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math214:hw13 [2020/04/26 23:58] 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 |