MATH 541 LECTURE I

Atlases and differential structures on a manifold

A manifold of dimension n is a topological space, M, which looks locally like an euclidian space ℝ^{n}, that is, ∀x ∈ M there is a neighborhood of x, U ⊂ M, homeomorphic to ℝ^{n}. The homeomorphism ϕ: U → ℝ^{n} is called a chart or a coordinate system around x. To avoid too exotic examples, it is required also that a manifold 1) Hausdorff, 2) has a countable basis of topology.

Exercises.

(1) Construct a non-Hausdorff locally euclidian space.

(2) Give an example of locally euclidian space with a non-countable basis.

(3) The same as in (2), but the space should be connected.

A differential structure in M allows to speak about differentiability of functions on M. Such a structure can be introduced if we fix an atlas of charts which agree with each other.

Recall that the class C^{k} is formed by those functions which are k times continuously differentiable. The class C^{∞} is formed by infinitely many times differentiable functions, the latter functions will be called smooth. C^{a} means analyticity, C^{0} means just continuity.

Two charts, ϕ_{i}: U_{i} → ℝ^{n}, are said to be C^{k}-compatible if the coordinates in these charts are related as C^{k}-functions in the common part of the domains. That is to say that the coordinate change map ϕ_{1,2} = ϕ_{2} ∘ ϕ_{1}^{−1} is a diffeomorphism of class C^{k} between the domains V_{i} = ϕ_{i}(U_{1} ∩ U_{2}) ⊂ ℝ^{n} The charts are automatically C^{k}-compatible if their domains U_{i} are disjoint.

A set of charts, A = {ϕ_{i}: U_{i} → ℝ^{n}}_{i∈I}, is called an atlas of class C^{k}, if the domains U_{i} cover M, and the charts of A are pairwise C^{k}-compatible.

Remarks.

(1) An atlas of class C^{k} belongs also to all rougher classes, C^{m}, m < k (m, k are from the ordered set 0, 1, 2, …, ∞, a).

(2) Using a similar approach with complex charts ϕ: U → ℂ^{n}, one can define complex manifolds. However, for complex functions differentiable=analytic (holomorphic), so there is no notion of class C^{k}. Complex manifolds can be only analytic !

Examples.

(1) ℝ^{n} has an atlas of class C^{a} with a unique chart, defined by the identity map id: ℝ^{n} → ℝ^{n}.

(2) sphere S^{n} has an atlas of class C^{a} consisting of a pair of charts, each of which is a stereographical progection, S^{n} ∖ {x} → ℝ^{n}.

Typeset by AMS-TEX

(3) Projective space ℝP^{n} has a C^{a}-atlas with n charts, ϕ_{i}: U_{i} → ℝ^{n}, ϕ_{i}([x_{0} : ⋯: x_{n}]) = (\frac{x_{0}}{x_{i}} , …, \frac{x_{n}}{x_{i}}), where U_{i} ⊂ ℝP^{n} is the complement of the hyperplane x_{i} = 0 (\frac{x_{i}}{x_{i}} is omitted in the right hand side).

Assume that A is a C^{k}-atlas fixed for a manifold M. A function f: M → ℝ is called differentiable of class C^{k} (with respect to A) if its representation, f ∘ ϕ_{i}^{−1} : ℝ^{n} → ℝ in any chart, ϕ_{i}: U_{i} → ℝ^{n}, of A is differential of class C^{k}. Similarly a map f: M_{1} → M_{2} between two differential manifolds is differentiable of class C^{k}, if in any charts it is defined by differentiable functions. More precisely, for any charts ϕ: U → ℝ^{n} of A_{1} and ψ: V → ℝ^{m} of A_{2} the composition ψ ∘ f ∘ ϕ^{−1} is differentiable of class C^{k} in the domain where this composition is defined (that is in ϕ(U ∩ f^{−1}(V))).

The above map f is called a diffeomorphism of class C^{k} if it is a homeomorphism and f together with f^{−1} belong to the class C^{k}.

Two C^{k}-atlases A_{1}, A_{2} are said to be C^{k}-equivalent if their charts are pairwise C^{k}-compatible. Note that this is the same as to say that the union A_{1} ∪A_{2} is again a C^{k}-atlas.

Exercises.

(1) Check that the above relation is really equivalence.

(2) Show that every C^{k}-equivalence class or atlases contains a unique maximal atlas (the latter is called sometimes the differential structure on a manifold M.

(3) Show that a pair of atlases A_{i}, i = 1, 2, in M are C^{k}-equivalent if and only if the identity map is a C^{k}-diffeomorphism (M, A_{1}) → (M, A_{2}).

(4) Show that for the above equivalent C^{k}-atlases, the sets, ℂ^{k}(M, A_{i}), of differentiable functions coincide. Show the converse.

Raising of the differentiability class of a manifold

Theorem. For any C^{k}-atlas A, 0 < n < ∞, on a manifold M there exists a C^{∞}-atlas A′ on M, which is C^{k}-compatible with A.

