from $p$ to $q_j=\gamma (t_j)$ with length ${\ell }_j=d(p,q_j)<t_j$ and initial unit vector $v_j$. Continuity gives ${\ell }_j\to t_0$.

The unit tangent sphere is compact, so after a subsequence $v_j\to v$. Smooth dependence of the Riemannian exponential map gives ${\exp }_p(t_{0}v)=q$. The geodesic with initial vector $v$ has length $t_0=d(p,q)$ up to $q$, hence minimizes. If $v\ne \dot{\gamma }(0)$, this is the required second minimizing geodesic.

Suppose instead $v=\dot{\gamma }(0)$ and $q$ is not a conjugate point. Then the differential of ${\exp }_p$ at $t_0\dot{\gamma }(0)$ is nonsingular. The inverse function theorem makes ${\exp }_p$ injective on a neighborhood of this vector. For large $j$, both ${\ell }_j{v}_j$ and $t_j\dot{\gamma }(0)$ belong to that neighborhood and exponentiate to $q_j$. They must be equal, implying ${\ell }_j=t_j$, a contradiction. Thus either a distinct minimizing geodesic exists or $q$ is conjugate. In the latter case it is the first conjugate point, since part (a) excludes any earlier one on a minimizing segment. We have proved the cut-point dichotomy for complete Riemannian manifolds.

Here the relevant concept is a Riemannian cut point, not the topological cut point obtained by disconnecting a space through removal of a point.

Expand a Jacobi field in the parallel orthonormal eigenbasis: $J(t)={\sum }_i{j}_i(t)e_i(t)$. Part (b) reduces its equation to the scalar equations $j_i^{\prime \prime }+{\lambda }_i{j}_i=0$. DefineFor arbitrary initial values $J(0)={\sum }_i{a}_i{e}_i$ and $D_{t}J(0)={\sum }_i{b}_i{e}_i$, the complete solution isA point at parameter $T>0$ is a conjugate point of $p$ precisely when a nonzero Jacobi field vanishes at both $0$ and $T$. The first condition makes every $a_i=0$. The second requires $b_i{S}_{{\lambda }_i}(T)=0$ for every $i$, with at least one nonzero $b_i$. For $T>0$, the factors for zero and negative eigenvalues never vanish; positive eigenvalues give exactly the sine zeros. ThereforeThe corresponding points are $\gamma (T)$. If several positive eigenspaces have a zero at the same parameter, the conjugate multiplicity is the sum of their dimensions. This also shows that a locally symmetric Riemannian manifold has no conjugate points along a given geodesic when that geodesic's Jacobi curvature operator has no positive eigenvalues.

A stationary black hole behaves mechanically like an equilibrium thermodynamic system. Its horizon area A, asymptotic energy E, angular momentum J and electric charge Q are linked by the laws of black-hole mechanics. Let ${\chi }^a=t^a+{\Omega }_H{\varphi }^a$ generate its Killing horizon, with the time translation normalized at infinity; its surface gravity is defined by ${\chi }^b{\nabla }_b{\chi }^a=\kappa {\chi }^a$ on that horizon. We use $c=1$ and keep G explicit. The horizon electrostatic potential ${\Phi }_H$ is measured relative to infinity.

The Zeroth law of black-hole mechanics states that kappa is constant on each connected stationary horizon, under the Einstein equations and suitable energy/regularity assumptions. It parallels the zeroth law of thermodynamics, which makes temperature uniform in equilibrium. Kappa need not be constant in a dynamical geometry. For an asymptotically flat static hole, local observers see gravitationally redshifted temperatures; the equilibrium quantity comparable to kappa is the temperature referenced to infinity.

The First law of black-hole mechanics relates nearby stationary solutions:The same leading relation describes a small physical process which perturbs a horizon and subsequently settles to equilibrium, with the appropriate hypotheses on fluxes and final state. The work terms correspond to rotational and electromagnetic work. The structural analogy with the first law of thermodynamics, $dE=TdS+\text{work}$, identifies area as an entropy-like variable and surface gravity as a temperature-like variable. Classically this fixes their product in the first law, rather than their separate normalizations.

The second law of black-hole mechanics, or black-hole area theorem, states that future event-horizon area does not decrease classically. Its essential geometric ingredient is the Raychaudhuri equation for affinely parametrized null generators. In four dimensions a hypersurface-orthogonal congruence obeysThe null energy condition and Einstein equations make the last term nonpositive. If the expansion became negative, the inequality would focus the generators to a conjugate point in finite affine parameter. Under future completeness and predictability assumptions, that would conflict with their remaining on an achronal event horizon. Thus $\vartheta \ge 0$, and the area element, satisfying $d\log dA/d\lambda =\vartheta$, cannot decrease. For a merger the final area is at least the sum of the initial component areas. These assumptions matter: the theorem is a classical statement, not a prohibition of quantum evaporation.

