We shall see shortly that the Lie derivative corresponds to an- an-other simple binary operation on vector fields, the Lie bracket

We shall see shortly that the Lie derivative corresponds to an-other simple binary operation on vector fields, the Lie bracket.

1We remark that if M is compact, the interval (ap, bp) can be extended to (−^{∞, ∞})and φt will be a diffeomorphism from M to itself. In general, a vector field is called complete if it has this property.

7. VECTOR FIELDS AND FROBENIUS THEOREM 31

Definition 1.34. Let X, Y be C^∞ vector fields on U. The Lie bracket is defined by[X, Y] :=XY−YX. That is,

[X, Y]_pf =X_p(Y(f))−^Yp(X(f)).

It seems that[X, Y]is a second order operator on C^∞(U). In fact, it is still of the first order:

Proposition 1.35. [X, Y]is a C^∞ vector field on U. The Lie bracket has the following basic (Lie algebra) properties:

Proposition 1.36. Let X, Y be C^∞ vector fields. Then (1) (Anti-symmetry)[X, Y] = −[^{Y, X}]

(2) (Jacobi identity)[[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0 Now we can prove the important formula:

Theorem 1.37 (Lie, 1870). (L_VW)_p = [V, W]_p

PROOF. Let V generate φ_tand W generate ψ_s. Choose a test func-tion h∈ ^C∞p. By definition,

(L_VW)_ph We may extend these concepts to some higher dimensional cases.

Definition 1.38. A k dimensional distribution D on a manifold M is a choice of a k dimensional subspace D(p)of T_pM for each p∈ ^M.

We say D is C^∞ if for every p ∈ M, there is a neighborhood U such that D is spanned by k C^∞vector fields X₁, X2, . . . , X_kon U.

The question in higher dimensions becomes: for a given distribu-tion D near p, does there exists a k dimensional submanifold S with p ∈S such that TqS=D(q)for all q∈ ^S?

If so, S is called an integral manifold of D passing through p.

Clearly a necessary condition for this integrability is the following:

Definition 1.39. D is called involutive if [X, Y] ∈ ^{D for all X, Y} ∈ ^D.

Here a vector field X ∈ ^{D meansX}p ∈ ^D(p)for all p.

This turns out to be sufficient at least locally:

Theorem 1.40 (Frobenius Integrability, 1877). If D is involutive, then for all p ∈ M there exists a maximal integral manifold S_p ⊂ ^{M passing} through p such that T_qS_p=D(q)for all q ∈ ^Sp.

7. VECTOR FIELDS AND FROBENIUS THEOREM 33

As in the case of integral curves, S_p ⊂ M needs not be closed.

That is S_pmay not be with the induced topology.

The proof is based on two lemmas. To state the first one, we need Definition 1.41. For a smooth map f : S → M and a smooth vector field X∈ ^C∞(TS), the tangent map d f = f_∗sends X to

The next lemma characterizes coordinate vector fields:

Lemma 1.42. If X1, . . . , X_k are k vector fields near a point p ∈ ^{M that}

7. VECTOR FIELDS AND FROBENIUS THEOREM 33

As in the case of integral curves, S_p ⇢ M needs not be closed.

That is S_pmay not be with the induced topology.

The proof is based on two lemmas. To state the first one, we need Definition 1.41. For a smooth map f : S ! M and a smooth vector field X 2 ^C•(TS), the tangent map d f = f_⇤ sends X to

The next lemma characterizes coordinate vector fields:

Lemma 1.42. If X1, . . . , X_k are k vector fields near a point p 2 ^{M that}

Let c(t, s) = ψ_−s ◦^φ−t ◦^ψs ◦^φt(p). Since L_VW = 0 implies φ_−t∗W =W, directly differentiation shows that

∂

∂sc(t, s) = −^Wψ_−s◦φ_−t◦ψs◦φt(p) +ψ_−s∗◦^φ−t∗(W_ψs◦φt(p))

=−^Wc(t,s)+W_c(t,s) =0

Similarly, ∂c(t, s)/∂t =0. So c(t, s) = p for all well-defined t, s.

Now, suppose X_igenerates φⁱfor i =1, 2, . . . , k near p. We define c(t¹, t², . . . , t^k) :=φ¹_t1◦^φ2_t²◦ · · · ◦^φk_tk.

