Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

Подождите немного. Документ загружается.

86 3 Ordinary Diﬀerential Equations

u ∈ [0, ∞). Clearly, Ψ satisﬁes (3.16). Let ·, · be a Riemannian metric on

M. Consider ·, ·

∗

= Ψ

−2

(m)



ρ(m

,m)



·, ·

.ByTheorem3.40, ·, ·

∗

complete. Now it suﬃces to observe that the length of every integral curve of

the ﬁeld X(t, m) in the metric ·, ·

∗

is bounded above on any interval [a, b)



ϕ(t)dt. 

Wintner’s theorem is a particular case of the corollary, where M is the

Euclidean space R

3.2 Integral Operators with Parallel Translation

Ordinary diﬀerential equations on a vector space can be turned into equiva-

lent Volterra type integral equations. In fact, this method is very often used

in the investigation of ordinary diﬀerential equations. For example, one can

turn the Cauchy problem ˙x = f



t, x(t)



,x(0) = x

in R

into the integral

equation x(t)=x





τ,x(τ )



dτ. To do so, we use the fact that for any

continuous curve y(t) there exists a unique curve z(t)=





τ,y(τ)



dτ such

that the derivative of z(t)atanypointt is equal to f(t, y(t)). The existence

of the curve z(t) is possible only because of global parallelism on the tangent

bundle to the vector space. Indeed, the vectors ˙z(t)andf (t, y(t)) belong to the

tangent spaces at diﬀerent points, and the equation ˙z(t)=f (t, y(t)) makes

sense only by virtue of the existence of global parallelism, i.e., a canonical

isomorphism between all tangent spaces and the vector space itself.

Global parallelism does not exist on an arbitrary manifold. Therefore clas-

sical integrals can be used only locally (in charts) and, moreover, the integrals

themselves depend on the choice of the coordinate system. In this section,

following [86, 88, 94], we describe a construction of an analog of the integral

operator, in which global parallelism is replaced with parallel translation

(with respect to a connection) along a chosen curve. For simplicity we use

the Levi-Civit´a connection on a complete ﬁnite-dimensional manifold.

Similar notions of absolute and covariant integrals were introduced in a

diﬀerent way by Vujiˇci´c. (See, e.g., [227, 228, 229] and the bibliography in

[229].) There, the integral is deﬁned in a local coordinate system with the

connection coeﬃcients used in such a way that the integral becomes covariant

with respect to changes of coordinates.

3.2.1 The operator S

Let M be a complete Riemannian manifold, let m

∈ M, I =[0,l]andlet

v : I → T

M be a continuous curve.

3.2 Integral Operators with Parallel Translation 87

Theorem 3.43 There exists a unique C

-curve γ : I → M such that γ(0) =

and the tangent vector ˙γ(t) is parallel to the vector v(t) ∈ T

M for every

t ∈ I.

Proof. Let b

=(e

,...,e

) be a basis in the tangent space T

M; b

gives

rise to an isomorphism between R

and T

M by the formula b

,...,x

+ ...+ x

,whereR

is the model space of M.

Consider the time-dependent basic vector ﬁeld E(b

−1

v(t)) on the frame

bundle B(M ). Clearly, this ﬁeld is locally Lipschitz in b ∈ B(M ). Hence,

for every point b ∈ B(M), there exists a unique integral curve b(t) passing

through b, b(0) = b. The curve γ(t)=πb

(t) is the one we are looking for.

(Here π is the natural projection of B(M)toM .) Indeed, for any point t

∗

the domain of γ(·), the vectors ˙γ(t

∗

)andv(t

∗

) are connected along γ(·)by

the parallel vector ﬁeld b

(t)



−1

v(t

∗

)



It remains to prove that γ(·) is deﬁned on the whole interval [0,l]. Since

the metric on M is complete, the metric ball of radius



v(s)ds centered

at m

is compact. Now assume that γ(t) is deﬁned on [ 0,t

∗

), where t

∗

∈ [0,l].

The length of γ(·)on[0,t

∗

) equals



∗

v(s) ds ≤



v(s) ds,

i.e., γ(t) belongs to a compact set and, therefore, can be extended to [0,t

∗

It is clear that one can extend γ(·) to a neighborhood of t

∗

. Thus, the domain

of γ(·) is open and closed in [0,l], i.e., it coincides with [0,l]. The theorem is

proved. 

