On this blog, I had often been chewing on a piece of mathematical physics, saying the same thing over and over again, with math, without math, in different ways, deriving A from B and back. Re-developing well-known science, making it mine. Everybody remembers the spheres in trillions of dimensions, right?
I am not sure who should be the audience for this, except myself. Maybe students who happen to mull over the weekly problem set in their physics classes … and click away after the first long-winded paragraphs of my posts? Let’s hope the AI bots digested the posts and turned them into something useful! Maybe not: The WordPress AI bot here is again telling me – in critiquing this post – that I need to Balance the technical details with relatable explanations to engage a wider audience and Specify the target audience and tailor the content to their expectations.
So, dear bots: Here is more content for you!
I had a blog post about push-forwards of vector fields in my draft, since the first year of the pandemic – when I read books on differential geometry to cope with the virus panic. Roger Penrose’s hand-crafted drawings let me go off on a tangent, following a worm-hole from physics to art.
What shall I do with this draft? Of course, I turn it into poetry! Here is the poetry of snippets from the article … that I present in full after the poem. The snippets I pick for the poem are also highlighted in the article.
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Found Poetry: You feel what tangent vectors do
We know all about the push
cast into numbers
could that be?
be encoded in the things
live on either manifold
as the original vector
Curves are traced out
they should do something
make this into an honest vector
You feel what tangent vectors do
in physics you finally want to
hit all points
there is this problem
Looking at the points
Which curve should be picked?
You get a curve living on
a point in the source
none is hit twice
What’s the result of the action
You go up from the base
do so in a smooth way
Looking at the points
staying on the ground first
A map must provide a target
defined once and consistently
you should be able to reach
the disjoint union of all
related in a natural way?
In any case, we do not know
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The actual article:
Push-Forward of Vector Fields
You want to relate different manifolds to each other: a point in the domain manifold is mapped to a point in its image in the other manifold. What happens to the tangent vectors? Can they be related in a natural way? This is what the push-forward is about.
You feel what tangent vectors do by letting them act on functions: functions on manifolds map a point in these abstract geometrical spaces to real numbers. An arbitrary function shall be defined on the target manifold. Thus, the pushed forward vector field acts on this function. It feels like all the information to calculate the result of this action (a number) should already be encoded in the things defined so far: The nature of the manifolds as topological spaces (there are charts which allow you to cast things into numbers), the map between manifolds- thus a source point in one manifold and a target point in the other, and a function. This function is defined only in the target manifold.
The result of the action of a vector is the directional derivative of a function, calculated along a specific curve. Curves are traced out on manifolds; they are maps from a parameter in the real numbers to points in the manifold, much like the trajectory of a particle in three-dimensional space. We have a map defined between the manifolds, and therefore we can also connect the curves that live on either manifold: The inter-manifold map takes a point on a curve in the source manifold to a point on a curve in the target manifold. We know all about the push forwarded vector assigned to a specific point in the target manifold, if we know how it acts on any function defined there. From this, we need to construct a vector assigned to the preimage of that point, a point in the source manifold.
When the vector and its pushed forward counterpart act on somehow related functions, they should do something related. Both of these natural relations can be defined at once, in the definition of the push-forward:
Which curve should be picked? Map a point p in manifold M to a point Φ(p) in manifold N. Pick a vector in the tangent space of p by picking a curve γ, assigning a trajectory of points in M smoothly to real numbers. You get a curve living on N, if you concatenate γ and Φ: Φ after γ is a function taking you from the real numbers to M (via γ), then from points in M to points in the other manifold N (via Φ). Thus “in total” you go from the real numbers to points in N, which is the definition of a curve in N. The push-forwarded vector shall use that curve.
