Pullback and Pushforward in Manifolds: Why Do We Do It?

[observer1]

In my ignorance, when first learning, I just assumed that one pushed a vector forward to where a form lived and then they ate each other.

And I assumed one pulled a form back to where a vector lived (for the same reason).

But I see now this is idiotic: for one does the pullback and pushforward between two DIFFERENT manifolds. But forms and vectors live on the cotangent and tangent space of the SAME manifold.

So, that being the case... WHY does one do this?

I can sort of intuit why ones does this for functions (pulling back a function to a simpler place).

But for forms? Why does one WANT to pull back a form?
Why does one WANT to push a vector forward?

Examples in dynamcis are most welcome (if possible).

 

[Orodruin]

One of the more intuitive examples would be the induced metric on a submanifold. Embedding a manifold in a higher-dimensional one will induce a metric on the embedded space which is the pullback of the metric tensor in the higher-dimensional one. Example: The embedding of Earth's surface in $\mathbb R^3$.

For the same reason, you can push vectors from the submanifold to the embedding manifold.

 

[observer1]

One of the more intuitive examples would be the induced metric on a submanifold. Embedding a manifold in a higher-dimensional one will induce a metric on the embedded space which is the pullback of the metric tensor in the higher-dimensional one. Example: The embedding of Earth's surface in $\mathbb R^3$.

For the same reason, you can push vectors from the submanifold to the embedding manifold.

OK, now that is interesting. And I get it. But that is an obscure example. Can you provide one from, say, dynamics?

I mean, in the case of continuum mechanics, and the stress tensor, I can "intuit" (as I become more adept at this -- which I am not, right now), that the two manifolds in question could be the original and deformed configuration, and one wants to see what happens to vectors between them. (I can't imagine how FORMS would come into play, but I can wait). Do you have any examples like that? Dynamics? Continuum Mechanics, Fluid Mechanics, Fracture Mechanics?

 

[Orodruin]

But that is an obscure example.

In which sense is this an obscure example? Embeddings of manifolds are among the more hands-on things you can do. If you want dynamics, just take a particle moving on a sphere. The configuration space is then a sphere - a submanifold of $\mathbb R^3$ and the inertia tensor is just the induced metric (multiplied by the particle mass).

 

[observer1]

In which sense is this an obscure example? Embeddings of manifolds are among the more hands-on things you can do. If you want dynamics, just take a particle moving on a sphere. The configuration space is then a sphere - a submanifold of $\mathbb R^3$ and the inertia tensor is just the induced metric (multiplied by the particle mass).

Ah... now I see... thank you!