Differential Geometry | Introduction

Discord:
/ discord
Feel free to join and discuss!
I introduce the topic of differential geometry. It is a very broad subject, so this is a very loose introduction.
Talked about first are the beginnings of dual vector spaces (covectors, 1-forms), hyperbolic geometry (I didn't really get into projective geometry just yet), some other linear algebra prerequisites, and some discussion of spherical geometry and Gaussian curvature. Understanding basic linear algebra, vectors, and matrix theory is viewed here as a prerequisite. Some prerequisite understanding of calculus and infinitesimals is also necessary.
Mentioned in passing is manifold theory, the metric tensor, Riemannian and pseudo-Riemannian manifolds (pseudo-Riemannian manifolds are where the metric is not necessarily positive-definite, like in the case of special relativity). Albert Einstein's special relativity and general relativity are very, very loosely discussed (mainly just the relevance to differential geometry is pointed out). Note that relativity is an important application of differential geometry.
Also mentioned in passing is the great Felix Klein's Erlangen program, which seeks to classify geometric spaces by their (group theoretic) symmetry properties. Klein's quart
The object featured in the thumbnail is Klein's quartic. From Wikipedia: "In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 168 × 2 = 336 automorphisms if orientation may be reversed." It is a complicated object exhibiting G2 symmetry (connecting it with the octonion). The study of this highly symmetric object is dear to my heart, and may actually be an important key to future unification physics (see John Baez's discussions online and on CZcams regarding this object). One might argue that it isn't directly connected with differential geometry, but I would definitely debate that. I wanted to feature this object of great symmetry and beauty. It can fill one with wonder, particularly when you watch an animation of it turning inside out. I think it is important and close to the work of Élie Cartan, who played an important role in the subjects of differential geometry and classifications in Lie theory.

[@pushkar-tm5wi]

thanks.....
appreciate your hard work

[@UViolet87]

It's my pleasure