Need help understanding the pushforward

  • Thread starter Identity
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In summary, the two functions defined are F^* and F_{*p}. F^* maps a function f on C^\infty(F(p)) to f o F on C^\infty(p). F_{*p} takes a vector X at p and sends it to a vector F_{*p}X at F(p), which acts on the germ of a function f at F(p) by evaluating X on f o F. The term "germ" refers to a local topological structure, and the functions are defined in terms of this structure.
  • #1
Identity
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In my notes, the following two functions are defined:

Suppose [itex]M^m[/itex] and [itex]N^n[/itex] are smooth manifolds, [itex]F:M \to N[/itex] is smooth and [itex]p \in M[/itex]. We define:
[tex]F^*:C^\infty (F(p)) \to C^\infty (p)\ ,\ F^*(f) = f \circ F[/tex]
[tex]F_{*p}: T_pM \to T_{F(p)}N\ ,\ [F_{*p}(X)](f) = X(F^*f) = X(f \circ F)[/tex]

I understand the first function, [itex]F^*[/itex]; it maps [itex]f[/itex], a function on [itex]C^\infty(F(p))[/itex], to [itex]f \circ F[/itex], a function on [itex]C^\infty(p)[/itex].

However, I don't understand the second one, [itex]F_{*p}[/itex]. Since [itex]X(f) \in T_pM[/itex], it follows that [itex]f \in C^\infty (p)[/itex]. But then how is
[tex][F_{*p}(X)](f) = X(F^*f)[/tex]
defined? After all, in the definition of [itex]F_{*p}(X)[/itex], [itex]f[/itex] is a function on [itex] C^\infty (p)[/itex], not [itex]C^\infty(F(p))[/itex], so how can we evaluate [itex]F^*f[/itex]?
 
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  • #2
X(f) isn't in TpM -- X is.

[itex]F_\ast[/itex] takes X to [itex]F_\ast(X)[/itex]. The question now is, what is [itex]F_\ast(X)[/itex]? We want it to be an element of TF(p)N, i.e., it should be a point derivation at F(p) on N, i.e., you need to know how to evaluate Fp(X) at smooth germ f at F(p).
 
  • #3
So F*p takes a vector X at p (derivation on the germs of smooth function at p) and sends it to a vector F*pX at F(p) (derivation on the germs of smooth function at F(p)). So what is this vector F*pX then? How does it act on a germ f at F(p)? This is what the formula is telling you: it says (F*pX)(f) is just X(F*f). And this makes sense, since F*f =f o F is indeed a germ of functions at p.
 
  • #4
Sorry, I've never heard of the term 'germ' before, can you explain please?
 
  • #5
Identity said:
Sorry, I've never heard of the term 'germ' before, can you explain please?

A germ is essentially just a local topological structure.
 
  • #6
Identity said:
In my notes, the following two functions are defined:

Suppose [itex]M^m[/itex] and [itex]N^n[/itex] are smooth manifolds, [itex]F:M \to N[/itex] is smooth and [itex]p \in M[/itex]. We define:
[tex]F^*:C^\infty (F(p)) \to C^\infty (p)\ ,\ F^*(f) = f \circ F[/tex]
[tex]F_{*p}: T_pM \to T_{F(p)}N\ ,\ [F_{*p}(X)](f) = X(F^*f) = X(f \circ F)[/tex]

I understand the first function, [itex]F^*[/itex]; it maps [itex]f[/itex], a function on [itex]C^\infty(F(p))[/itex], to [itex]f \circ F[/itex], a function on [itex]C^\infty(p)[/itex].

However, I don't understand the second one, [itex]F_{*p}[/itex]. Since [itex]X(f) \in T_pM[/itex], it follows that [itex]f \in C^\infty (p)[/itex]. But then how is
[tex][F_{*p}(X)](f) = X(F^*f)[/tex]
defined? After all, in the definition of [itex]F_{*p}(X)[/itex], [itex]f[/itex] is a function on [itex] C^\infty (p)[/itex], not [itex]C^\infty(F(p))[/itex], so how can we evaluate [itex]F^*f[/itex]?

Analytically, a tangent vector at a point,p, on a manifold is a linear operator that acts on differentiable functions defined in an open neighborhood of p. A function,g, on N composed with F is a function on M. So a tangent vector at p now acts on the composition of g with F. But this may also be viewed at an action on g at F(p).
 
  • #7
Thanks everyone :)
 
  • #8
Identity said:
Sorry, I've never heard of the term 'germ' before, can you explain please?

I assumed that by [itex]C^{\infty}(p)[/itex] you mean the set of real-valued functions f that are defined and smooth on some neighborhood U of p, modulo the equivalence relations according to which f~g iff f and g coincide on some small nbhd of p.

If so, then the elements of [itex]C^{\infty}(p)[/itex] are called germs of smooth functions.
 
  • #9
Just to make sure I've got it, in

[tex][F_{*p}(X)](f) = X(f\circ F)[/tex]
[itex]f[/itex] is kind of placeholder, in the sense that the [itex]f[/itex] on the LHS is an arbitrary function in [itex]C^\infty(p)[/itex] and the [itex]f[/itex] on the RHS is an arbitrary function in [itex]C^\infty (F(p))[/itex]So on the left and right sides of the equation, [itex]f[/itex] does not represent functions with the same germs. I think this is where I got confused.
 
  • #10
Identity said:
Just to make sure I've got it, in

[tex][F_{*p}(X)](f) = X(f\circ F)[/tex]
[itex]f[/itex] is kind of placeholder, in the sense that the [itex]f[/itex] on the LHS is an arbitrary function in [itex]C^\infty(p)[/itex] and the [itex]f[/itex] on the RHS is an arbitrary function in [itex]C^\infty (F(p))[/itex]

The f in the LHS is the same as the f in the RHS, and in both case, it is a function in [itex]C^\infty (F(p))[/itex]. Indeed, it better be so that f o F is in [itex]C^\infty(p)[/itex] so that X(f o F) makes sense!
 

What is the pushforward?

The pushforward, also known as the pushforward map or pushforward operator, is a mathematical concept used in differential geometry and physics. It is a way to map vectors from one tangent space to another, typically defined by a smooth differentiable mapping between manifolds. In simpler terms, it is a way to transport vectors from one point to another on a curved surface.

What is the purpose of the pushforward?

The pushforward is used to relate tangent spaces at different points on a manifold. This is important in many areas of mathematics and physics, such as in the study of curves and surfaces, where it allows for the calculation of derivatives and other geometric quantities. It also plays a crucial role in the theory of differential equations and the study of dynamical systems.

How is the pushforward related to the pullback?

The pushforward and the pullback are related by the concept of a dual map. The pullback is the dual map of the pushforward, meaning that it maps differential forms from one cotangent space to another. In other words, while the pushforward maps vectors from one tangent space to another, the pullback maps differential forms from one cotangent space to another.

What are some applications of the pushforward?

The pushforward has many applications in mathematics and physics. It is used in the study of curves and surfaces, differential equations, dynamical systems, and Lie groups. It also plays a crucial role in the theory of relativity and other areas of theoretical physics.

How is the pushforward calculated?

The pushforward is calculated using the Jacobian matrix, which is a matrix of partial derivatives of the mapping between manifolds. The Jacobian matrix is then applied to the vector that is being pushed forward, resulting in a new vector in the target tangent space. In certain cases, the pushforward can also be calculated using the Lie derivative, which is a special case of the Jacobian matrix for vector fields.

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