Suppose that M, N are smooth manifolds and that M is connected. Let be differentiable so that for all . Show that f must be constant. Anyone got any hint? Why do we need M to be connected?
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If we don't require M to be connected, then we can let and
Ok, I see. Now how to start proving this?
Use proof by contradiction. Suppose that and show that this leads to a non-zero derivative.
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