We check that its coordinate tangent vectors are precisely X1, . . . , X_k: c_∗(∂/∂t¹) = ^∂

∂t¹φ¹_t1 ◦ · · · ◦^φ_t^kk =X₁

c(t¹,...,t^k). And since every φⁱ_ti commutes with each other,

c_∗(∂/∂tⁱ) = ^∂

∂tⁱ φⁱ_ti ◦^φ_t¹1◦ · · ·^cφi_ti· · · ◦^φk_tk =X_ic

(t¹,...,t^k).

Let S := {^c(t¹, . . . , t^k) | ^tj ∈ ^Ij, ∀ ^j } ^{where I}j is chosen to be small enough such that every φ_t^j_j is well-defined for t^j ∈ ^Ij. We see that c_∗ is injective and then c is an immersion into M. By Lemma 1.16, there exists a chart (U, x) near p such that xⁱ(c(t¹, . . . , t^k)) = tⁱ for i = 1, . . . , k and xⁱ(c(t¹, . . . , t^k)) = 0 for i = k+1, . . . , m. Hence S∩U is a coordinate slice{^xj =0| ^j =k+1, . . . , m}^{on M.}  PROOF OF FROBENIUS THEOREM. We fix an arbitrary point p ∈ M. Since D(p) ⊂ ^TpM is a k dimensional subspace, we can select a chart(U, x) at p with D(p) = ^R∂₁|p, . . . , ∂_k|p

and construct the projection π : U ⊂ ^M → ^Rk onto the first k coordinates. Then there exists a smaller neighborhood U⁰ 3 p such that

D ∼=π_∗D =^R

 ∂

∂x¹, . . . , ∂

∂x^k



, ∀^q ∈^U0.

For each i = 1, . . . , k, let X_i ∈ D be the vector field lifted from

∂/∂xⁱ, i.e. π_∗(X_i) =∂/∂xⁱ. Then they are π-related and 0=

 ∂

∂xⁱ, ∂

∂x^j



= [π_∗(X_i), π_∗(X_j)] = π_∗[X_i, X_j]⇐⇒ [^Xi, X_j] = 0.

8. EXISTENCE, UNIQUENESS AND SMOOTH DEPENDENCE OF ODE 35 8. EXISTENCE, UNIQUENESS AND SMOOTH DEPENDENCE OF ODE 35

T_pM

The above lemma then implies that there exists an integral manifold S passing through p such that T_qS=D(q)for all q2 ^S.

Since p2 M is arbitrary, the union of all integral manifolds is the whole manifold M. Also for any two integral manifolds S and S⁰, if S\^S0 6= ^{∆ then S}[^S0is also an integral manifold. We conclude that there is a maximal integral manifold S_ppassing through p. ⇤

8. Existence, uniqueness and smooth dependence of ODE Now we go back to the ODE system:

8<

:

X⁰(t) = F(X(t)), F 2 ^C1(O, Rⁿ)a vector field on O⇢^Rn, X(0) = x0,

and assume the Picard–Lindel¨of theorem that there exists a unique continuous function f(t, x₀)satisfying the equation for t2 ^{J, a} max-imal interval for the existence of solutions, and with f(0, x₀) = x0.¹

It is obviously that _∂t^∂f(t, x0)exists. So, our goal is to discuss: the smooth dependence of the solution f(t, x0)on its initial value x0.

It turns out that f(t, x0) is C¹ in x0. Moreover, an iterative argu-ment then implies the C^kcase as stated in Theorem1.32.

1In this section we work on Rⁿ entirely and the symbols X, Y, Z etc. will be used to denote points in Rⁿ. This should not be confused with the same symbols in the last section which denote vector fields on a manifold.

The above lemma then implies that there exists an integral manifold S passing through p such that T_qS=D(q)for all q ∈^S.

Since p∈ M is arbitrary, the union of all integral manifolds is the whole manifold M. Also for any two integral manifolds S and S⁰, if S∩^S0 6= ^{∅ then S}∪^S0is also an integral manifold. We conclude that there is a maximal integral manifold S_ppassing through p. 

8. Existence, uniqueness and smooth dependence of ODE Now we go back to the ODE system:





X⁰(t) = F(X(t)), F∈ ^C1(O, Rⁿ)a vector field on O⊂^Rn, X(0) =x0,

and assume the Picard–Lindel¨of theorem that there exists a unique continuous function φ(t, x0)satisfying the equation for t ∈ ^{J, a} max-imal interval for the existence of solutions, and with φ(0, x₀) =x0.²

It is obviously that _∂t^∂φ(t, x0)exists. So, our goal is to discuss: the smooth dependence of the solution φ(t, x0)on its initial value x0.

