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The differential of the exponential map and conjugacy along a geodesic

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), let $p \in M$ , let $E_{p} \subseteq T_{p} M$ be the exponential domain and $\exp_{p} : E_{p} \to M$ the exponential map of The geodesic initial-value problem and the exponential map of the metric-selected connection, and identify $T_{h} (T_{p} M)$ with $T_{p} M$ by the canonical isomorphism for every $h \in T_{p} M$ (The differential of a smooth map).

(a) The differential in terms of Jacobi fields. Let $v \in T_{p} M$ , let $s_{1} > 0$ satisfy $s_{1} v \in E_{p}$ , and let $u \in T_{p} M$ . Let $γ : [0, s_{1}] \to M$ , $γ (t) : = \exp_{p} (t v)$ (well defined by star-shapedness of $E_{p}$ ), and let $J_{u}$ be the unique Jacobi field along $γ$ with $J_{u} (0) = 0$ and $D_{t} J_{u} (0) = u$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (c), Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (b)). Then $d (\exp_{p})_{t v} (t u) = J_{u} (t) (0 \leq t \leq s_{1});$ in particular $d (\exp_{p})_{s_{1} v} (s_{1} u) = J_{u} (s_{1})$ and, in the special case $s_{1} = 1$ , $v \in E_{p}$ , $d (\exp_{p})_{v} (u) = J_{u} (1)$ .

(b) Conjugacy as a kernel condition. With the notation of (a), $d (\exp_{p})_{s_{1} v}$ is injective if and only if no nonzero Jacobi field along $γ$ vanishes at $t = 0$ and at $t = s_{1}$ . Moreover $\ker d (\exp_{p})_{s_{1} v} = s_{1} {u \in T_{p} M : J_{u} (s_{1}) = 0}$ , so $\dim \ker d (\exp_{p})_{s_{1} v}$ equals the multiplicity of the conjugate instant $s_{1}$ whenever $γ (s_{1})$ is conjugate to $p$ along $γ$ (Jacobi fields, geodesic variations and conjugate instants along a geodesic clause (d)).

(c) Conjugate instants of a geodesic. Let $γ : I \to M$ be a smooth geodesic with $γ (0) = p$ , and let $s_{1} > 0$ with $s_{1} \in I$ and $s_{1} γ^{'} (0) \in E_{p}$ . Then $γ (s_{1})$ is conjugate to $γ (0) = p$ along $γ$ if and only if $d (\exp_{p})_{s_{1} γ^{'} (0)}$ is not injective.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold with metric-selected connection, a point $p$ , the exponential domain $E_{p}$ and map $\exp_{p}$ , a vector $v \in E_{p}$ and a vector $w \in T_{p} M$ .

[L1]

$E_{p}$ is open and star-shaped about $0$ ; for every $h \in E_{p}$ the curve $t \mapsto \exp_{p} (t h)$ on an interval containing $[0, 1]$ is the geodesic $γ_{p, h}$ with $γ_{p, h} (0) = p$ , $γ_{p, h}^{'} (0) = h$ , and $\exp_{p} (t h) = γ_{p, h} (t)$ ; the map $(t, p, h) \mapsto γ_{p, h} (t)$ is smooth on its domain (The geodesic initial-value problem and the exponential map of the metric-selected connection clauses (a)-(d)).

[L2]

For a smooth two-parameter map $Φ (r, s)$ , the variation field $\partial_{r} Φ$ of a family of geodesics is a Jacobi field along each longitudinal geodesic, the two covariant derivatives satisfy $D_{s} \partial_{r} Φ = D_{r} \partial_{s} Φ$ whenever the longitudinal curves are geodesics, and the differential of a smooth map is computed by the chain rule (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clauses (a)-(b), The differential of a smooth map, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[L3]

A Jacobi field along a geodesic is determined by its value and covariant derivative at one parameter; the initial-value map $J \mapsto (J (s_{0}), D_{s} J (s_{0}))$ is a linear isomorphism onto $T_{γ (s_{0})} M \oplus T_{γ (s_{0})} M$ (The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields clause (c)).

[L4]

A field $J$ along a geodesic is a Jacobi field when $D_{t}^{2} J = R (γ^{'}, J) γ^{'}$ ; conjugacy at a pair of instants means the existence of a nonzero Jacobi field vanishing there, and the multiplicity is the dimension of the space of such fields (Jacobi fields, geodesic variations and conjugate instants along a geodesic clauses (b) and (d)).

[L5]

The canonical identification $T_{h} (T_{p} M) ≅ T_{p} M$ of a vector space with its tangent space at a point, and the linearity of the differential in its second argument (The differential of a smooth map).

