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 $$ a_{ij}(t) = \la R(\dot \gamma(t), e_i(t)) \dot\gamma(t), e_j(t) \ra $$ Prove that the Jacobi equations can be written as $$ \ddot f_i(t) + \sum_{j} a_{ij}(t) f_j(t) = 0, \quad \forall i=1,\cdots, n $$

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 $|\dot \gamma(t)|=1$. Show that the Jacobi equation 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 $\partial_s \alpha_s(t) = J(t)$.

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 there exists $p \in M$ with $\nabla_Y X|_p=0$ for all $Y \in T_p M$, then $X=0$ on $M$.

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