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Home , Differentiable curve, Differentiable manifold, Immersion (mathematics), Local diffeomorphism

Seminar: Introduction to Riemannian Differentiable Carlos Sotillo Rodríguez

In the following Chapter we are going to define the main concepts as well as the basic results. From now on, differentiable will always signify of class C ∞

Definition 1: A differentiable of n is a M and a family of injective mappings x : U Rn M (where U is an of Rn), such that: α α ⊂ → α [ 1. xα(Uα) M α = T 1 1 2. for any pair α,β with x (U ) x (U ) W , the sets x− (W ) and x− (W ) are open α α β β = 6= ; α β n 1 sets of R and the mappings x− x are differentiable. β ◦ α

3. The family {(Uα,xα)} is maximal relative to conditions 1 and 2.

The pair (U ,x ) (or the mapping x ) with p x (U ) is called a parametrization (or system α α α ∈ α α of coordinates) of M at p, xα(uα) is then called a coordinated neighborhood at p. A family satisfying 1 and 2 is called a differentiable structure. The condition 3 is included for technical reasons. Indeed, given a differentiable structure, we can complete it to a maximal one. Therefore, with a certain abuse of lenguage, we can say that a is a set with a differentiable structure.

n m Definition 2: Let M1 and M2 (where n,m are the of M1,M2 respectively). A map- ping φ : M M is differentible at p M if given a parametrization y : V Rm M at ϕ(p) 1 → 2 ∈ 1 ⊂ → 2 there exists a parametrization x : U Rn M at p such that ϕ(x(U)) y(V ) and the map- ⊂ → 1 ⊂ ping 1 n m y− ϕ x : U R R ◦ ◦ ⊂ → 1 is differentiable at x− (p). ϕ is differentiable on an open set of M1 if it is differentiable at every point of this open set (it follows from condition 2 that this definition does not depend on the 1 parametrization).The expression y− ϕ x is called the expression of ϕ in the parametriza- ◦ ◦ tions x and y.

Definition 3: Let M be a differentiable manifold. A differentiable α :( ²,²) M − → is called a (differentiable) in M. Suppose that α(0) p M and let D be the set of = ∈ functions on M that are differentiable at p. The vector to the curve α at t 0 is a = function α0(0) : D R given by → ¯ d(f α)¯ α0(0)(f ) ◦ ¯ , f D = dt ¯t 0 ∈ =

1 A at p is the tangent vector at t 0 of some curve α :( ²,²) M with α(0) p. = − → = The set of all tangent vectors to M at p will be indicated by Tp M. ∂ ∂ Further, the set Tp M is a vector with assiciated {( )0, ,( )0}. It is immediate ∂x1 ··· ∂xn that the basis depends on the parametrization, but not the linear structure.

Proposition 4: Let M n and M m be differentiable manifolds and let ϕ : M M be a diffe- 1 2 1 → 2 rentiable mapping. For every p M and for each v T M , choose a ∈ 1 ∈ p 1 α :( ²,²) M with α(0) p,α0(0) v. Take β ϕ α. The mapping dϕ : T M T M − → 1 = = = ◦ p p 1 → ϕ(p) 2 given by dϕ (v) β0(0) is a linear mapping that does not depend on the choice of α. p = Definitions 5:

The linear mapping dϕp is called the differential of ϕ at p.

Let M and M be differential manifolds. A mapping ϕ : M M is a 1 2 1 → 2 if it is differentiable, bijective and its inverse is differentiable. ϕ is said to be a local diffeomisphism at p M if there exist neighborhoods U of p and V of ϕ(p) such that ∈ 1 ϕ : U V is a diffeomorphism. → Theorem 6: Let ϕ : M n M n be a differentiable mapping and let p M be. dϕ : T M 1 → 2 ∈ 1 p p 1 → Tϕ(p)M2 is an if and only if ϕ is a at p.

Definition 7: Let M m and N n be differentiable manifolds. A differentiable mapping ϕ : M → N is said to be an if dϕ : T M T N is injective for all p M. If in , ϕ p p → ϕ(p) ∈ is a onto ϕ(M) N, where ϕ(M) has the subspace induced from ⊂ N, we say that ϕ is an . If M N and the inclusion i : M N is an embedding, we ⊂ ⊂ say that M is a of N.

Proposition 8: Let ϕ : M n M m, n m, be an immersion of the differentiable manifold 1 → 2 ≤ M into the differentiable manifold M . For every point p M , there exists a neighborhood 1 2 ∈ 1 V M1 of p such that the restriction ϕ V : V M2 is an embedding. ⊂ | → Definition + Example 9 (The tangent ): Let M n be a differentiable manifold and let TM : {(p,v): p M,v T M}. We are going to provide the set TM with a differentiable = ∈ ∈ p structure. α α Let {(Uα,xα)} be a maximal differentiable structure on M. Denote by (x1 , ,xn ) the coordi- ∂ ∂ ··· nates of Uα and by { ∂xα , , ∂xα } the associated bases to the tangent spaces of xα(Uα). For 1 ··· n every α, define y : U Rn TM α α × → by n α α α α X ∂ n yα(x1 , ,xn ,u1, ,un) (xα(x1 , ,xn ), ui α ), (u1, ,un) R ··· ··· = ··· i 1 ∂xi ··· ∈ = n [ We are going to show that {(Uα R , yα}) is a differentiable structure on TM. Since xα(Uα) × α =

