Are the Two Equations for Push-Forwards in Differential Geometry Equivalent?

[dyn]

Hi. For a diffeomorphism between 2 manifolds Φ : M → N with a tangent vector v in M I have the following equation for the push-forward of v
( Φ_*v)f = v( Φ^*f) where Φ^* is the pull-back. I understand this equation but i have also come across the following equation for the push-forward
(Φ_*v)f = (Φ^-1)^*v (Φ*f ) . Surely these 2 equations are not the same. I'm confused.

 

[andrewkirk]

I assume that ##v## is a vector in a tangent space of ##M## and ##f## is a one-form in a cotangent space of ##N##.
Then the RHS of the second equation doesn't seem to make any sense. ##(\Phi^{-1})^*## needs as argument a one-form in the relevant cotangent space of ##M##, but what it is given ##v(\Phi^*f)##, which is a scalar. So the RHS is undefined - meaningless.

Where did you see that second formula? Perhaps it is a typo.

 

[Orodruin]

So the RHS is undefined - meaningless.

You can use the pullback of a one-form to define the pullback of an arbitrary p-form. For a 0-form this is rather uninteresting though so maybe not what was intended.

 

[Ben Niehoff]

I'm fairly sure what's intended is that ##v## is a vector on ##M## and ##f## is a function on ##N##. The action of ##v## on ##f## (if they were both on ##M##, which they are not) is defined by

$$v(f) \equiv df (v)$$
(or alternatively, that's the definition of ##df##, depending on which notions you've decided are more fundamental).

If ##f## is a function on ##N## given by ##f : N \to \mathbb{R} ; y \mapsto f(y)## for ##y \in N##, then the pullback ##\Phi^* f : M \to \mathbb{R}## is defined, for ##x \in M##, via

$$(\Phi^* f)(x) \equiv f(\Phi(x))$$
Now, as for your confusion about the equations. ##\Phi_* v## should be a vector field living on ##N##, and hence ##(\Phi_* v)(f)## should be a function living on ##N##. However, in your first equation, ##v(\Phi^* f)## is clearly a function on ##M##, not ##N##. Therefore, it is your second equation which is correct:

$$(\Phi_* v)(f) \equiv (\Phi^{-1})^* (v(\Phi^* f))$$
because now both sides of the equation live on ##N##. One must use ##(\Phi^{-1})^*## rather than ##\Phi_*##, because functions out of a space (as ##f## is) must be pulled back rather than pushed forward.

An interesting question arises when perhaps ##\Phi^{-1}## doesn't exist (for example, when ##M## has smaller dimension than ##N##, and ##\Phi## is an embedding). In this case ##\Phi_* v## is not defined on all of ##N##, but only on the portion of ##N## on which ##\Phi## is invertible. That is, ##\Phi_* v## is only defined on the image of ##\Phi##.

 

[andrewkirk]

##\Phi_* v## should be a vector field living on ##N##, and hence ##(\Phi_* v)(f)## should be a function living on ##N##. However, in your first equation, ##v(\Phi^* f)## is clearly a function on ##M##, not ##N##. Therefore, it is your second equation which is correct:

I suppose it depends on whether ##v## is a vector field or just a single vector in a single tangent space at point ##p\in M##. From the way the question is worded - not mentioning vector fields - I feel drawn to assume the latter.

In that case ##v(\Phi^* f)## is a scalar in the overarching field ##F##, being the directional derivative in direction ##v## of the scalar function ##(\Phi^* f):M\to F##, at point ##p##. The equation then asserts that that is equal to ##(\Phi_*v)f##, which is the directional derivative in direction ##\Phi_*v## of the scalar function ##f:N\to F##, at point ##\Phi(p)\in N##.

I don't know whether that equation is valid, but it is well-defined as an equality between two elements of the same field.

I think more context of the problem is needed to make a clear interpretation.

 

[dyn]

The vector v is a tangent vector in the manifold M. Of the 2 equations I quoted does one refer to a function in M and one refer to a function in N ? Do functions exist only in specific manifolds ?

 

[dyn]

I forgot to say ; thanks for all your replies.

 

[andrewkirk]

The vector v is a tangent vector in the manifold M. Of the 2 equations I quoted does one refer to a function in M and one refer to a function in N ? Do functions exist only in specific manifolds ?

Assuming that ##f## represents a function and not a one-form in this context, which seems likely, then both equations refer to both a function ##f## on ##N## and a function ##(\Phi^*f)## on ##M##. The latter is the 'pullback' of the former and, as Ben pointed out above, is defined by

$$(\Phi^*f)(p)=f(\Phi(p))$$

where ##p\in M##.