The third law of black-hole mechanics is an unattainability statement: a nonextremal hole cannot be driven to zero surface gravity in a finite physical process satisfying the usual bounded-stress and regularity conditions. This parallels the unattainability form of the third law of thermodynamics. Extremal stationary solutions exist, so the law concerns how an initially nonextremal hole approaches them. The stronger entropy form of Nernst's theorem does not transfer unchanged: extremal holes can have finite area and entropy, depending on their conserved charges.

Quantum field theory turns the mechanical analogy into a physical thermodynamic identification. Hawking radiation is thermal at infinity to leading semiclassical order, withThese give $T_H\delta S_{\mathrm{BH}}=\kappa \delta A/(8\pi G)$ exactly, so the mechanical first law becomes the thermodynamic first law with Bekenstein-Hawking entropy. Scattering through the exterior adds greybody factors to the emitted spectrum but leaves this temperature identification intact. Equilibrium with an external bath and isolated evaporation are different physical states of the field.

During black-hole evaporation, the area can decrease because the semiclassical stress tensor need not satisfy the classical null energy condition. The proposed generalized second law applies instead to $S_{\mathrm{gen}}=S_{\mathrm{BH}}+S_{\mathrm{outside}}$, asserting nondecrease of the combined entropy. A Schwarzschild hole illustrates an unusual thermodynamic property: $T_H\propto E^{-1}$, so its heat capacity is negative. It becomes hotter as it loses energy and is unstable in a simple canonical heat bath, while entropy and energy laws still describe an isolated system. The quantum temperature, area entropy and generalized second law supply the correspondence; the mechanical laws alone do not establish a thermal emission spectrum or a microscopic entropy count.

Use the curvature convention $\mathcal{R}$ of Solution 1, and put $T=\dot{\gamma }$. On continuous piecewise smooth perpendicular vector fields with zero endpoint values, the Riemannian index form is the symmetric bilinear formA Jacobi field is a smooth field $J$ along $\gamma$ satisfying $D_t^{2}J+\mathcal{R}(J,T)T=0$. The point $\gamma (t_0)$ is a conjugate point to $\gamma (0)$ along this geodesic when there is a nonzero Jacobi field with $J(0)=J(t_0)=0$. Such a field is perpendicular to $T$, because $⟨J,T⟩$ is affine and vanishes at both endpoints.

The conjugate-point criterion for the Riemannian index form, stated without proof, isAt a first conjugate endpoint the form is nonnegative but has a nonzero nullspace, consisting of endpoint-vanishing Jacobi fields. A conjugate point strictly inside gives a negative direction. This explicitly distinguishes the non-strict and strict senses of positivity in the question; in terminology where “positive definite” already means strict positivity, the second line is the positive-definiteness condition.

Choose $n-1$ perpendicular orthonormal fields $E_1,\dots ,E_{n-1}$ by parallel transport along $\gamma$, and put $V_i(t)=\sin (\pi t/L)E_i(t)$. They are admissible andThe tangent has unit length. The assumed Ricci curvature bound and the two elementary sine-square and cosine-square integrals give the sine index-form bound for positive Ricci curvature:Since $n\ge 2$, at least one of the admissible $V_i$ therefore has $I(V_i,V_i)<0$.

If the Riemannian manifold is complete, the Hopf-Rinow theorem supplies a minimizing geodesic between any two points. Its Riemannian index form is nonnegative: any smooth fixed-endpoint variation has nonnegative second variation of geodesic energy, since a length minimizer of constant speed also minimizes energy on a fixed parameter interval by the Cauchy-Schwarz inequality. The same conclusion for piecewise smooth fields follows by smooth approximation, or by piecewise variations. The negative direction above excludes a minimizing segment longer than $\pi /\sqrt{\kappa }$. Taking the supremum over pairs yields the diameter conclusion of the Bonnet-Myers theorem:

The completeness assumption is essential. An explicit incomplete positively curved strip with infinite diameter isThe map $(u,v)\mapsto (\cos u\cos v,\cos u\sin v,\sin u)$ is a local isometry to the unit sphere, so $K=1$ and, in dimension two, $\mathrm{Ric}=g$. Thus the required Ricci curvature bound holds with $n=2$, $\kappa =1$. The unit-speed meridian $t\mapsto (t,0)$ reaches the missing boundary $u=\pi /4$ in finite time, so the strip is geodesically incomplete. On the other hand every curve joining $(0,0)$ to $(0,A)$ hassince $\cos u\ge 1/\sqrt{2}$ throughout the strip. Therefore its intrinsic diameter is infinite, in particular larger than $\pi$. This is the unwrapped strip with $v\in \mathbb{R}$, not a strip with longitude identified modulo $2\pi$; that distinction prevents spherical shortcuts from invalidating the example.