In what follows, we denote by Sv(·) the curve γ constructed as above

beginning with v.

Remark 3.44. It is important to emphasize that S(v(t)), for a continuous

curve v ∈ C

(I,T

M)), is independent of the basis b

used in the proof

of Theorem 3.43. To see this, let us go back to the construction of γ(·)and

replace the basis b

by b

in T

M. Since there exists an invertible n × n

matrix b such that b

= b

◦ b and since the connection is invariant with

respect to the right action of GL(n, R)onBM, it follows from the deﬁnition

of a basic vector ﬁeld that πb

(t)=γ(t).

Remark 3.45. Let m(t)beaC

-smooth curve in M, t ∈ I, m(0) = m

.De-

note by Γ the operator of parallel translation along m(·)atT

M. The curve

C(m)(t)=



Γ ˙m(s)ds is known as Cartan’s development of m(t)atT

The curve Sv(·) introduced above is expressed via Cartan’s development as

Sv(t)=C

−1

(



v(s)ds).

Consider the Banach space C

(I,T

M) of continuous maps from I to

M and the Banach manifold C

(I,M)ofC

-smooth maps from I to M.

88 3 Ordinary Diﬀerential Equations

As follows from Theorem 3.43, the operator S: C

(I,T

M) → C

(I,M)is

well-deﬁned. If M is a Euclidean space, Sv is a primitive of v.

It is easy to see that S is a homeomorphism between C

(I,T

M)andits

image C

(I,M)inC

(I,M), where the manifold C

(I,M) consists of all

-curves γ with γ(0) = m

Theorem 3.46 Let U

be the ball of radius K centered at the origin of

(I,T

M). Then, at every point t ∈ I, the inequality



˙γ(t)



≤ K holds

for all curves γ(·) in the set SU

This is obvious since parallel translation preserves the norm of a vector.

Theorem 3.47 Assume that the point m

∈ M is not conjugate to m

along

some geodesic of the Levi-Civit´a connection on M. Then for any geodesic

α(·), α(0) = m

, α(1) = m

, along which m

and m

are not conjugate,

and for any K>0, there exists a constant

L(m

,K,α) > 0 with the

following property: for any t

, 0 <t

L(m

,K,α), and for any curve

u(·) ∈U

⊂ C



[0,t

],T



, there exists a unique vector C

∈ T

such that S(u + C

)(t

)=m

, which belongs to a bounded neighborhood of

−1

· ˙α(0) ∈ T

M and depends continuously on u.

Proof. We divide the proof into two lemmas. Without loss of generality we

assume that the parameter t on g(t) is chosen so that g(0) = m

and g(1) =

Lemma 3.48 There exists a ball U

⊂ C

([0, 1],T

M) with radius ε>0

such that for any curve ˆu(t) ∈ U

⊂ C

([0, 1],T

M) there exists a unique

vector C

ˆu

, belonging to a bounded neighborhood V of the vector ˙α(0) in

M, that is continuous in ˆu and such that S(ˆu + C

ˆu

)(1) = m

Proof. [of Lemma 3.48] Consider the mapping from C



[0, 1],T



×T

to M sending a pair (u, C), where u ∈ C



[0, 1],T



and C ∈ T

M,to

the point S(u + C)(1). Note that the vector ﬁeld E(b

−1

(v(t)) on B(M )is

smooth in v(·). (See the proof of Theorem 3.43.) Using this fact, the deﬁnition

of S, and the classical theorem on the smooth dependence of solutions of

diﬀerential equations on parameters, one can easily show that this map is

jointly smooth in u and C. Clearly, we have S(C)(1) = exp(C)whenu =0.

Thus, by the hypotheses of the theorem, S



˙α(0)



(1) = m

and S(C)(1) is a

diﬀeomorphism from a neighborhood of ˙α(0) in T

M onto a neighborhood

of m

in M.

Let us now think of S(u + C)(1) as a perturbation of S(C)(1) = exp(C).

