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 be a Riemannian manifold of negative curvature, a geodesic on , and a Jacobi field along with . If , then for all .
Indeed, we consider , which equals
I claim that this second derivative is negative, which will follow if we show that
But here we can use the Jacobi equation and the antisymmetry of the curvature tensor to turn into
(The last inequality is from the assumption of negative curvature.)
This proves the claim.
Now there are arbitrarily small with because , so in particular there must be arbitrarily small with . 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 is a regular map. Now if
where is the Jacobi field along the geodesic with . This is nonzero by what has just been proved, which establishes the claim and the Cartan-Hadamard theorem.