math214:hw13
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| Both sides previous revision Previous revision Next revision | Previous revision | ||
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math214:hw13 [2020/04/29 10:17] 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 | ||
| - | $$ a_{ij}(t) = \la R(\dot \gamma(t), e_i(t)) \dot\gamma(t), | + | $$ a_{ij}(t) = \la R( e_i(t), \dot \gamma(t)) \dot\gamma(t), |
| Prove that the Jacobi equations can be written as (Thanks to Mason to point out a sign error in the original eqn) | Prove that the Jacobi equations can be written as (Thanks to Mason to point out a sign error in the original eqn) | ||
| $$ \ddot f_j(t) + \sum_{i} f_i(t) a_{ij}(t) | $$ \ddot f_j(t) + \sum_{i} f_i(t) a_{ij}(t) | ||
| Line 19: | Line 19: | ||
| 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/hw13.1588180634.txt.gz · Last modified: 2020/04/29 10:17 by pzhou
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