Comparison Theory
Throughout, manifolds and maps are smooth unless stated otherwise. On a Riemannian manifold, \(\nabla\) denotes the Levi–Civita connection. The needed local constructions start with coordinate charts.
A curvature bound estimates the separation of nearby geodesics. Write \(T=\gamma'\) for a unit-speed geodesic and \(P_t:T_{\gamma(0)}M\to T_{\gamma(t)}M\) for parallel transport. In constant curvature \(k\), decompose \(v=v^\top+v^\perp\) relative to \(T(0)\). The Jacobi field with \(J(0)=0,\ J'(0)=v\) is
Indeed, the tangential Jacobi equation is \(j''=0\), while the normal equation is \(j''+kj=0\). Thus the familiar formulas \(\sin(t)P_tv,\ tP_tv,\ \sinh(t)P_tv\) for the unit sphere, Euclidean space, and hyperbolic space apply to normal initial vectors. They express the initial separation and subsequent convergence on the sphere, linear separation in Euclidean space, and exponential separation in hyperbolic space. The results below compare variable curvature with these scalar model equations; see also Cheeger and Ebin (1975).
1. Rauch Comparison Theorem
We begin with comparison against scalar model solutions, then prove Rauch’s theorem by comparing index forms. This separates the explicit model calculations from the argument that applies to general manifolds. In what follows, for \(\delta \in \mathbb R\), let
Note that these are the solutions to the Jacobi equation for constant sectional curvature \(\delta\). As usual, let \(\gamma(t)\) be a geodesic on \(M\) parametrized by arclength, and \(J(t)\) denote a Jacobi field along \(\gamma(t)\).
Theorem 1. Let \(K\leq\delta\) along the unit-speed geodesic \(\gamma\), and let \(J\) be a Jacobi field. Assume either \(J\perp T\) or \(\delta\geq0\). Put \(a=|J(0)|,\ b=(|J|)'(0+)\), and \(f_\delta=ac_\delta+bs_\delta\). When \(J(0)=0\), the right derivative is \(b=|J'(0)|\). Suppose \(f_\delta>0\) on \((0,\tau)\). Then
and \(|J(t)|\geq f_\delta(t)\). The inequalities without quotients extend to finite endpoints where the fields are defined.
Proof. Set \(h=|J|\) on an interval on which \(J\neq0\). Our curvature convention is \(J''+R(T,J)T=0\), so
The first numerator is nonnegative by Cauchy–Schwarz. For a normal field the second is nonnegative by the curvature bound. For a field with a tangential component, \(R(T,J)T=R(T,J^\perp)T\), and \(\langle R(T,J)T,J\rangle\leq\delta|J^\perp|^2\leq\delta|J|^2\) when \(\delta\geq0\).
Let \(W=h'f_\delta-hf_\delta'\). The scalar model equation gives
The initial limit of \(W\) is zero. If \(a>0\), \(h/f_\delta\to1\); if \(a=0\), positivity of the model forces \(b>0\), and \(J(t)=tP_tJ'(0)+O(t^2)\), so the same limit holds. Consequently \(h/f_\delta\geq1\) and is nondecreasing up to the first possible zero of \(J\). Such a zero strictly before \(\tau\) would contradict \(h\geq f_\delta>0\) by continuity. This proves the result on the whole interval. Multiplying \(W\geq0\) by \(h\) gives the first displayed inequality, and continuity gives the endpoint assertions. ◻
We can also work with manifolds with a lower bound on their sectional curvature.
Theorem 2. Let \(\gamma\) be a unit-speed geodesic with \(\lambda\leq K\leq\mu\), and suppose \(J(0)\) and \(J'(0)\) are linearly dependent. Assume that \(J\) is normal, or that \(\lambda\leq0\leq\mu\). Write
Suppose \(s_r>0\) on \((0,\tau)\) and \(f_r\geq0\) on \([0,\tau)\). Then \(|J(t)|\leq f_\lambda(t)\) for \(0\leq t<\tau\), and at a finite endpoint by continuity.
The condition on \(f_r\) matters: even in a flat space, \(J(t)=(1-t)P_tv\) would otherwise give the false bound \(|1-t|\leq1-t\) after \(t=1\). For a field with a tangential component, the condition \(0\in[\lambda,\mu]\) ensures that the curvature operator's zero tangential eigenvalue lies in the same spectral interval.
Proof. Use parallel transport to identify the tangent spaces with one Euclidean space, and put \(A(t)Z=R(T,Z)T\). On the normal space, \(A\) is self-adjoint with eigenvalues in \([\lambda,\mu]\); on the whole tangent space this remains true under \(\lambda\leq0\leq\mu\). Hence \(\|rI-A(t)\|\leq\eta\).