Proof

technique · direct

1.1L1given

(The two-parameter family.) Fix $v \in T_{p} M$ , $u \in T_{p} M$ and $s_{1} > 0$ with $s_{1} v \in E_{p}$ , and put $Φ (r, s) : = \exp_{p} (s (v + r u))$ for $s \in [0, s_{1}]$ and $∣ r ∣$ small. This is well defined and smooth: $E_{p}$ is open and star-shaped by [L1], for small $∣ r ∣$ the vector $v + r u$ is close to $v$ , hence $s_{1} (v + r u) \in E_{p}$ , and then $s (v + r u) = \frac{s}{s_{1}} (s_{1} (v + r u)) \in E_{p}$ for $0 \leq s \leq s_{1}$ . For each $r$ , $s \mapsto Φ (r, s)$ is the geodesic with initial tangent $v + r u$ at $s = 0$ by [L1]; its tangent field is $T : = \partial_{s} Φ$ , and $T (0, r) = v + r u$ .

2.1step 1.1L1L2algebra

(The variation field is Jacobi.) By [L2], $J (s) : = \partial_{r} Φ (0, s)$ is a Jacobi field along $s \mapsto Φ (0, s) = \exp_{p} (s v) = γ (s)$ , and $D_{s} J = D_{r} T$ . Its initial data are $J (0) = \partial_{r} Φ (0, 0) = 0$ because $Φ (r, 0) = \exp_{p} (0) = p$ for all $r$ , and $D_{s} J (0) = D_{r} T (0, 0) = \partial_{r} (v + r u) ∣_{r = 0} = u$ , because the curve $r \mapsto Φ (r, 0)$ is constant and hence $D_{r}$ at $s = 0$ is the ordinary derivative.

3.1step 2.1L2L5algebra

(Value at the parameter $t$ .) For $0 \leq t \leq s_{1}$ , by [L2] and the chain rule, $J (t) = \partial_{r} (\exp_{p} (t (v + r u))) ∣_{r = 0} = d (\exp_{p})_{t v} (t u)$ , the derivative of $r \mapsto t (v + r u)$ at $r = 0$ being $t u$ .

4.1step 3.1L3algebra

(Identification with $J_{u}$ .) By step 2.1, $J$ is a Jacobi field along $γ$ with initial data $(0, u)$ ; by the uniqueness and linearity of the initial-data correspondence [L3], $J = J_{u}$ . Combining with step 3.1, $d (\exp_{p})_{t v} (t u) = J_{u} (t)$ for every $t \in [0, s_{1}]$ ; at $t = 0$ both sides vanish. The cases $t = s_{1}$ and $s_{1} = 1$ are the stated special cases, so (a) holds.

5.1step 4.1L3L4algebra

(Kernel and conjugacy.) By step 4.1 with $t = s_{1}$ , $d (\exp_{p})_{s_{1} v} (s_{1} u) = J_{u} (s_{1})$ ; since $u \mapsto s_{1} u$ is a linear isomorphism, $\ker d (\exp_{p})_{s_{1} v} = {s_{1} u : J_{u} (s_{1}) = 0}$ . By [L3], $u \mapsto J_{u}$ is a linear isomorphism from $T_{p} M$ onto the Jacobi fields along $γ$ vanishing at $0$ (those with initial value $0$ ), so the kernel is nonzero exactly when some nonzero Jacobi field vanishes at $0$ and $s_{1}$ , and its dimension, namely $\dim {u : J_{u} (s_{1}) = 0}$ , is the multiplicity by [L4]. This proves (b).

6.1step 5.1L1L4algebra∎

(Geodesics starting at $p$ .) Let $γ : I \to M$ be a geodesic with $γ (0) = p$ and let $s_{1} γ^{'} (0) \in E_{p}$ . By [L1], $γ (s) = \exp_{p} (s γ^{'} (0))$ for those $s \in I$ with $s γ^{'} (0) \in E_{p}$ , and $s_{1}$ is such a value; so the claim is the special case of step 5.1 with $v = γ^{'} (0)$ . This proves (c) and completes the proof.

Remarks

No Riemannian positivity is used. The statement and proof use only the smooth exponential map, the geodesic-variation equation and the initial-value theory of Jacobi fields. Positivity of the metric on a normal bundle enters only in the separate index-form lemma The normal index form before the first conjugate instant is positive definite.
The base point. Formula (a) compares the differential at $t v$ with a Jacobi field along the geodesic from $p$ ; the factor $t$ in the argument is essential and is what makes the kernel computation of (b) exact. The convention $v \in E_{p}$ ensures that the entire radial segment $[0, v]$ is in the exponential domain, so no completeness of $(M, g)$ is assumed.

The differential of the exponential map and conjugacy along a geodesic

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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