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--- math214:final [2020/05/07 01:15]
pzhou
+++ math214:final [2026/02/21 14:41] (current)
@@ Line 35: / Line 35: @@
 . (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  such that for any $p \in M$, $v \in T_p M$, the geodesics with initial condition $(p,v)$ exists for inifinite long time? Describe your construction explicitly.
+  * (10 pt) Can you construct a geodesically complete metric ((For any point $p \in M$, the exponential map exists for the entire $T_p M$, https://en.wikipedia.org/wiki/Geodesic_manifold )) on $M = \R^3 \RM K$? i.e. a metric  such that for any $p \in M$, $v \in T_p M$, the geodesics with initial condition $(p,v)$ exists for inifinite long time? Describe your construction explicitly.
   * (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 $\gamma(0)=\gamma(1)=p$.

Lecture Notes