Lecture Notes

This is our last homework.

In this week, we discussed the variational approach of geodesics, mentioning the first and second variation formula, and then Jacobi field equation.

1. (2pt) Jacobi Equation in a nice coordinate. Let $\gamma: [0,1] \to M$ be a geodesic. Let $e_1, \cdots, e_n$ be parallel, orthonormal tangent vectors along $\gamma$, i.e. $e_i(t) \in T_{\gamma(t)} M$ and $\nabla_{\dot \gamma(t)} e_i(t)=0$, and $\la e_i(0), e_j(0) \ra = \delta_{ij}$ (enforced at one time, satisfied for all time). Let $J(t)$ be a Jacobi field along $\gamma$, with coefficients $$ J(t) = \sum_i f_i(t) e_i(t). $$ Define coefficients $$ a_{ij}(t) = \la R( e_i(t), \dot \gamma(t)) \dot\gamma(t), e_j(t) \ra $$ Prove that the Jacobi equations can be written as (Thanks to Mason to point out a sign error in the original eqn) $$ \ddot f_j(t) + \sum_{i} f_i(t) a_{ij}(t) = 0, \quad \forall j=1,\cdots, n $$

2. (2pt) Jacobi equation in constant sectional curvature. Let $M$ be a manifold with constant sectional curvature $K$ (recall the definition of sectional curvature on page 168 in [Ni]). Let $\gamma$ be a normalized geodesic, i.e $|\dot \gamma(t)|=1$. Let $J(t)$ be a Jacobi field, normal to the curve. (Thanks to Helge for pointing out this) Show that the Jacobi equation of $J$ in a parallel orthonormal basis (as in problem 1) become $$ \ddot f_i(t) + K f_i(t) = 0, \quad \forall i=1,\cdots, n $$

3. (2pt) Let $M$ be a Riemannian manifold, $\gamma: [0,1] \to M$ be a geodesic, $J(t)$ be a Jacobi field. Prove that there exists a family of geodesics $\alpha_s(t)$, for $s \in (-\epsilon, +\epsilon)$, such that $\alpha(0,t) = \gamma(t)$ and $\alpha_s(t)$ are geodesics, and $\partial_s|_{s=0} \alpha_s(t) = J(t)$.

4. (3pt) Let $\gamma: [0,1] \to M$ be a geodesic, and let $X$ be a Killing vector field on $M$, i.e a vector field whose flow induces isometry on $M$. Show that

• The restriction of $X(\gamma(s))$ of $X$ to $\gamma(s)$ is a Jacobi field along $\gamma$.
• If $M$ is connected, and there exists $p \in M$ with $\nabla_Y X|_p=0$ for all $Y \in T_p M$, and $X|_p=0$, then $X=0$ on $M$.

5. (1pt) Read Theorem 5.2.24 in [Ni]. Sketch the idea of the proof.