Suppose thatM,Nare smooth manifolds and thatMis connected. Let be differentiable so that for all .

Show thatfmust be constant.

Anyone got any hint? Why do we need M to be connected?

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- October 7th 2010, 06:15 AMEinStoneDerivative 0 => constant function
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? - October 7th 2010, 06:52 AMDefunkt
If we don't require M to be connected, then we can let and

- October 7th 2010, 10:18 AMEinStone
Ok, I see. Now how to start proving this?

- October 8th 2010, 07:15 AMHallsofIvy
Use proof by contradiction. Suppose that and show that this leads to a non-zero derivative.