- #1
conquest
- 133
- 4
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=[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
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