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Rauch comparison theorem second form

Statement

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $(M, g)$ be a Riemannian manifold of dimension $n \geq 2$ , let $T > 0$ and $γ : [0, T] \to M$ be a unit-speed geodesic, and let $J \neq 0$ be a normal Jacobi field along $γ$ whose initial data have scalar shape: $D_{t} J (0) = λ J (0) for some λ \in R .$ Let $Y$ be the normal Jacobi tensor with $Y (0) = id$ and $Y^{'} (0) = λ id$ , that is, $Y (t) w$ is the normal Jacobi field with value $w$ and covariant derivative $λ w$ at $t = 0$ , and let $t_{f} \in (0, T] \cup {+ \infty}$ be the first singular time of $Y$ in $[0, T]$ , with $t_{f} = + \infty$ if none occurs; this is the first focal time of the scalar initial shape on this segment. Suppose every sectional curvature of $M$ in a plane containing $\dot{γ} (t)$ is at least $k$ , for every time under consideration. Let $M_{k}$ be the simply connected space form of constant sectional curvature $k$ , let $γ_{k}$ be a unit-speed geodesic in $M_{k}$ , and choose a linear isometry $ι : T_{γ (0)} M \to T_{γ_{k} (0)} M_{k}$ carrying $\dot{γ} (0)$ to ${\dot{γ}}_{k} (0)$ . Put $u : = J (0)$ and $u_{k} : = ι u$ ; let $Φ$ be parallel transport along $γ_{k}$ and set $J_{k} (t) : = f_{k} (t) Φ_{t} u_{k}, f_{k} : = {cs}_{k} + λ {sn}_{k} .$ Then:

$f_{k} (t) > 0$ for every $t \in [0, T]$ with $t < t_{f}$ , so the model field $J_{k}$ does not vanish before $t_{f}$ on this segment;
$∣ J (t) ∣ \leq ∣ J_{k} (t) ∣$ for every $t \in [0, T]$ with $t < t_{f}$ , and the inequality extends to $t = t_{f}$ by continuity when $t_{f} \leq T$ .

The comparison makes no claim beyond $t_{f}$ : the tensor $Y$ is singular there, and the fact that this particular field $J$ may remain nonzero does not extend the estimate.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1], the manifold $(M, g)$ of dimension $n \geq 2$ , $T > 0$ and the unit-speed geodesic $γ : [0, T] \to M$ , the nonzero normal Jacobi field $J$ with $D_{t} J (0) = λ J (0)$ , the normal Jacobi tensor $Y$ with $Y (0) = id$ , $Y^{'} (0) = λ id$ , its first singular time $t_{f}$ on $[0, T]$ , and the constant-curvature- $k$ model with field $J_{k} = f_{k} Φ u_{k}$ under the isometry $ι$ of the statement.

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the sectional-curvature, constant-curvature and full curvature-symmetry interfaces. The Jacobi-field and parallel initial-value suppliers require no choice; the local matrix argument adds none.

[F1]

Riccati comparison for scalar initial shape (Riccati comparison for scalar initial shape): for a finite-dimensional real inner product space $E$ , continuous self-adjoint families $R_{1} \geq R_{2}$ and $C^{2}$ solutions $Y_{i}$ of $Y_{i}^{''} + R_{i} Y_{i} = 0$ with $Y_{i} (0) = id$ , $Y_{i}^{'} (0) = λ id$ , the operators $S_{i} = Y_{i}^{'} Y_{i}^{- 1}$ satisfy $S_{1} \leq S_{2}$ before the first singularity of $Y_{1}$ ; if $R_{2} = k id$ and $f = {cs}_{k} + λ {sn}_{k}$ , then $Y_{2} = f id$ , $f > 0$ on $[0, t_{1})$ and $∣ Y_{1} (t) w ∣ \leq f (t) ∣ w ∣$ for all $w \in E$ and $0 < t < t_{1}$ .

[F2]

Radial Jacobi data (Radial Jacobi tensor, Radial riccati equation, Jacobi field): in the parallel trivialization of the normal bundle along $γ$ the normal Jacobi fields with normal initial data satisfy the matrix equation $y^{''} + R_{γ} y = 0$ , where the curvature symmetries of Algebraic symmetries of the Riemann tensor, under [A1], make $R_{γ} (t)$ the self-adjoint endomorphism $w \mapsto P_{t}^{- 1} (R (P_{t} w, \dot{γ} (t)) \dot{γ} (t))$ of the normal space at $γ (0)$ . The radial Jacobi tensor $A$ has $A (0) = 0$ , $D_{t} A (0) = id$ and generates the fields with $J (0) = 0$ .