Let \(V''+rV=0\) have the initial values of \(J\). Linear dependence of the initial vectors gives \(V=f_re\) for a fixed unit vector \(e\), with the zero field handled separately. Our sign assumption gives \(|V|=f_r\). Variation of constants, applied to \(H=J-V\), gives
Since the kernel is nonnegative for \(0\leq u\leq t<\tau\),
Thus \(g(0)=g'(0)=0\), and
Now \(f=f_\lambda-f_r\) solves \(f''+\lambda f=\eta f_r\) with zero initial values. The kernel \(s_\lambda\) is also nonnegative on \([0,\tau)\): if \(\lambda>0\), its first positive zero is no earlier than that of \(s_r\), since \(\lambda\leq r\); if \(\lambda\leq0\), it has no positive zero. Therefore
It follows that \(|J|\leq |V|+|H|\leq f_r+g\leq f_\lambda\). This also supplies explicitly the scalar comparison step in the auxiliary-ODE argument. ◻
The Rauch comparison theorem, in its original form, is the following statement on comparing Jacobi fields.
Theorem 3 (Rauch comparison). Let \(\dim M=d\leq\dim\widetilde M\), and let \(\gamma,\widetilde\gamma\) be unit-speed geodesics on \([0,a]\). Let \(J,\widetilde J\) be Jacobi fields with
Suppose \(\widetilde\gamma(t)\) is not conjugate to \(\widetilde\gamma(0)\) for \(0<t\leq a\), and that every radial plane along \(\widetilde\gamma\) has sectional curvature at least that of every radial plane along \(\gamma\) at the same time. Then \(|\widetilde J(t)|\leq|J(t)|\).
If their common initial normal derivative is nonzero and equality holds at some \(t_0>0\), then
At \(t=0\), interpret the planes using the nonzero normal parts of the initial derivatives. Purely tangential fields give equality of lengths without any curvature rigidity. The same length comparison holds for equal nonzero constant speeds by a common change of parameter; for zero speed the fields are linear in a fixed tangent space and the assertion is immediate.
Proof. The tangential parts are \(t\alpha T\) and \(t\alpha\widetilde T\), with the same scalar \(\alpha\). Subtract them; their squared lengths agree, so it suffices to compare the normal parts. If their common initial norm is zero, uniqueness for the Jacobi equation makes both normal parts zero. Otherwise \(\widetilde J\neq0\) on \((0,a]\).
Work initially before the first possible zero of \(J\). Fix \(b\) in this interval and set
Integration by parts in the Jacobi equation yields
where
There is a parallel linear isometric embedding \(\Phi\) of the normal bundle of \(\gamma\) into that of \(\widetilde\gamma\), chosen so that \(\Phi U(b)=\widetilde U(b)\). To construct it, choose orthonormal frames at \(b\) starting with these unit vectors and extend by parallel transport. It preserves derivatives and lengths. Curvature comparison and the index lemma give, respectively,
The first inequality uses the same endpoint values and the absence of conjugate points on the higher-curvature geodesic. Hence
The ratio tends to one as \(b\downarrow0\), by the common nonzero initial norm. It is therefore at least one and nondecreasing. A first zero of \(J\) would contradict this lower bound, so the comparison holds throughout \((0,a]\). Restore the equal tangential squared lengths to conclude.
If equality holds at \(t_0\), the normal ratio is identically one up to \(t_0\). Both index inequalities at \(b=t_0\) are equalities. Strictness of the index lemma for a zero-endpoint difference gives \(\Phi U=\widetilde U\) on the whole segment. Moreover,
The continuous integrand is nonnegative and \(U(t)\neq0\) for \(t>0\); it must vanish pointwise. This proves the asserted curvature equality, including the initial limiting planes by continuity. ◻
Proposition 4. Let \(M\) be a complete Riemannian manifold of dimension at least two, with \(0<L\leq K\leq H\). Along a unit-speed geodesic starting at \(p\), the first positive conjugate time to \(p\), denoted \(\tau_p\), satisfies
These are lengths along the geodesic. One may restart the assertion at any point of the geodesic; it does not assert a lower bound for successive members of the list of all points conjugate to one fixed base point.
Proof. The lower bound comes from comparing \(M\) to \(\mathbb S^d (H)\), i.e. \(d\)-sphere of constant sectional curvature \(H\). Take some \(p \in M\). Let \(\gamma : [0,a] \to M\) be a normalized geodesic in \(M\) with \(\gamma(0) = p\), and let \(J\) be a Jacobi field along \(\gamma\) with \(J(0) = 0\) and \(\langle J, \gamma' \rangle = 0\). To start the comparison, choose \(\widetilde p \in \mathbb S^d (H)\) and a normalized geodesic \(\widetilde \gamma : [0,a] \to \mathbb S^d (H)\) such that \(\widetilde \gamma (0) = \widetilde p\). Similarly, let \(\widetilde J\) be a Jacobi field along \(\widetilde \gamma\) such that \(\widetilde J(0) = 0\), \(\langle \widetilde J, \widetilde \gamma' \rangle = 0\), and \(|J'(0)| = |\widetilde J'(0)|\). By the usual argument, \(\widetilde \gamma\) does not have a conjugate point to \(\widetilde \gamma(0)\) on the interval \(t \in (0,\frac{\pi}{\sqrt{H}})\), therefore, by the Rauch comparison theorem (Theorem 3), we have that \(|J(t)| \geq |\widetilde J(t)| > 0\) on \(t \in (0,\frac{\pi}{\sqrt{H}})\). Therefore, the distance from \(p\) to its first conjugate point is no less than \(\frac{\pi}{\sqrt{H}}\).
