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5. (10 pt) Let $(M, g)$ be a Riemannian manifold, a closed geodesic is a geodesic $\gamma: [0, 1] \to M$ such that $\gamma(0)=\gamma(1)$ and $\dot \gamma(0) = \dot \gamma(1)$. | 5. (10 pt) Let $(M, g)$ be a Riemannian manifold, a closed geodesic is a geodesic $\gamma: [0, 1] \to M$ such that $\gamma(0)=\gamma(1)$ and $\dot \gamma(0) = \dot \gamma(1)$. | ||
- | * (7 pt) Let $M$ be a genus $g\geq 1$ surface ((see wiki https:// | + | * (7 pt) Let $M$ be a genus $g\geq 1$ surface, smooth, compact without boundary, orientable 2-dimensional manifold |
* (3 pt) Let $M = S^2$ and $g$ any smooth Riemannian metric. Is it true that there is always exists a closed geodesic on $M$? Try your best to give an argument. | * (3 pt) Let $M = S^2$ and $g$ any smooth Riemannian metric. Is it true that there is always exists a closed geodesic on $M$? Try your best to give an argument. | ||
6. (15 pt) Let $K \In \R^3$ be a knot, that is, a smooth embedded submanifold in $\R^3$ diffeomorphic to $S^1$. | 6. (15 pt) Let $K \In \R^3$ be a knot, that is, a smooth embedded submanifold in $\R^3$ diffeomorphic to $S^1$. | ||
- | * (10 pt) Can you construct a geodesically complete metric on $M = \R^3 \RM K$? i.e. a metric | + | * (10 pt) Can you construct a geodesically complete metric |
* (5 pt) Assume such metric exists, prove that for any point $p \in M$, there are infinitely many distinct geodesics $\gamma: [0,1] \to M$ with $\gamma(0)=\gamma(1)=p$. | * (5 pt) Assume such metric exists, prove that for any point $p \in M$, there are infinitely many distinct geodesics $\gamma: [0,1] \to M$ with $\gamma(0)=\gamma(1)=p$. | ||