Book on Curvature wrong or am I confused

1. Jan 11, 2012

conquest

Hi,

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=$\partial_x$ + y$\partial_y$ 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=$\partial_x$ (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

2. Jan 11, 2012

morphism

You're right, of course. The book should say "[...] if and only if Xf=0 on N [...]".

3. Jan 12, 2012

conquest

Ok thank you, I should probably doubt myself less!

4. Jan 12, 2012

quasar987

Lee has errata for all his books in his web site.

5. Jan 12, 2012

zhentil

Lee's also wonderful about answering emails with questions about his books. I spotted an error in that book and sent him an email, and he had emailed me back and posted the erratum within 24 hours.

6. Jan 12, 2012

conquest

Oh that's awesome! so next time I should check the website first and then e-mail!