Manifolds and differential forms are foundational concepts of differential topology, and connections are a foundational concept of differential geometry. They are mathematical building blocks used in modern physics, essentially enabling the transfer of multivariable calculus to arbitrary curved surfaces (without relying on an explicit embedding into Euclidean space). I think the joke is that physics students don’t typically learn the details of these building blocks, rather just the relevant results, and get confused when they’re emphasized.
For a tl;dr about the concepts mentioned:
A manifold is a curve, surface, or higher-dimensional object which locally resembles Euclidean space around each point (e.g. the surface of a sphere is a 2D manifold; tiny person standing on a big sphere perceives the area around them to resemble a flat 2D plane).
Differential forms are “things that can be integrated over a manifold of the corresponding dimension.” In ordinary calculus of 1 variable, that’s a suitably regular function (e.g. a continuous function), and we view such a function f(x) as a differential form by writing it as “f(x) dx.”
A connection is a way of translating local tangent vectors from one point on a manifold to another in a parallel manner, i.e. literally connecting the local geometries of different points on the manifold. The existence of a connection on a manifold enables one to reason consistently about geometric concepts on the whole manifold.
The existence of a connection on a manifold enables one to reason consistently about geometric concepts on the whole manifold.
The geometry of the cosmos itself. Tracing good ol’ fashioned circles and triangles with the full extent of the visible universe and even beyond. This stuff blows my mind, even just the mere fact that we’re doing it, let alone the fact that we’re getting such incredible, counter-intuitive results.
Picture yourself having a time machine, going back to visit Euclid or Pythagoras, or even Kepler or Galileo, and blowing their mind with four words: The Geometry Of Spacetime.
Not only does time itself have a geometry, it must curve and contract to accommodate the absolute speed of light… nuts I tell ya.
Then there are at least four spatial dimensions, but there may be as many as eleven. To think that Copernicus thought epicycles were weird, wait till he gets a load of THIS!
Spinoza? Meet quantum entanglement with no hidden variables! “Oy vey!”
Is there a way to learn general relativity WITHOUT those concepts? My curriculum made sure to introduce all that before going into GR, didn’t know that wasn’t common. Guess that was my point of confusion.
That’s what I thought, which is why I don’t get the meme. Who is it talking about? Who has learned about GR but gets confused by manifolds? How else would they have learned GR?
I do not recall well, but connections are sections of a suitable bundle, right? I have to eventually learn this but eh, seems too tedious. Do I need to take an entire class to really know it?
You might be thinking of a [connection of an affine bundle](https://en.wikipedia.org/wiki/Connection_(affine_bundle). You could learn it through classes (math grad programs usually have a sequence including general topology, differential topology/smooth manifolds, and differential geometry) or just read some books to get the parts you need to know.
Manifolds and differential forms are foundational concepts of differential topology, and connections are a foundational concept of differential geometry. They are mathematical building blocks used in modern physics, essentially enabling the transfer of multivariable calculus to arbitrary curved surfaces (without relying on an explicit embedding into Euclidean space). I think the joke is that physics students don’t typically learn the details of these building blocks, rather just the relevant results, and get confused when they’re emphasized.
For a tl;dr about the concepts mentioned:
A manifold is a curve, surface, or higher-dimensional object which locally resembles Euclidean space around each point (e.g. the surface of a sphere is a 2D manifold; tiny person standing on a big sphere perceives the area around them to resemble a flat 2D plane).
Differential forms are “things that can be integrated over a manifold of the corresponding dimension.” In ordinary calculus of 1 variable, that’s a suitably regular function (e.g. a continuous function), and we view such a function f(x) as a differential form by writing it as “f(x) dx.”
A connection is a way of translating local tangent vectors from one point on a manifold to another in a parallel manner, i.e. literally connecting the local geometries of different points on the manifold. The existence of a connection on a manifold enables one to reason consistently about geometric concepts on the whole manifold.
The geometry of the cosmos itself. Tracing good ol’ fashioned circles and triangles with the full extent of the visible universe and even beyond. This stuff blows my mind, even just the mere fact that we’re doing it, let alone the fact that we’re getting such incredible, counter-intuitive results.
Picture yourself having a time machine, going back to visit Euclid or Pythagoras, or even Kepler or Galileo, and blowing their mind with four words: The Geometry Of Spacetime.
Not only does time itself have a geometry, it must curve and contract to accommodate the absolute speed of light… nuts I tell ya.
Then there are at least four spatial dimensions, but there may be as many as eleven. To think that Copernicus thought epicycles were weird, wait till he gets a load of THIS!
Spinoza? Meet quantum entanglement with no hidden variables! “Oy vey!”
Is there a way to learn general relativity WITHOUT those concepts? My curriculum made sure to introduce all that before going into GR, didn’t know that wasn’t common. Guess that was my point of confusion.
Not really, you need to have a basic understanding at least
That’s what I thought, which is why I don’t get the meme. Who is it talking about? Who has learned about GR but gets confused by manifolds? How else would they have learned GR?
I do not recall well, but connections are sections of a suitable bundle, right? I have to eventually learn this but eh, seems too tedious. Do I need to take an entire class to really know it?
You might be thinking of a [connection of an affine bundle](https://en.wikipedia.org/wiki/Connection_(affine_bundle). You could learn it through classes (math grad programs usually have a sequence including general topology, differential topology/smooth manifolds, and differential geometry) or just read some books to get the parts you need to know.
You lost a bracket in your link