2 M and (dx ) (Rn) T M, q U , we have only to prove the condition 2. Now let α q = xα(q) ∈ α \ (p,v) y (U Rn) y (U Rn) ∈ α α × β β × Then (p,v) (x (q ),dx (v )) (x (q ),dx (v )), = α α α α = β β β β where q U ,q U ,v ,v Rn. Therefore, α ∈ α β ∈ β α β ∈ 1 1 1 1 y− y (q ,v ) y− (x (q ),dx (v )) ((x− x )(q ),d(x− x )(v )). β ◦ α α α = β α α α α = β ◦ α α β ◦ α α 1 Since x− x is differentiable, d(x x ) is as well. It follows that condition 2 is verified. β ◦ α β ◦ α Example 10 (Regular surfaces in Rn): A subset M k Rn is a regular of dimension ⊂ k n if for every point p M k there exists a neighborhood V of p in Rn and a mapping ≤ ∈ x : U M V of an open set U Rk onto M V such that: → ∩ ⊂ ∩ x is a differentiable homeomorphism.

(dx) : Rk Rn is injective for all q U. q → ∈ It can be proved that if x : U M k and y : V M k are two parametrizations with x(U) → 1 →1 1 ∩ y(V ) W , then the mapping h x− y : y− (W ) x− (W ) is a diffeomorphism. We give = 6= ; = ◦ → a sketch of the proof: Let (u , ,u ) U and (v , ,v ) Rn, and write x in these coordina- 1 ··· k ∈ 1 ··· n ∈ tes as x(u , ,u ) (v (u , ,u ), ,v (u , u )) 1 ··· k = 1 1 ··· k ··· n 1 ··· k From condition 2 we can suppose that

∂(v1, ,v ) ··· k (q) 0 ∂(u , ,u ) 6= 1 ··· k n k n Extend x to a mapping F : U R − R given by × → F (u1, ,uk ,tk 1, ,tn) (v1(u1, ,uk ), ,vk (u1, ,uk ),vk 1(u1, ,uk ) ··· + ··· = ··· ··· ··· + ··· tk 1, ,vn(u1, ,uk ) tn) + + ··· ··· + n k where (tk 1, ,tn) R − . It is clear that F is differentiable and the restriction of F to U + ··· ∈ × {(0, ,0)} coincides with x. By a calculation, we obtain that ··· ∂(v1, ,vk ) det(dFq ) ··· (q) 0 = ∂(u , ,u ) 6= 1 ··· k Now we are under the conditions of the , which guarantees the exis- 1 tence of a neighborhood Q of x(q) where F − exists and is differentiable. By the continuity of y, there exists a neighborhood R V of r such that y(R) Q. Note that the restriction of h 1 ⊂ ⊂ to R, h R F − y R is a composition of differentiable mappings. Thus h is differentiable at r , | = 1 ◦ | 1 hence in y− (W ). A similar argument would prove that h− is differentiable as well, proving the assertion.

Definition 11: Let M be a differentiable manifold. We say that M is orientable if M admits a differentiable structure {(Uα,xα)} such that:

3 for every pair α,β with xα(Uα) xβ(Uβ) W , the differential of the change of coor- 1 ∩ = 6= ; dinates x− x has positive determinant. β ◦ α In the opposite case, we say that M is non-orientable. If M is orientable, a choice of a diffe- rentiable structure satisfying th condition from above is called as orientation of M.

Example 12 (Discontinuous of a ): We say that a group G acts on a differentiable manifold M if there exists a mapping ϕ : G M M such that: × → 1. For each g G, the mapping ϕ : M M given by ϕ (p) ϕ(g,p), p M, is a diffeo- ∈ g → g = ∈ , and ϕ Id . e = M 2. If g ,g G, ϕ ϕ ϕ . 1 2 ∈ g1g2 = g1 ◦ g2 Frequently, when dealing with a single action, we set g p : ϕ(g,p). = We say that the action is properly discontinuous if every p M has a neighborhood U M ∈ ⊂ such that U g(U) for all g e. When G acts on M, the action determines an equivalence ∩ = ; 6= relation on M, in which p p if and only if p g p for some g G. Denote the quotient ∼ 1 ∼ 2 2 = 1 ∈ space of M by this equivalence relation by M/G. The mapping π : M M/G,p Gp will be → 7→ called the projection of M onto M/G. Now let M be a differentiable manifold and let G M M be a properly discontinuous action × → of a group G on M. We are going to show that there is a differentiable structure on M/G such that π : M M/G is a local diffeomorphism. → For each point p M choose a parametrization x : V M at p so that x(V ) U, where U M ∈ → ⊂ ⊂ is a neighborhood of p such that U g(U) ,g e. Because of that π U is injective, hence ∩ = ; 6= | y π x : V M/G is injective. The family {(V, y)} clearly covers M/G. Now we are going to = ◦ → show that this family is a differentiable structure; let y π x : V M/G and y π x : 1 = ◦ 1 1 → 2 = ◦ 2 V2 M/G be two mappings with y1(V1) y2(V2) and let πi be the restriction of π to → ∩ 6=1 ; 1 x (V ), i 1,2. Let q y (V ) y (V ) and let r x− π− (q). Let W V be a neighborhood i i = ∈ 1 1 ∩ 2 2 = 2 ◦ 2 ⊂ 2 of r such that (π x )(W ) y (V ) y (V ). Then the restriction to W is given by 2 ◦ 2 ⊂ 1 1 ∩ 2 2 1 1 1 (y1− y2) W x1− π1− π2 x2 ◦ | = ◦ ◦ ◦ 1 1 1 Therefore, it is enough to show that π− π is differentiable at p π− (q). Let p π− 1 ◦ 2 2 = 2 1 = 1 ◦ π2(p2). Then p1 and p2 are equivalent in M, hence there is a g G such that g p2 p1. It 1 ∈ = follows easily that the restriction (π1− π2) x2(W ) coincides with the diffeomorphism ϕ x2(W ), 1 ◦ | | which proves that π− π is differentiable at p and so we have constructed a differentiable 1 ◦ 2 2 structure on M/G.