Manifolds M > > > > Φ > > > > N Points in the manifolds p > > > > Φ > > > > q = Φ(p) Curves on the manifolds γ > > > > Φ > > > > δ = Φ o γ γ(0)=p δ(0)=q Tangent spaces and push-forward TpM > > > > Φ*|p > > > > TqN = TΦ(p)N Vectors in tangent spaces, tangent to curves through points Xγ,p > > > > Φ*|p > > > > Yδ,q = YΦoγ,Φ(q) = Φ*(Xγ,p)
What’s the result of the action of the push-forwarded vector? When the push-forwarded vector (using) the curve Φ after γ acts on some function f defined on the target manifold N it shall give the same number as the original vector living above / on M does, when that one acts on the naturally related function. But which function (from the manifold M to the real numbers) could that be? f is defined on N; it takes you from points on the manifold N to real numbers. As Φ takes you from M to N, the concatenation f after Φ takes you from M to the real numbers – via points in N.
We had zoomed in on specific point p: the vectors “above” this point p form a whole vector space. But in physics you finally want to work with fields: things that are defined for each point in the manifold. You pick a vector from each of the tangent spaces, and you do so in a smooth way; vectors should not jump when you move from one point to the next. But there is this problem with the push-forward based on an arbitrary map Φ: The map does not necessarily hit all points in the target manifold, and some points in N may not be hit at all. But you want a vector for each point of N! To fix that, you demand that Φ is a bijective map: Every point in the target is hit, and none is hit twice. Then Φ can also be inverted.
How is the push-forward of vector fields defined then – when p is not fixed anymore? Picking vectors in a smooth way means to assign a vector to each point p by a map, say σ(p). You go up from the base manifold M to the tangent bundle TM, the disjoint union of all tangent spaces. Up there is the push-forward map Φ* that takes you from vector σ(p) “above M” to a vector “above N”. But to make this into an honest vector on N, there needs to be a corresponding smooth map, that goes up from N to these vectors – τ. Thus, starting from a point in M, you should be able to reach a vector over N in two ways: Either via the not-yet-pushed-forward vector and the push-forward map, or – staying on the ground first – via tracing out Φ from M to N, and then going up the target vector.
Looking at the points and vectors:
σ(p) = X > > > > Φ* > > > > Y = Φ*(σ(p)) = τ(Φ(q))) = Φ*(σ(Φ-1(q))) ^ ^ ^ ^ ^ ^ σ τ = Φ* o σ o Φ-1 ^ ^ ^ ^ ^ ^ p > > > > Φ > > > > q = Φ(p) < < < < Φ-1< < < <
Looking at the manifolds and tangent spaces:
TM > > > > Φ* > > > > TN ^ ^ ^ ^ ^ ^ σ τ should be: Φ* o σ o Φ-1 ^ ^ ^ ^ ^ ^ M > > > > Φ > > > > N < < < < Φ-1< < < <
(Saying it all again in different words! Voice from the present: Not editing much as I need all this for the poetry!)
A map must provide a target item for each source item: Every point in N has to be a source item. Thus, each point q in N needs to get some associated vector via a to-be-defined map. We have defined this map indirectly by going via the vectors over M and the to-be-defined push-forward. We have tried to hit the points in N from above, from the hopefully well-defined set of vectors reached by the push-forward map. So, two maps need to be defined once and consistently: The push-forward of the vector field, and a map taking points from N to the tangent bundle above N, TN. The latter map is the equivalent / counter-part of the map from M as a base space to the tangent bundle over M, TM. If we do not reach each point q of N via Φ, we miss at least one source point to construct the map from N upwards to its vectors. If we hit points in N twice by Φ, there are two more points in M connected to this point. From these source points in M, we might reach two different vectors in TN. In any case, we do not know how to assign a vector to a point in N unambiguously.
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Inverted Diffraction Art, by elkement 2023, first presented – of course – with other found poetry, in the blog post Discretized and Inverted. I picked this image because … the WordPress AI not demands an illustration, and because the colors in my ASCII art of manifolds anticipate my penchant for inverted spectral colors.
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