It turns out that φ(t, x0)is C¹in x0. Moreover, an iterative argu-ment then implies the C^kcase as stated in Theorem1.32.

2In this section we work on Rⁿ entirely and the symbols X, Y, Z etc. will be used to denote points in Rⁿ. This should not be confused with the same symbols in the last section which denote vector fields on a manifold.

Historically there exists two different proofs of this theorem, one goes through a classical method by estimates (c.f. [HSD13]) and the other makes use of the inverse function theorem on Banach spaces.

Below we follow the first method closely.

Suppose there are two solutions X(t) and ˜X(t) with the given initial data x0and x0+z0. The key point is to estimatek^X(t)− ^˜X(t)k in terms of x0and z0. We consider the variational equation:

(*)





U⁰(t) = A(t)U(t), U(0) = z0

where A(t) = F⁰(X(t)), which is C⁰dependent on t. The idea is that, when z0 is small, X(t) +U(t) should approximate ˜X(t) with initial data x0+z0. In fact, this comes from the intuition that if F is C², then the solution to the variational problem is just the first order term of the Taylor expansion for ˜X(t)’s in z0.

Proposition 1.43. Let U(t, ξ)be the flow of (*), i.e. U(0, ξ) = ξ, x0+ξ ∈ O, and Y(t, ξ)be the flow of X⁰(t) = F(X(t))with Y(0, ξ) = ξ. Then

kξlimk→⁰

k^Y(t, ξ)−^X(t)−^U(t, ξ)k

k^ξk =0

uniformly on an interval of existence J.

Assuming the proposition, the theorem follows immediately.

Theorem 1.44. If F ∈ ^Ck, then the flow φ(t, x0) of the ODE system X⁰(t) = F(X(t)), X(0) = x0is C^kas well.

PROOF. By the proposition,

φ(t, x₀+ξ)−^φ(t, x₀) =Y(t, ξ)−^X(t) =U(t, ξ) +o(|^ξ|)^. Note that from solving the linear system U⁰(t, ξ) = A(t)U(t, ξ)with U(0, ξ) = ξ, we see that

U(t, ξ) = e^A(t)ξ

is linear in ξ. Hence D2φ(t, x0)ξ =U(t, ξ)and φ(t, x)is C¹in x. This proves the theorem for the case k =1.

8. EXISTENCE, UNIQUENESS AND SMOOTH DEPENDENCE OF ODE 37

Back to the variational equation (*), we get d

dt(D2φ(t, x0)) = F⁰(φ(t, x0))D2φ(t, x0)

with D2φ(0, x0) = idRⁿ. Then by induction, F ∈ ^Ck implies that

φ(t, x)is C^k in x. 

PROOF OF PROPOSITION. We rewrite the differential equations into integral equations as:

X(t) =x0+ Z _t

0 F(X(s))ds, Y(t, ξ) =x0+ξ+

Z _t

0 F(Y(s, ξ))ds, U(t, ξ) =ξ+

Z _t

0 F⁰(X(s))U(s, ξ)ds.

By the Taylor expansion, we have an estimate:

k^Y(t, ξ)−^X(t)−^U(t, ξ)k

≤ Z _t

0 k^F(Y(s, ξ))−^F(X(s))−^F0(X(s))U(s, ξ)k^ds

≤ Z _t

0

|^F0(X(s))kk^Y(s, ξ)−^X(s)−^U(s, ξ)k

+k^R(X(s), Y(s, ξ)−^X(s))k^ds where R is the first order remainder term.

We use the Gronwall’s inequality to deal with the iteration of dif-ference appearing in the integral.

Exercise 1.21 (Gronwall’s inequality, an easy version). If u ∈ ^C1[0, d], u > 0 and u satisfies u(t) ≤ ^c+R_t

0Ku(s)ds for some positive con-stants c, K. Then u≤^{ceKt on}[0, d].

