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.

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.