math214:hw13
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math214:hw13 [2020/04/26 23:58] pzhou |
math214:hw13 [2020/04/29 10:24] pzhou |
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| Define coefficients | Define coefficients | ||
| $$ a_{ij}(t) = \la R(\dot \gamma(t), e_i(t)) \dot\gamma(t), | $$ a_{ij}(t) = \la R(\dot \gamma(t), e_i(t)) \dot\gamma(t), | ||
| - | 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) |
| - | $$ \ddot f_i(t) + \sum_{j} a_{ij}(t) | + | $$ \ddot f_j(t) + \sum_{i} f_i(t) |
| - | 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 $|\dot \gamma(t)|=1$. |
| $$ \ddot f_i(t) + K f_i(t) = 0, \quad \forall i=1,\cdots, n $$ | $$ \ddot f_i(t) + K f_i(t) = 0, \quad \forall i=1,\cdots, n $$ | ||
| - | 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 $\alpha_s(t)$, | + | 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 $\alpha_s(t)$, |
| 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.txt · Last modified: 2020/04/29 10:34 by pzhou
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