math214:hw13
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math214:hw13 [2020/04/26 23:58] pzhou |
math214:hw13 [2020/04/27 10:18] pzhou |
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| $$ a_{ij}(t) = \la R(\dot \gamma(t), e_i(t)) \dot\gamma(t), | $$ 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 | ||
| - | $$ \ddot f_i(t) + \sum_{j} a_{ij}(t) | + | $$ \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 $|\dot \gamma(t)|=1$. Show that the Jacobi equation in a parallel orthonormal basis (as in problem 1) become | ||
math214/hw13.txt · Last modified: 2020/04/29 10:34 by pzhou
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