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math214:hw13 [2020/04/26 23:58]
pzhou
math214:hw13 [2020/04/29 10:34] (current)
pzhou
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 $$ J(t) = \sum_i f_i(t) e_i(t). $$ $$ J(t) = \sum_i f_i(t) e_i(t). $$
 Define coefficients 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) 
-$$ \ddot f_i(t) + \sum_{j} a_{ij}(t) f_j(t) = 0, \quad \forall i=1,\cdots, n $$+$$ \ddot f_j(t) + \sum_{if_i(t) a_{ij}(t)  = 0, \quad \forall j=1,\cdots, n $$
  
-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 $|\dot \gamma(t)|=1$. Show that the Jacobi equation in a parallel orthonormal basis (as in problem 1) become+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
 $$ \ddot f_i(t) + K f_i(t) = 0, \quad \forall i=1,\cdots, n $$ $$ \ddot f_i(t) + K f_i(t) = 0, \quad \forall i=1,\cdots, n $$
  
  
-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 $\partial_s \alpha_s(t) = J(t)$. +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  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 $\gamma$.    * 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 $Y \in T_p M$, 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 $M$. 
  
 5. (1pt) Read Theorem 5.2.24 in [Ni]. Sketch the idea of the proof. 5. (1pt) Read Theorem 5.2.24 in [Ni]. Sketch the idea of the proof.
math214/hw13.1587970692.txt.gz · Last modified: 2020/04/26 23:58 by pzhou