Apologies for some initial bugs in the formulas–I have now corrected them.

Today I will prove the Cartan-Hadamard theorem.

Nonvanishing of Jacobi fields

The key lemma is that (nontrivial) Jacobi fields do not vanish.

Lemma 1 Let {M} be a Riemannian manifold of negative curvature, {\gamma: [0,M]} a geodesic on {M}, and {J} a Jacobi field along {\gamma} with {J(0)=0}. If {\frac{D}{dt}V(t)|_{t=0} \neq 0}, then {J(t) \neq 0} for all {t > 0}.

Indeed, we consider { \frac{d^2}{dt^2} \left \langle J(t), J(t)\right \rangle}, which equals

\displaystyle 2\frac{d}{dt} \left \langle \frac{D}{dt} J(t), J(t) \right \rangle = 2\left \langle \frac{ D^2}{dt^2} J(t), J(t)\right \rangle + 2 \left| \frac{D}{dt} V(t) \right|^2.

I claim that this second derivative is negative, which will follow if we show that

\displaystyle \left \langle \frac{ D^2}{dt^2} J(t), J(t)\right \rangle \geq 0.

 But here we can use the Jacobi equation and the antisymmetry of the curvature tensor to turn {\left \langle \frac{ D^2}{dt^2} J(t), J(t)\right \rangle } into

\displaystyle \left \langle R(\dot{\gamma}(t), J(t)) \dot{\gamma(t)}, J(t) \right \rangle = -\left \langle R( J(t),\dot{\gamma}(t) ) \dot{\gamma(t)}, J(t) \right \rangle \geq 0.

 (The last inequality is from the assumption of negative curvature.)

This proves the claim.

Now there are arbitrarily small {t} with { \left \langle J(t), J(t) \right \rangle \neq 0} because {\frac{D}{dt} J(t)|_{t=0} \neq 0}, so in particular there must be arbitrarily small {t} with { \frac{d}{dt} \left \langle J(t), J(t) \right \rangle > 0}. In particular, this derivative is always positive. This proves the claim.

I followed Wilkins in the proof of this lemma.

Proof of the Cartan-Hadamard theorem

By yesterday’s post, it’s only necessary to show that {\exp_p} is a regular map. Now if {X,Y \in T_p(M)}

\displaystyle d(\exp_p)_X(Y) = J(1)

 where {J} is the Jacobi field along the geodesic {\gamma(t) = \exp_p(tX)} with {J(0)=0, \frac{D}{dt} J(t)|_{t=0} = Y}. This is nonzero by what has just been proved, which establishes the claim and the Cartan-Hadamard theorem.

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