Lecture 4: Differentiable Manifolds (International Winter School on Gravity and Light 2015)
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users cite it as the source of specific claims about atlases and differentiability on manifolds that they then ask the community to unpack.
Recommended by 0 people across 0 different months, 2015 to 2020.
What it is
Lecture 4 of Frederic Schuller's 24-lecture general relativity course from the 2015 International Winter School on Gravity and Light. It introduces differentiable manifolds, explaining why a topological manifold alone cannot support differentiability and how compatible smooth atlases fix this, including the Whitney result on smooth sub-atlases.
What people said
Cites RagingBull · 2015-10 · Mathematics Stack Exchange · on “Defining differentiability on a topological manifold”
I am currently watching a series of lectures which are a part of International Winter School on Gravity and Light 2015 (by Prof. Frederick P. Schuller). In one of the lectures on differentiable manifolds he asks- Is the structure of a topological manifold enought to talk about differentiability of curves on the manifold?Can you construct from it a notion of differentiability? And he answers in the negative saying that the structure of a topological manifold is not good enough to define a notion of differentiability. Why is this true? What is missing in the structure of a topological manifold that is preventing us from defining differentiability on it? Here is the video lecture link. [this video]
Cites BigM · 2016-12 · Mathematics Stack Exchange · on “Understanding a theorem of Whitney”
While watching a video [this video] (min 34)on differentiable manifolds the lecturer mentioned a theorem of Whitney which sounded striking to me. The theorem is as follows Any $C^k{\geq1}$ atlas of a topological manifold contains a $C^\infty$ atlas. What exactly the theorem says? Can one elaborate a little bit on this?
Cites Jacques Smeets · 2020-10 · Mathematics Stack Exchange · on “How to select a smooth atlas out of many to change a topological manifold into a smooth manifold”
On youtube I am following a video series on general relativity (link: [Google][video] The relevant part for this question is at time stamp $33.00$ (about $3$ minutes). I repeat here this a part: It starts with a list showing the different types of differentiable atlases that can be linked to a topological manifold: $C^k$ : $k$ -times continuously differentiable. $D^k$ : $k$ -times differentiable, don't need to be continuous differentiable. $C^\infty$ : infinity times continuously differentiable. This is called smooth . $C^\omega$ : analytic functions (can be Taylor expanded). $\mathbb{C}^{\infty}$ : dim $M = 2n$ for integer $n$ , they satisfy the Cauchy-Riemann equations pairwise. This gives