The Christoffel Symbols In Riemannian Geometry

The illustrious Christoffel Symbols are requisite to any study of curved surfaces, but can their abstract nature be made more concrete and physical? Here our exploration of the daunting subject of differential geometry continues, as with a host of vivid and visceral 3D animations we attempt to explore the nature of these mathematical objects, whilst striving to bring the wonderful world of Riemannian Geometry to life!

We highly recommend watching "An Introduction to Curvilinear Coordinates" before viewing this video:

If you're completely new to the subject of the Christoffel Symbols, try starting here:

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Henry Lindner
Antsu Sausanen
Michael O'Connor
James Bretz
Jordan Rosario

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Video Contents:
00:00 - Introduction
00:50 - Curvilinear Coordinate Recap
02:17 - Basis Vectors & Christoffel Symbols: Physical Intuition
06:49 - Basis Vectors & Christoffel Symbols on a Curved Manifold
13:17 - Extrinsic Solution of a 2-Sphere
14:49 - Metric Tensor & Intrinsic Method
19:15 - Levi-Civita Constraints; Christoffel Equation Derivation & Interpretation
27:04 - Example Problem/Intrinsic Solution of a 2-Sphere
31:20 - Global vs. Local Flatness/Conclusion

[dialectphilosophy]

Regarding the 2-sphere Christoffel calculations, there is a visual TYPO at 30:54: cotv is the value of the Gamma u-u-v christoffel symbol, as shown in the top half of the video, not Gamma v-u-v, as shown in the lower half, or as shown again during the ending title screen of the video. Indeed, the cotv and zero values should have their places swapped in the lower half of the screen at 30:54. Our apologies for this mistake, and thank you to our commenters for pointing it out. One of these days we'll make it through a Christoffel video without some sort of indices typo!

[DanielKRui]

What a masterclass! You have changed the course of Riemannian geometry history for the better with this video. Such concepts have never before been so readily accessible

[generalCross]

I was drowning in my GR course because they don't give a visual explanation of what the Christoffel Symbol tells you. But now I am saved!! Thank you from the bottom of my heart <3

[krzysztofwos]

Incredible. The connection between parametric space and the manifold is the question of what observers are. That's the crux of the quantum mechanics x general relativity puzzle. We are so close to seeing the full picture, and it is beautiful.

[benjamingross]

This series is fantastic. It's been a very long time since I took GR and I'm currently teaching multivariable calc and it just brings it back with a clarity that is very satisfying.

[TheodoreGelber]

When I watch these science, engineering and math videos, the cleverness of these inventions and discoveries is amazing. The focus on the subtle nature must be so intense that only a few humans could achieve it.

[snakeeva]

@dialectphilosophy Your channel doesn't get enough praise for what you have done. You have made understanding tangent spaces, manifolds, the metric tensor and other concepts in GR so easy and approachable unlike anywhere else online. I spend hundreds of hours, literally, watching lectures online over the years on GR with the Standford Lectures by Leonard Susskind being one of the best. And you have managed to condense the rigour given by Prof Susskind in 30 mins without sacrificing any of the abstract concepts or important points. Your channel is godsent and I hope it grows and gets the appreciation it so deserves. This is probably my third or fourth watch of this video and its not because I don't understand but instead because it is so good at its explanation that I can rewatch it to reinforce the revelations that you have imparted. Keep up the great job that you and your team are doing and I look forward to when your channel reaches the level of appreciation to comparable channels like 3Blue1Brown and Veritasium (your channel is already on that caliber in terms of the quality of your content) which have also done a tremendous job at science communication. I absolutely love this channel, every single video that your team has published is a gem!!

[-_Nuke_-]

Such a heavy subject handled with the OUTMOST of professionalism!
You guys are first and foremost masters of pedagogy, and storytelling... I wish more and more people find and watch your videos, because you are a very special channel on the entirety of YouTube! As always, I can't wait for more - I like exploring relativity with you guys - keep it up!

[asthmen]

Thank you so much — I'm finally starting to understand the GR lectures I attended in undergrad. ❤

[mobilephil]

This is pure magic. Wonderful, comprehensive presentation and careful explanation. Thank you for this. If only all teaching was this good.

[n-da-bunka]

By FAR the best video I have seen on this to date! Thanks

[howdynoah]

One of the best videos I've ever watched - genuinely. Thank you for ths!

[jul]

Wow! Talk about a riveting Saturday night show! Thank you for the outstanding quality of this video. Can't wait for the next episode.

[naysay]

wow this is a FANTASTIC explanation. so intuitive! indexing the peaks and troughs of the manifold to a flat surface is such a genius simple way of defining such a surface!

[MikeLeed]

Amazing, this video is a gold mine, thank you so much for bringing this material down to common folks like me.

[JahanaraBegum-yfzv]

this guy is truly great for teaching these deep and with great visualisation

[EvanMildenberger]

Here's hoping your channel blows up with success!!!

[richsalinas]

Another Great one. Congrats on breaking 100K subs. 🤜🤛🏼

[culpritgene]

This is truly an incredible visualization and top notch explanation!

[haniyasu]

Great vid, love the animations (and the sound effects haha)