I am trying to prove that there is a smooth curve that connects two points on a sphere. I want to prove this by using the Implicit Function Theorem. (I know a lot of other ways, but I want to practice this way). Can someone help me? Thanks.
2026-03-26 21:12:25.1774559545
Smooth curve that connects two points on a sphere
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Perhaps you could find a smooth function $f$ which satisfies $f(0,0,1)=0$ and $f(0,0,-1)=0$ and then claim that the solutions of the simultaneous equations $f(x,y,z)=0$ and $x^2+y^2+z^2=1$ define a nice curve. After all, by definition, the solutions lie on the unit sphere. Does the implicit function theorem justify asserting that the solutions define a nice curve (1 dimensional submanifold of $\mathbb{R}^3$)?