Therefore, the constant c=|^ξ|can be taken to be small and k^Y(t, ξ)−^X(t)k ≤ k^ξk +

Z _t

0 k^F(Y(s, ξ))−^F(X(s))k^ds

≤ k^ξk + Z _t

0 k^F0kk^Y(s, ξ)−^X(s)k^ds

Choose K large such that k^F0k < K on a small neighborhood. By Gronwall’s inequality, k^Y(t, ξ)−^X(t)k ≤ k^ξk^eKt. So, for any e > 0, we can choose ξ small such that the remainder term

k^R(X(s), Y(s, ξ)−^X(s))k ≤^ek^Y(s, ξ)−^X(s)k^. Denote g(t) =k^Y(t, ξ)−^X(t)−^U(t, ξ)kand rewrite

g(t)≤ Z _t

0 Kg(s) +ek^ξk^eKsds≤^ek^ξk^C+ Z _t

0 Kg(s)ds

for some bounded constant C depending on F and the existece inter-val J. By Gronwall’s inequality again, g(t) ≤^ek^ξk^{CeKt and hence}

k^Y(t, ξ)−^X(t)−^U(t, ξ)k k^ξk ≤^{e ˜C}

which is uniformly in t. 

9. Problems

1. ([War83] Ch.1 #10) Let M be a compact manifold of dimension n, and let f : M→^{Rnbe C∞}. Prove that f cannot everywhere be non-singular.

2. ([War83] Ch.1 #3) Let{^Uα}be an open cover of a manifold M. Prove that there exists a refinement{^Vα}such that Vα ⊂^Uαfor each α.

3. ([War83] Ch.1 #9) Let f : R²→^Rbe defined by f(x, y) =x³+xy+y³+1.

For which points p = (0, 0), p = (¹₃,¹₃), p = (−¹₃^,−¹₃)is f⁻¹ f(p)
an imbedded submanifold in R?

4. ([War83] Ch.1 #16) Let N⊂M be a submanifold. Let γ : (a, b)→ ^{M be a} C^∞ curve such that γ(a, b)⊂ N. Show that it is not necessarily true that

˙γ(t)∈ ^Nγ(t)for each t∈ (^{a, b}).

5. ([War83] Ch.1 #17) Prove that any C^∞vector field on a compact manifold is complete.

6. ([War83] Ch.1 #18) Prove that a C^∞map f : R² →^R1cannot be one-to-one.

7. ([War83] Ch.1 #23) A Riemannian structure on a differentiable manifold M is a smooth choice of a positive definite inner producth^, im on each tangent space Mm, smooth in the sense that whenever X and Y are C^∞ vector fields on M, thenh^{X, Y}i^{is a C∞}function on M. Prove that there exists a Riemannian structure on every differentiable manifold. You will

9. PROBLEMS 39

need to use a partition of unity argument. A Riemannian manifold is a differentiable manifold together with a Riemannian structure.

8. ([War83] Ch.1 #6) Prove that if ψ : M→ ^{N is C∞}, one-to-one, onto, and everywhere non-singular, then ψ is a diffeomorphism.

9. ([War83] Ch.1 #19) Supply the details of the equivalence of the Frobenius theorem1.40and the classical version:

Remark 1.45 (classical Frobenius theorem). Let U and V be open sets in R^m and Rⁿ respectively. We use coordinates r₁, . . . , r_m on R^m and s₁, . . . , snon Rⁿ. Let

b : U×^V→ ^M(n, m)

be a C^∞ map of U×V into the set of all n×m real matrices, and let (r₀, s₀)∈^U×^{V. If}

∂b_iβ

∂rγ −^∂b_∂r^iγ

β

+

∑

n j=1

∂b_iβ

∂s_j b_jγ−^∂b_∂s^iγ

j b_jβ



=0

(i=1, . . . , n; γ, β=1, . . . , m) on U×V, then there exist neighborhoods U₀of r₀in U and V₀of s₀in V and a unique C^∞map

α: U₀×^V0→^V such that if

αs(r) =α(r, s) (s ∈^V0, r∈^U0) then

αs(r₀) =s, dα_sr =b(r, α(r, s)) for all(r, s)∈^U0×^V0.

10. ([War83] Ch.1 #20) Let ϕ : N → ^{M be C∞}, and let X be a C^∞vector field on N. Suppose that dϕ(X(p)) = dϕ(X(q))whenever ϕ(p) = ϕ(q). Is there a smooth vector field Y on M which is ϕ-related to X?

在文檔中 Differential Geometry Chin-Lung Wang (頁 30-40)