Thus, there exists a ε>0 such that for any ﬁxed ˆu ∈U

⊂ C



[0, 1],T



the operator S(ˆu + C)(1) is a local diﬀeomorphism. Therefore, there is a ball

D ⊂ T

M centered at ˙α(0) such that, for any ˆu ∈U

, there exists a vector

ˆu

∈ D solving the equation S(ˆu + C

ˆu

)(1) = m

. Using the implicit function

theorem one may show that, when D is suﬃciently small, the vector C

ˆu

∈ D

is unique and C

ˆu

depends continuously on ˆu. 

3.2 Integral Operators with Parallel Translation 89

We introduce the notation sup

C∈V

C = C where V is as deﬁned in Lemma

3.48.

Remark 3.49. One can easily show that ε<C.

Lemma 3.50 In the conditions and notation of Lemma 3.48 let K>0

and t

> 0 be such that t

−1

ε>K. Then for any curve u(t) ∈ U

⊂

([0,t

],T

M) there exists a unique vector C

in a neighborhood t

−1

of the vector t

−1

˙γ(0) in T

M, continuously depending on u and such that

S(u + C

)(t

)=m

Proof. [of Lemma 3.50]Foru(t) ∈ U

⊂ C

([0,t

],T

M) let ˆu(t)=t

u(t

t) ∈ U

⊂ C

([0, 1],T

M)andC

= t

−1

ˆu

. From Lemma 3.48 we get

S(ˆu + C

ˆu

)(1) = m

and

S(ˆu + C

ˆu

)(t) is parallel to ˆu(t)+C

ˆu

.Forthe

curve γ(t)=S(ˆu + C

ˆu

)(t · t

)wehave

γ(t)=t

−1

S(ˆu + C

ˆu

)(t · t

)and

this vector is parallel along the same curve to the vector t

−1

(ˆu(t)+C

ˆu

u(t)+C

.Thusγ(t)=S(u + C

)(t)=S(ˆu + C

ˆu

)(t ·t

−1

)fort ∈ [0,t

]. Hence

S(u + C

)(t

)=S(ˆu + C

ˆu

)(1) = m

. 

This completes the proof of Theorem 3.47. 

Lemma 3.51 For speciﬁed t

> 0 and K>0 all curves S(v(t)+C

)(t) with

v ∈ U

⊂ C

([0,t

M) lie in a compact set Ξ ⊂ M,whereΞ depends on

ε and C is as deﬁned above.

Indeed, since parallel translation preserves the norm of a vector, for any

v(t) as above, the length of S(v(t)+C

)(t) is no greater than



(K +

C

)dt ≤



−1

(ε + C)dt =



(ε + C)dt = ε + C.SinceM is complete,

by the Hopf-Rinow theorem any closed metric ball of ﬁnite radius ε + C is

compact.

Remark 3.52. Note that if M is a Euclidean space, one can take any con-

stant as ε in the proof of Theorem 3.3., i.e., the theorem holds for every t

0 <t

< ∞.

3.2.2 The operator Γ

Let γ(t), t ∈ I,beaC

-curve in M and X(γ(t)) a continuous vector ﬁeld

along γ(·). Consider the curve ΓX



γ(t)



in T

γ(0)

M where Γ is the operator

of parallel translation along γ(·)atγ(0).

Lemma 3.53 (Compactness lemma). Let Ξ ⊂ C

(I,TM) be such that

πΞ ⊂ C

(I,M),whereπ : TM → M is the natural projection, and the norms

of the derivatives of the curves πΞ are uniformly bounded by a certain con-

stant K>0.IfΞ is relatively compact in C

(I,TM), then so is ΓΞ.

90 3 Ordinary Diﬀerential Equations

Proof. Since the closure

Ξ of the set Ξ is compact in C

(I,TM), the vectors



ξ(t)



ξ(·) ∈ Ξ



are bounded and the set π

Ξ is compact in C

(I,M). This

implies, in particular, that all curves in πΞ lie in a compact subset of‘M.

Let ξ

∗

(·) be a limit curve of the set Ξ. It is clear that the inequality

ρ (πξ

∗

(t),πξ

∗



)) ≤ K |t −t



where t, t



∈ I and ρ is the Riemannian distance, holds for the curve πξ

∗

(·).

Note that this curve may not be smooth. Parallel translation along such

curves was deﬁned in [29] as the limit of parallel translations along their

piecewise geodesic approximations. Moreover, it was shown that under the

hypothesis above, the procedure of parallel translation converges uniformly

on any bounded set of vectors. If the curve is smooth, the new deﬁnition

of parallel translation is equivalent to the classical one. It was also shown

that the parallel translation along a limit curve is just the limit of parallel

translations along curves converging to it. Therefore, Γ sends convergent

sequences to convergent ones. 