If A′′ is any other such a C^{∞}-atlas on M, then it defines essentially the same (C^{∞}-diffeomorphic) differential structure on M.

This result shows that there is no specific difference in the topology of manifolds of different classes of differentiability C^{k}, 1 ⩽ n ⩽ ∞ (existence of C^{1}-atlas implies existence and essentially uniqueness of C^{∞}-atlas). That is why we will be dealing in what follows only with smooth (C^{∞}) manifolds.

Remark. On the contrary, topological manifolds are in general not smoothable, and smooth manifolds cannot be made analytic. So, we will exclude classes C^{0} and C^{a} from the further considerations.

Different but diffeomorphic differential structures

Example. Any homeomorphism f: ℝ → ℝ gives a chart and thus defines a certain smooth structure in ℝ. It differs from the standard one, unless f is a diffeomorphism.

For example, consider an atlas A_{1} = {x^{3}} formed by a uniques chart ℝ → ℝ, x ↦ x^{3}. It is not compatible with the standard chart in ℝ, the identity map x ↦ x, and thus A_{1} is not compatible with the standard atlas A_{0} = {x}.

Exercise. (1) Show that the algebra of smooth functions C^{k}(ℝ, A_{1}) with respect to the atlas A_{1} is a subalgebra of the usual algebra C^{k}(ℝ, A_{0}) of smooth functions on ℝ.

(2) Show that line ℝ with the differential structures defined by A_{1} is diffeomorphic to the standard ℝ.

Remark. We conclude that A_{0} and A_{1} define distinct differential structures on ℝ, which are however equivalent up to a diffeomorphism. Such a distinction is not very crucial from the point of view of differential topology. It is much more essential to understand if the differential structures on a manifold are equivalent up to diffeomorphis.

For example, it turns out that any manifold of dimension ⩽ 3 has essentially (that is up to a diffeomorphism) unique differential structure. Beginning from the dimension 4 it is no longer true. On one hand, manifolds may have no smooth structure at all. On the other hand, there may be many such structures, essentially non-equivalent.

This phenomenon was discovered by J.Milnor in 1950s, who constructed famous 27 “exotic” 7-sphere (that is essentially non-equivalent differential structures on S^{7}).

Implicit presentation of differential manifolds

A subset L ⊂ M in a smooth manifold M of dimension m is called a (smooth) l-dimensional submanifold of M if for any point x ∈ M there exists a chart ϕ: U → ℝ^{m} around x, such that ϕ(U ∩ L) = ℝ^{l}. This means that l looks in this coodinate system like a subspace ℝ^{l} of ℝ^{m}.

Note that the restriction of ϕ is a chart on L mapping V = U ∩ L to ℝ^{l}.

Exercise. Check that these charts agree. Define a differential structure on L. Show that the inclusion L → M is a smooth map.

According to the Whitney embedding theorem, any smooth manifold can be considered lying in ℝ^{N} (for sufficiently big N), that is diffeomorphic to a submanifold of ℝ^{N}.

Submanifolds in a manifold M most often appear as the locus L = f^{−1}(y), where y ∈ N is a regular value of some smooth mapping f: M → N m = dim M ⩾ n = dim N. We recall that x ∈ M is called a regular point if the rank of the differential d_{x}f has maximal rank, n, and critical points otherwise. If f^{−1}(y) contains a critical point, then y ∈ N is called critical value. Otherwise (if f^{−1}(y) consists of regular values or empty) y is called a regular value .

Example. Consider f: ℝ^{m} → ℝ^{n}, x = (x_{1}, …, x_{m}) f = (f_{1}, …, f_{n}), and L = f^{−1}(0). At a regular point the Jacobi matrix (∂f_{i}/∂x_{j}) has maximal rank n. By the implicit function theorem, there exists a neighborhood U ⊂ ℝ^{m} (say, a ball) around x such that U ∩L has 1-1 projection to one of the coordinate (m−n)-planes in ℝ^{m}, and the remaining n coordinates express as smooth functions of the chosen (m − n) coordinates.

The projection to this (m − n) plane provides a chart L ∩ U → ℝ^{m−n}.

Exercise. Prove that such charts are compatible. Prove that L is a smooth manifold in case of arbitrary M.

Exercise. Prove that the smooth structure on S^{n} = {x_{0}^{2} + ⋯ + x_{n}^{2} = 0} ⊂ ℝ^{n+1} defined implicitly coincides with the structure defined earlier by stereographic projection.

Homework. In the following examples determine if the set is a smooth manifold. Find its dimension, the local coordinate systems.

(1)

\{ \begin{matrix}3x + y − z + u^{2} & = a \\ x + y + 2z + u & = 0 \\ 2x + 2y − 3z + 2u & = 0 \end{matrix}

(2) The Lie groups SL(n, ℝ) ⊂ ℝ^{n×n} SL(n, ℂ) ⊂ ℂ^{n×n}

(3) Lie groups O(n) and U(n).

(4) Stiefel manifold S_{n,k} formed by all k-frames in ℝ^{n}.