Differential Geometry: The Intrinsic Point of View #SoME3

#SoME3

Chapters:
0:00 Intro
2:19 How much does a curve ... curve?
3:56 Gaussian Curvature
7:14 Local Isometries
7:38 The Punchline
8:25 Intrinsic vs. Extrinsic
9:40 How does this apply to us?

By explaining the contents of Gauss' Theorema Egregium, I provide an overview of the "intrinsic point of view" in differential geometry: that is, from the point of view of living on a surface (or a manifold, more generally). I compare this point of view to that of a flatlander. Ultimately, this point of view applies more to our existence than we may think...

[ianmathwiz]

0:00 "If you've ever scraped the surface of mathematics"

I see what you did there.

[SnapstickGamer]

One of the best somepi vids I’ve seen. Thank you!

[lunaticluna]

this is really well made!! great and understandable explanation, and it's really interesting!

[dobariyanaitikdineshbhai]

Great video, keep making more such stuff.

[tamir]

I would love to see more of all kinds of interesting connections in geometry like this made simple. This was great👌

[lowerbound]

You sparked the interest🔥🔥🔥🔥🔥🔥 Great work, thanks!

[jursamaj]

I disagree with that last sentence. The fact that a space has curvature does *not* mean it is imbedded in a higher-dimensional space. The curvature *can* simply be intrinsic to the space.

[protocol]

What happens with an expanding surface? Imagine living on an expanding sphere with a finite speed of light. The path light takes follows a path through the bulk while simultaneously tracing an apparent path along the surface. Things will never appear to be quite where they were when the light was emitted or where they are when it is received.

[se]

Great video! Now to share a bunch of tangential philosophical asides that have always bothered me:

How could flat-landers possibly “meausure” lengths and angles if all their instrumentation was already curved? If you think about this, it makes no sense. If you lived on the surface of a sphere, you wouldn’t be able to construct the tangent plane which is necessary for determining the angle between two adjoining curves.

We can measure angles on the surface of the earth by walking in “straight lines” then turning, but these straight lines are characterized through projections and connections of the tangent planes. Which would mean it would be impossible to determine what the “straightest path” was on a curved surface without first imbedding it in some higher dimension, or at least defining a way to construct a tangent surface.

Likewise in GR curvature is measured via the acceleration of nearby free-falling bodies. But this acceleration is measured relative to idealized rigid bodies—these rigid bodies form the idea of the “tangent” space which defines the connections which allows for determining curvature.

If you truly “lived” in a curved surface, you could never know it, because all your measuring instruments would be warping too. Hence, some notion of euclidean geometry must always be prior to any non-euclidean geometry.

That’s my take at least!

[frankjohnson]

I was a bit confused by a points towards the end of the video about general relativity. Does the Gaussian curvature of a point in space depend on the matter in its vicinity or is it constant? If the former, when scientists say they're trying to measure the curvature of the universe, are they referring to the underlying curvature of space (which is distinct from the local curvature of a point)?

[AdrianBoyko]

Are you familiar with the book “Turtle Geometry”?