Take the normalizationThe first variation of geodesic energy says that the critical points on the fixed-endpoint path space are exactly the affinely parametrized geodesics. The kernel of the second variation of geodesic energy, or Riemannian index form, is the space of Jacobi fields vanishing at both endpoints. Hence a geodesic is a nondegenerate critical point exactly when that space is zero, or equivalently when its terminal endpoint is not a conjugate point to its initial endpoint along the given geodesic.

The Morse index theorem states that the index is the sum of the multiplicities of all conjugate points strictly inside the parameter interval:The multiplicity at the terminal endpoint gives nullity rather than an additional contribution to the index. These statements use the usual Sobolev $H^1$ completion of the path space; smooth paths have the same homotopy type.

First suppose $p\ne q$ as well as $p\ne -q$, and put $\theta =d(p,q)\in (0,\pi )$. There is a unique great circle through these two points. Every critical path runs along it with constant speed, possibly passing around it extra times. Ifthen a complete enumeration without repetitions isIts length is $L=|\ell |$. Equivalently the two positive length lists areThere are no other critical paths, because a nonconstant geodesic of the round sphere is a constant-speed great circle and its plane must contain $p,q$.

To compute conjugate points and indices of round-sphere geodesics, the round unit sphere has constant sectional curvature $1$. Along a geodesic of speed $L$, a normal Jacobi field in a parallel direction satisfies $y^{\prime \prime }+L^{2}y=0$. A field with $J(0)=0$ is therefore a constant multiple of $\sin (Lt)$ in each of the $n-1$ normal directions. Its tangential component satisfies $y^{\prime \prime }=0$ and contributes no endpoint-vanishing field. Consequently the conjugate points occur at $Lt=j\pi$, each with multiplicity $n-1$. Since neither length list contains an integer multiple of $\pi$, all the displayed critical paths are nondegenerate, and the Morse index theorem gives

The permitted case $p=q$ requires separate treatment. There is a constant geodesic, which has index and nullity zero. All other critical paths areFor each $r$, this is a family parametrized by $S^{n-1}$; both directions of traversal are included through $v$ and $-v$. The interior conjugate points number $2r-1$, and the endpoint is also conjugate. ThusThese are Morse-Bott critical manifolds for $n>1$, rather than nondegenerate critical points. For $n=1$ the same formulas give zero index and nullity, with the two isolated directions comprising $S^0$.

For the requested homology calculation, choose distinct nonantipodal endpoints. Concatenation with a fixed path back to the basepoint gives a homotopy equivalence between this fixed-endpoint path space and the based loop space $\Omega S^n$. We use the Morse cell-attachment theorem for geodesic energy: on a complete compact Riemannian manifold, the fixed-endpoint energy, when all its critical points are nondegenerate, gives a CW complex of the same homotopy type with one cell of dimension equal to the index of each critical point. One can obtain this theorem from finite-dimensional broken-geodesic approximations and ordinary Morse theory; critical energy values tend to infinity here.

The two index lists interleave to give exactly one cell in every dimension $j(n-1)$, $j\ge 0$. If $n>2$, these dimensions are separated by at least two. The cellular homology groups therefore have zero boundary maps, since no occupied cell dimension has an occupied dimension one lower. With integer coefficients,This determines the graded abelian groups. The argument does not need an identification of the multiplication on loop-space homology.

Finally apply the Freudenthal suspension theorem: for an $(n-1)$-connected based CW complex $X$, $n\ge 2$, the suspension homomorphismis an isomorphism for $k\le 2n-2$ and a surjection for $k=2n-1$. A sphere $S^n$ is $(n-1)$-connected, and its reduced suspension of a topological space is homeomorphic to $S^{n+1}$. HenceThe isomorphism is the suspension map. Its stable range is also reflected by the Morse theory cell structure of $\Omega S^{n+1}$: after its bottom $n$-cell, the next positive-dimensional cell has dimension $2n$. For the stated homology problem $n>2$, all the connectivity hypotheses apply.

Suppose the closed trapped surface $T$ were not wholly in the black hole. The stated consequence of strong asymptotic predictability then supplies a point $p\in {\dot{J}}^{+}(T)\cap {\mathcal{I}}^{+}$ lying on a future null generator $\gamma$ orthogonal to $T$, with no point conjugate to $T$ before $p$. Both future null expansions at $T$ are negative. The null focusing theorem therefore forces the expansion along $\gamma$ to diverge after finite affine parameter, producing a conjugate point before the generator reaches future null infinity. A null geodesic with such a point cannot continue to generate the achronal boundary ${\dot{J}}^{+}(T)$, contradicting the property of $p$. Hence no point of $T$ can communicate with future null infinity, and