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--- math214:hw13 [2020/04/26 22:15]
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+++ math214:hw13 [2020/04/29 10:34] (current)
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 In this week, we discussed the variational approach of geodesics, mentioning the first and second variation formula, and then Jacobi field equation.
-. 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
+. (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
-$$ a_{ij}(t) = \la R(\dot \gamma(t), e_i(t)) \dot\gamma(t), e_j(t) \ra $$
+$$ a_{ij}(t) = \la R( e_i(t), \dot \gamma(t)) \dot\gamma(t), e_j(t) \ra $$
-Prove that the Jacobi equations can be written as
+Prove that the Jacobi equations can be written as (Thanks to Mason to point out a sign error in the original eqn)
-$$ \ddot f_i(t) + \sum_{j} a_{ij}(t) f_j(t) = 0, \quad \forall i=1,\cdots, n $$
+$$ \ddot f_j(t) + \sum_{i} f_i(t) a_{ij}(t)  = 0, \quad \forall j=1,\cdots, n $$
-. 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
+. (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
 $$ \ddot f_i(t) + K f_i(t) = 0, \quad \forall i=1,\cdots, n $$
-. (3pt) 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 $\partial_s \alpha_s(t) = J(t)$.
+. (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 $\partial_s|_{s=0} \alpha_s(t) = J(t)$.
 . (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 $M$.
+  * 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 $M$.
+. (1pt) Read Theorem 5.2.24 in [Ni]. Sketch the idea of the proof.

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