Let $(M^n, m)$ be a Riemannian manifold. A Jacobi field along a geodesic $ \gamma$ is an element $c \in \Gamma_{ \gamma}(TM)$ such that

$$ \nabla_T( \nabla_T(c )) + R^{\nabla}( c, \gamma’)( \gamma’) = 0.$$

We prove that that the (vector) space $ \operatorname{Jac}(\gamma)$ of Jacobi fields along $ \gamma$ has dimension $2n$. We then show that the Jacobi field equation is the linearisation of the geodesic equation, and use this to prove the Gauss Lemma.

Remarks
  • For those of you intending to take the Differential Geometry II exam (or the summer session of the Differential Geometry I exam), there is now an “exam info” page on my forum.
  • As announced in lecture today, I will run some form of reading course/semester project/seminar next semester on Gauge Theory (since I didn’t have time to cover it in this course). This will run next semester—please sign up on the relevant page on my forum if you are interested.

Comments and questions?