math214:final
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| 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 $\gamma(0)=\gamma(1)=p$. | ||
math214/final.1588814116.txt.gz · Last modified: 2020/05/06 18:15 by pzhou
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