Proposition 13: With the notation from the previous example the manifold M/G is orientable if and only if there exists an orientation of M that is preserved by all the of G.

Definition 14: A vector field X on a differentiable manifold M is a correspondence that aso- ciates to each point p M a vector X (p) T M. In terms of mappings, X is a mapping of M ∈ ∈ p into the TM. The field is differentiable if the mapping X : M TM is diffe- → rentiable.

4 Considering the parametrization x : U Rn M we can write ⊂ → n X ∂ X (p) ai (p) = i 1 ∂xi = ∂ where each ai : U R is a function on U and { } is the basis asociated to x, i 1, ,n. It → ∂xi = ··· is clear that X is differentiable if and only if the functions ai are differentiable for some (and, therefore, for any) parametrization. Hence if f is a on M, we have the following expression X ∂ (X f )(p) ai (p) (p) = i ∂xi where f denotes the expression of f in the parametrization x. It is immediate that X is dif- ferentiable if and only if X f D for any f D, that is when X : D D. Observe that if ∈ ∈ → ϕ : M M is a diffeomorphism, v T M and f is a differentiable function in a neighbor- → ∈ p hood of ϕ(p), we have ¯ d ¯ (dϕ(v)f )ϕ(p) (f ϕ α)¯ v(f ϕ)(p) = dt ◦ ◦ ¯t 0 = ◦ = Now, we are able to see some results of vector fields.

Lemma 15: Let X and Y be differentiable vector fields on a differentiable manifold M. Then there exists a unique vector field Z such that, for every differentiable function f , Z f (XY = − YX )f .

The vector field Z is called the bracket of X and Y (Z [X ,Y ] XY YX ) and has the follo- = = − wing properties.

Proposition 16: If X ,Y and z are differentiable vector fields on M, a,b R and f ,g are diffe- ∈ rentiable functions, then: 1.[ X ,Y ] [Y , X ] = − 2.[ aX bY , Z ] a[X , Z ] b[Y , Z ] + = + 3. [[X ,Y ], Z ] [[Y , Z ], X ] [[Z , X ],Y ] 0 + + = 4.[ f X ,gY ] f g[X ,Y ] f X (g)Y gY (f )X = + − Theorem 17: Let X be a differentiable vector field on a differentiable manifold M, and let p M. Then there exist a neighborhood U M of p, an ( δ,δ), δ 0, and a cur- ∈ ⊂ − > ve t ϕ(t,q), t ( δ,δ), q U, is the unique curve which satisfies ∂ϕ X (ϕ(t,q)) and 7→ ∈ − ∈ ∂t = ϕ(0,q) q. The function ϕ : U M,q ϕ(t,q) is called the local flow of X . = t → 7→ Proposition 18: Let X ,Y be differentiable vector fields on a differentiable manifold M, let p M, and let ϕ be the local flow of X in a neighborhood U of p. Then ∈ t 1 [X ,Y ](p) l´ım [Y dϕt Y ](ϕt (p)) = t 0 t − →

5 Definition 19: [ 1. Locally finite: A family of open sets Vα M with Vα M is said to e locally finite if ⊂ α = every point p M has a neighborhood W such that W V for only a finite number ∈ ∩ α 6= ; of indices.

2. : The support of a function f : M R is the closure of the set of points where f → is different from zero.

3. Differentiable : We say that a family {fα} of differentiable functions fα : M R is a differentiable partition of unity if: → For all α, f 0 and the support of f is contained in a coordinate neighborhood α ≥ α V x (U ) of a differentiable structure {(U ,x )} of M. α = α α β β The family {Vα} is locally finite. X fα(p) 1, for all p M (this condition makes sense, because for each p M, α = ∈ ∈ f (p) 0 only for a finite number of indices). α 6= Theorem 20: A differentiable manifold M has a partition of unity if and only if every connec- ted component of M is Hausdorff and has a countable basis.

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