[F3]

Curvature form of the hypothesis (Sectional curvature, Riemann curvature four-tensor): for $w$ normal to $\dot{γ} (t)$ , $⟨ R_{γ} (t) w, w ⟩ = Rm (w, \dot{γ}, \dot{γ}, w) = \sec (w \land \dot{γ}) ∣ w ∣^{2}$ , so "every radial sectional curvature is at least $k$ " is exactly the Loewner bound $R_{γ} (t) \geq k id$ on the normal space.

[F4]

Existence, uniqueness and linearity (Jacobi field, Existence and uniqueness of jacobi fields from initial data): for prescribed $J (t_{0})$ , $D_{t} J (t_{0})$ there is exactly one Jacobi field along $γ$ ; consequently the map $w \mapsto Y (t) w$ is linear, the tensor $Y$ is the solution of the matrix initial value problem of [F1] with $R_{1} = R_{γ}$ on the normal space $N = {\dot{γ} (0)}^{⊥}$ .

[F5]

Parallel transport (Existence and uniqueness of parallel sections, Levi civita parallel transport preserves lengths angles and volume): parallel transport is an isometry and $Φ_{t} u$ is the unique parallel field with value $u$ at $0$ .

[F6]

Model functions and model space (Model functions solve the constant curvature jacobi equation, Comparison sine, cosine and cotangent functions, Constant sectional curvature and space form, Curvature tensor of constant sectional curvature): on the space form of constant curvature $k$ the curvature tensor is $R (X, Y) Z = k (g (Y, Z) X - g (X, Z) Y)$ ; and ${sn}_{k}^{''} + k {sn}_{k} = 0$ , ${cs}_{k}^{''} + k {cs}_{k} = 0$ , with ${sn}_{k} (0) = 0$ , ${sn}_{k}^{'} (0) = {cs}_{k} (0) = 1$ , ${cs}_{k}^{'} (0) = 0$ .

Proof

technique · direct: transport the given field to the normal space at $\gamma(0)$, exhibit the tensor equation $y''+R_\gamma y=0$ with initial data $\operatorname{id},\lambda\operatorname{id}$, and apply the in-run Riccati comparison for scalar initial shape against the scalar model $f_k$

1.1F2F3F4F5given

The tensor equation of the given field. [F2, F3, F4, F5, given] Let $N : = {\dot{γ} (0)}^{⊥}$ and, for each $t$ , identify $N_{t} : = {\dot{γ} (t)}^{⊥}$ with $N$ by parallel transport $P_{t}$ [F5]. By [F2] the transported field $y_{u} (t) : = P_{t}^{- 1} J (t)$ satisfies $y_{u}^{''} + R_{γ} y_{u} = 0$ with $R_{γ} (t)$ self-adjoint, and by the initial condition $y_{u} (0) = J (0) = u, y_{u}^{'} (0) = D_{t} J (0) = λ u .$ By [F3], the hypothesis of the theorem says $R_{γ} (t) \geq k {id}_{N}$ for every $t$ under consideration, in the Loewner order. By [F4] the prescription $Y (t) w : = P_{t}^{- 1} J_{w} (t)$ , where $J_{w}$ is the normal Jacobi field with value $w$ and derivative $λ w$ at $0$ defines a $C^{2}$ endomorphism-valued solution of the same matrix initial value problem, with $Y (0) = {id}_{N}, Y^{'} (0) = λ {id}_{N},$ and $y_{u} (t) = Y (t) u$ ; in particular $∣ J (t) ∣ = ∣ Y (t) u ∣$ for all $t$ , because parallel transport is an isometry [F5]. Moreover $Y (t)$ is singular exactly when some nonzero field of this scalar initial shape vanishes at $t$ , so $t_{f}$ is the first positive singular time of $Y$ ; and $u = J (0) \neq 0$ , since $u = 0$ would give $D_{t} J (0) = λ u = 0$ and hence $J \equiv 0$ by [F4].

