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GitHub Repository: sagemanifolds/IntroToManifolds
Path: blob/main/01Manifold_Def.ipynb
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Kernel: SageMath 9.6

1. Basic notions of topology

This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).

A topology on a set SS is a collection T\mathscr{T} of subsets containing both the empty set ∅ and the set SS such that T\mathscr{T} is closed under arbitrary unions and finite intersections, i.e.,

(i) if UaTU_a ∈ \mathscr{T} for all aa in an index set AA, then aAUaT\bigcup_{a∈A}U_a ∈ \mathscr{T},
(ii) if U1,...,UnTU_1, . . . ,U_n ∈ \mathscr{T}, then i=1nUiT.\bigcap_{i=1}^n U_i ∈ \mathscr{T}.

The elements of T\mathscr{T} are called open sets. The set SS with a topology will be called a topological space.

If AA is a subset of a topological space SS, then the subspace topology on AA is defined as TA={UA  UT}.\mathscr{T}_A = \{U ∩ A |\ \ U ∈ \mathscr{T}\}.

A neighborhood of a point pp in SS is an open set UU containing pp.

A subcollection B\mathcal{B} of a topology T\mathscr{T} on a topological space SS is a basis for the topology T\mathscr{T} if given an open set UU and point pUp\in U, there is an open set BBB ∈ \mathcal{B} such that pBUp ∈ B ⊂U. We also say that B\mathcal{B} generates the topology T\mathscr{T} or that B\mathcal{B} is a basis for the topological space SS. A collection B\mathcal{B} of open sets of SS is a basis if and only if every open set in SS is a union of sets in B\mathcal{B}.

If X,YX,Y are topological spaces a function f:XYf : X →Y is continuous if and only if the inverse image of any open set is open.
A continuous bijection f:XYf : X →Y whose inverse is also continuous is called a homeomorphism.

Topologigal manifold

A topological manifold MM of dimension nn is a topological space with the following properties:
(i) MM is Hausdorff, that is, for each pair p1,p2p_1,p_2 of distinct points of MM there exist neighborhoods V1,V2V_1,V_2 of p1p_1 and p2p_2, respectively such that V1V2=V_1∩V_2=∅.
(ii) Each point pMp∈M possesses a neighborhood VV homeomorphic to an open subset UU of RnR^n.
(iii) MM satisfies the second countability axiom, that is, MM has a countable basis for its topology.

For an nn-dimensional topological manifold each pair (U,φ)(U, φ), where UU is an open subset of MM and φ:Uφ(U)Rnφ : U → φ(U ) ⊂ R^n is a homeomorphism of UU to an open subset of RnR^n is called a coordinate map, chart or coordinate system and UU is a coordinate neighborhood.

For pU, p\in U,\ φ(p)φ( p) belongs to RnR^n, and therefore consists of nn real numbers that depend on pp. Thus φ(p)φ( p) is of the form

φ(p)=(x1(p),x2(p),...,xn(p)).φ( p) = (x^1 ( p), x^2 ( p), . . . , x^n ( p)).

Smooth manifold

A map ϕ=(ϕ1,,ϕm) \phi=(\phi^1,\ldots,\phi^m)\ from an open subset URnU\subset R^n to RmR^m is smooth on UU or belongs to C(U)C^\infty(U) if all partial derivatives α1++αnϕk(x1)α1(xn)αn,where  αi denote non-negative integers\frac{\partial^{\alpha_1+\ldots+\alpha_n} \phi^k} {\partial (x^1)^{\alpha_1}\ldots\partial (x^n)^{\alpha_n}}, \quad \text{where }\ \alpha_i \ \text{denote non-negative integers} are continuous on UU.

When two coordinate neighborhoods overlap, we have formulas for the associated coordinate change. The idea to obtain smooth manifolds is to choose a subcollection of coordinate neighborhoods so that the coordinate changes are smooth maps.

An nn-dimensional CC^\infty or smooth manifold is a topological manifold of dimension nn and a family of coordinate charts φα:UαRnφ_α : U_α → R^n defined on open sets UαMU_α ⊂ M, such that:

(i) the coordinate neighborhoods UαU_\alpha cover MM,

(ii) for each pair of indices α,βα, β such that W:=UαUβ,W := U_α \cap U_β \not= ∅, the overlap maps (transitions)

φβφα1:φα(W)φβ(W),φ_β ◦ φ_α^{-1} : φ_α (W ) → φ_β (W ),
φαφβ1:φβ(W)φα(W),φ_α ◦ φ_β^{-1} : φ_β (W ) → φ_α (W ),

are CC^\infty,

(iii) the family A={(Uα,φα)}A = \{(U_α , φ_α )\} is maximal with respect to (i) and (ii), meaning that if φ0:U0Rnφ_0 : U_0 → R^n is a chart such that φ0φ1φ_0 ◦ φ^{-1} and φφ01φ\circ φ_0^{-1} are CC^\infty for all φAφ \in A, then (U0,φ0)(U_0 , φ_0 ) is in AA.

Any family A={(Uα,φα)}A = \{(U_α , φ_α )\} that satisfies (i) and (ii) is called a CC^∞-atlas for MM. If AA also satisfies (iii) it is called a maximal atlas or a differentiable or smooth structure.

Given any atlas A={(Uα,φα)}A = \{(U_α , φ_α )\} on MM, there is a unique maximal atlas Aˉ\bar{A} containing it. In fact, we can take the set Aˉ\bar{A} of all maps that satisfy (ii) with every coordinate neighborhoods on AA. Clearly AAˉA ⊂ \bar{A}, and one can easily check that Aˉ\bar{A} satisfies (i) and (ii). Also, by construction, Aˉ\bar{A} is maximal with respect to (i) and (ii). Two atlases are said to be equivalent if they define the same differentiable structure.

Smooth functions and maps.

By C(M)C^\infty(M) we shall denote the family of smooth functions on a smooth manifold MM, i.e., functions ff, such that fϕ1f\circ\phi^{-1} is smooth on ϕ(U)Rn\phi(U)\subset R^n for every coordinate chart (U,ϕ)(U,\phi).

If MM, NN are smooth manifolds, then a map ψ:MN\psi: M\to N is smooth if for every pair of charts (U,φ)(U, φ) on MM and (V,χ)(V, χ ) on NN, the map χψφ1χ ◦ ψ ◦ φ^{−1} is smooth on φ(ψ1(V)U)Rn.φ(\psi^{-1}(V)\cap U)\subset R^n.

What's next?

Take a look at the notebook Examples of charts. Cartesian and spherical coordinates.