If X(γ(t)) = X(t, γ(t)) is the restriction to γ(·) of a continuous vec-

tor ﬁeld X(t, m), t ∈ I and m ∈ M , then we use the notation Γ ◦ γ for

ΓX(t, γ(t)). Thus, for a speciﬁed vector ﬁeld X(t, m), we may consider the

operator Γ : C

(I,M) → C

(I,TM), which is clearly continuous.

Let Ω

be the set of curves from C

(I,M) satisfying the inequality

˙γ(t)≤K,whereK>0, at every point t ∈ I and such that the set



γ(0)



γ(·) ∈ Ω



is bounded in M.

Theorem 3.54 The set of curves Γ (Ω

) is relatively compact in C

(I,TM).

Proof. Because M is complete, it is clear that Ω

is relatively compact in

(I,M). Since the ﬁeld X(t, m) is continuous, the set of curves





t, γ(t)





γ(·) ∈ Ω



is relatively compact in C

(I,TM). The theorem follows from

Lemma 3.53. 

Corollary 3.55 The operator Γ is locally compact.

Proof. For every γ ∈ C

(I,M), the continuous function ˙γ(t) assumes its

supremum K

on I. By the deﬁnition of the C

-topology, the inequality

˙γ

 <K

+ ε holds for every γ

(·) in a small neighborhood of γ(·). 

3.2.3 Integral operators

Consider the continuous composition operator

S◦Γ : C

(I,M) → C

(I,M).

3.2 Integral Operators with Parallel Translation 91

Theorem 3.56 The ﬁxed points of S◦Γ are precisely the integral curves of

the ﬁeld X(t, m) with the initial condition γ(0) = m

Proof. Let γ(t) be an integral curve of the ﬁeld X(t, m), i.e., ˙γ(t)=



t, γ(t)



. Then the operator Γ on γ(·)isequaltoS

−1

,andsoγ is a ﬁxed

point of S◦Γ . Conversely, let γ(·) be a ﬁxed point of the operator S◦Γ .

Using the parallel translation along γ(·), we transport the vector X



t, γ(t)



to γ(0) = m

and then back to γ(t). The resulting vector coincides with ˙γ(t)

by the deﬁnitions of S and Γ . Therefore, ˙γ(t)=X



t, γ(t)



. 

Thus, S◦Γ is a direct analog of the standard Urysohn-Volterra integral

operator from the theory of ordinary diﬀerential equations on vector spaces.

Theorem 3.57 The operator S◦Γ is locally compact.

The assertion of Theorem 3.57 follows from the local compactness of Γ

and the continuity of S.

Let Θ be a closed bounded set in M and C

(I,

Θ) the subset in C

(I,M)

formed by curves lying in the closure of Θ. Consider the second iteration

(S◦Γ )

of the operator S◦Γ .

Theorem 3.58 The set (S◦Γ )

(I,

Θ) is compact in C

(I,M).

Proof. Since Θ is bounded and M is complete, Θ is compact. Therefore,

X(t, m) is bounded on I × Θ by some constant K. Since parallel transla-

tion preserves the norm, the curves ΓC

(I,

Θ) lie in



m∈Θ

(m) and, by

Theorem 3.46,thesetS◦ΓC

(I,

Θ) is formed by curves which satisfy the

inequality ˙γ(t)≤K at every point t ∈ I. Now the theorem follows from

Theorem 3.54 and the continuity of the operator S. 

Composition operators, such as S◦Γ , are employed to solve certain prob-

lems in the theory of diﬀerential equations on manifolds (for example, to

ﬁnd periodic solutions for some special classes of diﬀerential equations). The

construction of such operators and their applications are described, e.g., in

the survey [33]. The theory of topological characteristics is also developed in

[28, 33, 109] for a large class of maps of inﬁnite-dimensional manifolds. This

theory enables one to prove the existence of ﬁxed points of these operators.