1.2F5F6given

The model field. [F5, F6, given] Let $f_{k} = {cs}_{k} + λ {sn}_{k}$ . By [F6], $f_{k}^{''} + k f_{k} = 0$ , $f_{k} (0) = 1$ , $f_{k}^{'} (0) = λ$ . Put $u_{k} : = ι u$ as in the statement. On the space form $M_{k}$ , $J_{k} (t) : = f_{k} (t) Φ_{t} u_{k}$ satisfies $D_{t} J_{k} = f_{k}^{'} Φ_{t} u_{k}, D_{t}^{2} J_{k} = f_{k}^{''} Φ_{t} u_{k} = - k f_{k} Φ_{t} u_{k} = - k J_{k},$ and by the constant-curvature tensor identity of [F6] together with the normality of $J_{k}$ (preserved by the isometric parallel transport [F5]), $R (J_{k}, {\dot{γ}}_{k}) {\dot{γ}}_{k} = k (g ({\dot{γ}}_{k}, {\dot{γ}}_{k}) J_{k} - g (J_{k}, {\dot{γ}}_{k}) {\dot{γ}}_{k}) = k J_{k} .$ Hence $D_{t}^{2} J_{k} + R (J_{k}, {\dot{γ}}_{k}) {\dot{γ}}_{k} = 0$ : the field $J_{k}$ is a normal Jacobi field with $J_{k} (0) = u_{k}, D_{t} J_{k} (0) = λ u_{k}, ∣ J_{k} (t) ∣ = ∣ f_{k} (t) ∣ ∣ u ∣,$ since $ι$ and parallel transport are isometries.

2.1F1step 1.1step 1.2given

Riccati comparison with the scalar model. [F1, step 1.1, step 1.2, given] Apply the Riccati comparison for scalar initial shape [F1] with $E : = N, R_{1} : = R_{γ} \geq k {id}_{N} = : R_{2}, Y_{1} : = Y, Y_{2} : = f_{k} {id}_{N} .$ By step 1.1 the family $Y_{1}$ is a $C^{2}$ solution of $Y_{1}^{''} + R_{1} Y_{1} = 0$ with the initial data ${id}_{N}, λ {id}_{N}$ ; by step 1.2 the family $Y_{2} = f_{k} {id}_{N}$ is a $C^{2}$ solution of $Y_{2}^{''} + k Y_{2} = 0$ with the same initial data; and the curvature families are continuous, self-adjoint and ordered by step 1.1 and [F1]. The lemma gives that the first singular time $t_{2}$ of $Y_{2}$ satisfies $t_{2} \geq t_{1} = t_{f}$ (that is, $f_{k} > 0$ on $[0, t_{f})$ , since $Y_{2} = f_{k} {id}_{N}$ is singular exactly at the zeros of $f_{k}$ and $f_{k} (0) = 1$ ), and that $∣ Y (t) u ∣ \leq f_{k} (t) ∣ u ∣ (0 \leq t \leq T, t < t_{f}) .$

3.1step 1.1step 1.2step 2.1given∎

The norm comparison. [step 1.1, step 1.2, step 2.1, given] For $0 \leq t \leq T$ with $t < t_{f}$ , steps 1.1, 1.2 and 2.1 combine to $∣ J (t) ∣ = ∣ Y (t) u ∣ \leq f_{k} (t) ∣ u ∣ = ∣ J_{k} (t) ∣,$ because $f_{k} (t) > 0$ there. If $t_{f} \leq T$ , both sides depend continuously on $t$ , so the inequality passes to the limit $t ↑ t_{f}$ ; this covers in particular the case that the model field itself vanishes at $t_{f}$ . No comparison is asserted at or beyond $t_{f}$ : the tensor $Y$ is singular at $t_{f}$ , the logarithmic and matrix arguments underlying [F1] stop there, and a particular solution $J (t)$ may remain nonzero without extending the estimate. The value $λ = 0$ (purely `parallel' initial shape) and the case $n = 2$ (one-dimensional normal space) are included; the initial vector $u$ is nonzero as shown in step 1.1. Every field used here is fixed by the initial data and the comparison invokes only the suppliers named above, so the inherited ${A C}_{ω}$ of [A1] is not drawn on beyond its declaration.

Source locator

Eschenburg §3 states Rauch II (printed p.13) as: solutions of $J_{i}^{''} + R_{i} J_{i} = 0$ with $J_{i}^{'} (0) = 0$ , equal initial norms and $λ_{-} (R_{1}) \geq λ_{+} (R_{2})$ satisfy $∥ J_{1} ∥ \leq ∥ J_{2} ∥$ up to the first zero of $J_{1}$ ; this is the $λ = 0$ case of the statement above, and the general scalar shape $λ$ is the matched-asymptotic case explained in the same section before Remark 3.2. The in-run Riccati comparison for scalar initial shape supplies the matrix comparison; the published model functions supply $f_{k}$ and its positivity interval.

Depends on

Used by

Rigidity in rauch comparison Proposition

Dependency tree · two levels

58 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J.-H. Eschenburg, Comparison Theorems in Riemannian Geometry (standard reference, not scraped)
Ved Datar, Lectures on Riemannian Geometry (2025) (standard reference, not scraped)