The upper bound comes from comparing \(\mathbb S^d(L)\) to \(M\). By way of contradiction, suppose \(\tau_p > \frac{\pi}{\sqrt{L}}\). We apply the Rauch comparison theorem with \(\mathbb S^d(L)\) as \(M\) and our manifold as \(\widetilde M\) (since \(\widetilde K = K_M \geq L = K_{\mathbb S^d(L)}\)). Then, \(|\widetilde J(t)| \leq |J(t)|\) for \(t \in (0, \frac{\pi}{\sqrt{L}}]\). Since \(\tau_p > \frac{\pi}{\sqrt{L}}\), \(\widetilde J(t)\) does not vanish on \((0, \frac{\pi}{\sqrt{L}}]\). In particular, \(0 < |\widetilde J(\frac{\pi}{\sqrt{L}})| \leq |J(\frac{\pi}{\sqrt{L}})|\). But \(J(t)\) on \(\mathbb S^d(L)\) vanishes strictly at \(\frac{\pi}{\sqrt{L}}\), so \(|J(\frac{\pi}{\sqrt{L}})| = 0\), which is a contradiction. ◻
Proposition 5. Let \(M\) and \(\widetilde M\) be differentiable manifolds. Suppose for all \(p \in M\) and \(\widetilde p \in \widetilde M\), \(\sigma \subset T_p M\) and \(\widetilde \sigma \in T_{\widetilde p} \widetilde M\), we have \(\widetilde K (\widetilde \sigma) \geq K(\sigma)\). Consider a linear isometry \(i: T_p M \to T_{\widetilde p} \widetilde M\). Let \(r > 0\) be such that the restriction \(\exp_p|_{B_r(0)}\) is a diffeomorphism, and \(\exp_{\widetilde p}|_{\widetilde B_r (0)}\) is a diffeomorphism onto its image. Let \(c:[0,a] \to \exp_p (B_r (0))\) be a differentiable curve, and let \(\widetilde c : [0,a] \to \exp_{\widetilde p} (\widetilde B_r (0))\) be defined as
Then, \(L(c) \geq L(\widetilde c)\).
Here “linear isometry” may mean an isometric embedding into a higher-dimensional tangent space. Require the exponential maps on both displayed balls to be diffeomorphisms onto their images, so the model radial segments have no conjugate points. These are normal neighborhoods; mere set-theoretic invertibility is not being used as a substitute for nonsingularity.
Proof. This follows from an application of the Rauch comparison theorem. To start, let \(c_0 : [0,a] \to B_r(0)\) be the unique curve in \(B_r(0)\) such that \(\exp_p (c_0 (s)) = c(s)\). This is possible because the restriction of \(\exp_p\) to \(B_r(0)\) is a diffeomorphism. Now, consider the rectangle \(\Gamma(s,t) : (s,t) \mapsto \exp_p (t c_0(s))\). Take the variation field \(V_s\) of \(\Gamma(s,t)\), which is a Jacobi field along \(\gamma_s : t \mapsto \exp_p (t c_0(s))\), with the initial conditions \(V_s(0) = 0\) and \(V_s (1) = c'(s)\). Furthermore, this implies
As such, \(\nabla_{\dif/\!\dif t} V_s(0) = c'_0 (s)\). Repeat the same arguments for the rectangle \(\widetilde \Gamma (s,t) = \exp_{\widetilde p} (t i (c_0 (s)))\): there exists a Jacobi field \(\widetilde V_s (t)\) along \(\widetilde \gamma_s : t \mapsto \exp_{\widetilde p} (t i (c_0(s)))\) such that \(\widetilde V_s (0) = 0\), \(\widetilde V_s (1) = \widetilde c' (s)\), and \(\nabla_{\dif/\!\dif t} \widetilde V_s(0) = i ( c'_0 (s))\). Because a linear isometry preserves lengths, it follows that \(| c'_0 (s) | = | i ( c'_0(s) ) |\): observe that
We are now ready to apply the Rauch comparison theorem: from a direct application, we get
Integrating both sides of (3) from \(0\) to \(a\) gives the desired result. ◻
If \(c_0(s)\neq0\), reparameterize both radial geodesics by their common speed before applying Theorem 3. If \(c_0(s)=0\), both differentials of the exponential maps at zero are the identity, so \(|\widetilde c'(s)|=|c'(s)|\) directly. Thus the pointwise speed comparison, and hence its integral, covers every parameter value.
Reading: Carmo (1992), Riemannian geometry.