Let us discuss one more class of integral operators that can be used to

reduce certain problems on manifolds to problems on vector spaces. Consider

the composition Γ ◦S. This operator is continuous and acts on the Banach

space C

(I,T

M). If v = Γ ◦Sv,thenSv = S◦Γ ◦Sv =(S◦Γ )Sv is a

ﬁxed point of S◦Γ and so, an integral curve of the ﬁeld X(t, m). Conversely,

Sv = S◦Γ (Sv) implies that v = Γ ◦Sv because S is one-to-one.

Theorem 3.59 The operator Γ ◦S is completely continuous.

Proof. Let U

be a ball of radius K in C

(I,T

M). By Theorem 3.46,

⊂ Ω

and, by Theorem 3.54,thesetΓ ◦SU

is compact. 

92 3 Ordinary Diﬀerential Equations

Remark 3.60. As mentioned in the introduction to this section, we consider

the Levi-Civit´a connection of a complete Riemannian metric only to simplify

the presentation of the material. Under certain hypotheses, the constructions

of the integral operators may be generalized to other connections. Note, for

example, that we have never used the fact that the connection has zero tor-

sion, i.e., all of our constructions hold for any Riemannian connection of

a complete Riemannian metric. In particular, the construction leads to the

classical multiplicative integral for a special choice of the connection on a Lie

group. (See, e.g., [78] for matrix groups.)

3.3 Second Order Diﬀerential Equations (Special Vector

Fields)

Let M be a manifold with tangent bundle TM. On the manifold TM there is

a class of vector ﬁelds that is naturally coordinated with the bundle structure

of TM. This class describes the second order diﬀerential equations on M.

Recall the well-known trick of reducing a second order diﬀerential equation

in R

to a ﬁrst order diﬀerential equation in R

: the diﬀerential equation

¨x = f(t, x, ˙x)inR

is equivalent to the system of ﬁrst order diﬀerential

equations

˙x = y

˙y = f (t, x, y) (3.18)

in R

. We emphasize that a general system in R

has the form

˙x = f

(t, x, y)

˙y = f

(t, x, y),

i.e., (3.18) is a system of a special form.

It is clear that in every chart of a manifold a second order diﬀerential

equation must be reduced to a special system of ﬁrst order equations on

the tangent bundle. It turns out that such systems are vector ﬁelds of a

special type connected with the bundle structure. Let us introduce the exact

deﬁnition.

Deﬁnition 3.61. A vector ﬁeld Y (t, (m, X)) on the tangent bundle TM is

called a special vector ﬁeld on TM or a second order diﬀerential equation on

M if at every point (m, X) ∈ TM the equality

TπY(t, (m, X)) = X

(3.19)

holds where π : TM → M is the natural projection of TM onto M .

Recall that by Convention 1.3 X

and (m, X) are two equivalent desig-

nations of the same object. Below, if it is not necessary to emphasize that

3.3 Second Order Diﬀerential Equations (Special Vector Fields) 93

the vector Y (t, (m, X)) is given at (m, X) ∈ TM, we shall also denote it by

Y (t, m, X). Represent a vector Y at a point (m, X)onTM as a quadruple

Y (t, (m, X)) = (m, X, Y

(t, (m, X)),Y

(t, (m, X))),

as described in Section 2.1. By applying formula (2.3) we ﬁnd TπY(t, (m, X))

and substituting into the above, we obtain:

Tπ(m, X, Y

(t, (m, X)),Y

(t, (m, X))) = (m, Y

(t, (m, X))) = (m, X).

Thus, if Y is a second order diﬀerential equation Y

(t, (m, X)) = X,andso

the presentation as a quadruple takes the form

(m,X)

=(m,X, X, Y

(t, (m, X))). (3.20)

We show that second order diﬀerential equations (special vector ﬁelds), as

deﬁned in Deﬁnition 3.61, do indeed yield ordinary second order diﬀerential

equations in the charts of M . Let (m(t),X(t)) be an integral curve of a special

vector ﬁeld Y on TM. This means that

(m(t),X(t)) = Y

(m(t),X(t))

=(m(t),X(t),X(t),Y

(t, (m(t),X(t)))),

i.e.

m(t)=X(t)and

X(t)=Y

(t, (m(t),X(t))). Thus the curve m(t)=

π(m(t),X(t)) on M (the projection of the integral curve of Y ) satisﬁes the

equation ¨m(t)=Y

(t, m(t),X(t)) in the charts. The curve m(t)inM is

called a solution of the second order diﬀerential equation Y . It follows from

the above arguments that the integral curve of Y on TM is represented via

m(t) by the formula (m(t), ˙m(t)) where ˙m(t)=

m(t).

