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I was doing some exercises from the book on curvature by Lee to buff up my differential geometry. I came a cross the following question and it seems to me the question isn't completely correct, but I'm not so good at differential geometry that I am confident. Maybe someone else is!

the question is:

Suppose N ⊂ M is an embedded submanifold.

If X is a vector ﬁeld on M , show that X is tangent to N at points

of N if and only if Xf = 0 whenever f is a smooth function on M that

vanishes on N.

What looks to be wrong is Xf only needs to vanish at points of N not all of M.

I came up with the example:

the vector field X=[itex]\partial_x[/itex] + y[itex]\partial_y[/itex] on M=ℝ² where the submanifold N is the real line (so set y to 0).

It seems that although at points of N X=[itex]\partial_x[/itex] (so at p \in N)

which is tangent to N.

the smooth function f(x,y)=y which vanishes on the real line has Xf=y so this only vanishes on N not on all of M.

So the question is is their something wrong with this reasoning or is the question wrong?

Thanks

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# Book on Curvature wrong or am I confused

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