Inculcation: smooth manifolds

parent:: inculcation

initial philosophy of manifolds §

Philosophy of manifolds I: to go out of open sets in ${\mathbb{R}}^n$

After studying submanifolds of ${\mathbb{R}}^n$, or just surfaces in ${\mathbb{R}}^2$, why do we want to think of arbitary sets as “manifolds”, as something we can do things we could do to smooth surfaces? Because then anything could be a manifolds.

Well not anything, but many sets can be attached with a smooth manifold structures. For example, $SO({\mathbb{R}}^3)$ (which is a group or a set of functions really!), spacetime in physics, . This helps us do analysis in more kinds of spaces, and generalise geometry equipped with differentiation.

architecture of manifolds §

Sadly, the definition and introductory theory of manifolds is “complete garbage” ¹

Architecture of manifolds I

We obtain a set attached with coordinates.

hence manifolds generalize coordinate systems §

motivation for the technologies §

There are three motivations for the definitions and ideas

• Analysis and geometry in the submanifolds of ${\mathbb{R}}^n$ or surfaces in ${\mathbb{R}}^3$
• Analysis in normed vector spaces
• Topology, algebraic topology

There are some absurdity in the definitions of objects which are very simple in the case of normed vector spaces

technology	in normed vector space $V$	in smooth manifold $M$
tangent vectors	(not essential) velocity of curves in $V$ passing though some $p\in V$ are just vectors in $V$	multiple ways to define it, easiest one is to consider velocities in a chart
tangent space	(not essential) $V$ itself because any $v\in V$ is a velocity of some curve
tangent bundle	(not essential) disjoint union of $V$ for all elements of $V$, hence $V\times V$	$TM:={\bigcup }_{p\in M}\{p\}\times T_{p}M$
tangent vector fields	a smooth function $V\to V$	a smooth function $M\to TM$ such that $p\in M$ is mapped to a vector in $v\in T_{p}M$ (called a smooth section of $TM$)
dual (tangent) space	$V^{\ast }:=\mathrm{Hom}{\mathsf{Vec}}_{\mathbb{R}}(V,\mathbb{R})$	$T_p^{\ast }M:=(T_{p}M{)}^{\ast }$
$(p,q)$-(tangent) tensor space	${\mathbf{T}}^{p,q}(V)$	${\mathbf{T}}^{p,q}(T_{p}M)$
tensor bundle	(not essential)	${\mathbf{T}}^{p,q}TM:={\bigcup }_{p\in M}\{p\}\times {\mathbf{T}}^{p,q}(T_{p}M)$
$k$-exterior (tangent) vectors	${\Lambda }^k(V)$
differential $k$-form	a smooth function $\omega :V\to {\Lambda }^k(V)$
integration of differential forms on chains

a first course just to construct the theory :( §

• book Lee - Smooth manifolds
• book Boothby - manifolds
• also uses the theory inside physics
  • lectures Frederic Schuller -International Winter School on Gravity and Light 2015
  • lectures Frederic Schuller - Lectures on the Geometric Anatomy of Theoretical Physics

tangents §

differential forms §

bundles on smooth manifolds §

Lie groups and smooth Lie group actions §

beyond the first course and uses §

• (semi)Riemannian geometry, GR
• Differential topology
  • Morse theory
  • Cobordism theory
• Knots
• Geometric classical mechanics (classical mechanics done right)
  • Lagrangian dynamics
  • Symplectic geometry and Hamiltonian dynamics
• Geometric classical field theory
• Gauge theory
• Geometric quantum mechanics
  • quantization (classical mechanics $\to$ quantum mechanics, proper)
• Smooth dynamical systems: vector fields on manifolds can give us ODEs Inculcation: ODEs

Footnotes §

1. evan-chen-napkin, chapter 45 ↩