Book on Curvature wrong or am I confused

[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

 

[morphism]

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

 

[conquest]

Ok thank you, I should probably doubt myself less!

 

[quasar987]

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

 

[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.

 

[conquest]

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