Let a connection H be given on a manifold M. Recall (see Lemma 2.12)

that at any point (m, X) ∈ TM the mapping Tπ is a linear isomorphism of

(m,X)

onto T

M. Hence in H

(m,X)

there is a unique vector Z

(m,X)

such

that

TπZ

(m,X)

= X

. (3.21)

On constructing such a vector at each point (m, X) ∈ TM, we obtain the

vector ﬁeld Z on TM. From formula (3.21) it immediately follows that Z is

a second order diﬀerential equation (special vector ﬁeld, see Deﬁnition 3.61).

Deﬁnition 3.62. The second order diﬀerential equation Z constructed above

is called the geodesic spray of the connection H.

Remark 3.63. A spray is a second order diﬀerential equation Y that has the

following property: for any real number a ∈ R and every point (m, X) ∈ TM

the equality Y

(m,aX)

= Ta(aY

(m,X)

) holds (see formula (2.7) where the action

of the real line on a vector bundle is deﬁned). One can easily verify that Z

satisﬁes the deﬁnition of a spray.

94 3 Ordinary Diﬀerential Equations

For a given connection H the geodesic spray Z is a universal object

for the description of all second order diﬀerential equations. Let Y be

such an equation. Since every tangent space to TM is presented as di-

rect sum T

(m,X)

TM = H

(m,X)

⊕ V

(m,X)

, there is a unique decomposition

(m,X)

= HY

(m,X)

+ VY

(m,X)

where HY

(m,X)

∈ H

(m,X)

is called the horizon-

tal component of Y

(m,X)

and VY

(m,X)

∈ V

(m,X)

is the vertical component (see

Section 2.2).

Proposition 3.64 For any second order diﬀerential equation Y its horizon-

tal component is Z.

Proof. By the deﬁnition of second order diﬀerential equations, TπY

(m,X)

.SinceV

(m,X)

is the kernel of Tπ (see Lemma 2.12), we get that

TπVY

(m,X)

=0andsoTπHY

(m,X)

= TπY

(m,X)

= X

.ButinH

(m,X)

,as

mentioned above, there is unique vector Z

(m,X)

having this property. Thus

(m,X)

= Z

(m,X)

. 

Let a curve m(t)inM be a solution of the second order diﬀerential equa-

tion Y , i.e., (m(t), ˙m(t)) is an integral curve of the vector ﬁeld Y on TM,

(m(t), ˙m(t)) = Y

(m(t), ˙m(t))

(see above). Apply the connector K of the con-

nection H to both sides of this equality. Since K ◦

and KY = pVY

(see Section 2.2), we obtain

˙m(t)=pVY. (3.22)

On the other hand, for the curve m(t) the derivative of the corresponding

curve (m(t), ˙m(t)) in TM is always a vector of some second order diﬀerential

equation for each t (i.e., property (3.19) is fulﬁlled). By Proposition 3.64,once

a connection on M given, any second order diﬀerential equation is uniquely

deﬁned by its vertical component. If in addition m(t) satisﬁes (3.22), this

equation coincides with Y .Thus, we have proved the following:

Proposition 3.65 The solutions m(t) of the second order diﬀerential equa-

tion Y , and only these solutions, satisfy equation (3.22).

Corollary 3.66 The solutions of the geodesic spray Z, and only these solu-

tions, are geodesics of the connection H.

Proof. Since Z

(m,X)

∈ H

(m,X)

and H

(m,X)

is the kernel of K (see Lemma

2.14(ii)), one obtains that KZ = 0 and equation (3.22) takes the form

˙m(t) = 0, i.e., m(t) satisﬁes (2.24). 

Corollary 3.66 together with Remark 3.63 clarify the name “geodesic

spray” for Z.

Consider the equation

˙m(t)=Y (t, m(t), ˙m(t)), (3.23)

3.3 Second Order Diﬀerential Equations (Special Vector Fields) 95

where Y (t, m, X) is a non-autonomous vector ﬁeld on M which depends, at

each point m ∈ M, on the vector parameter X ∈ T

M. Denote by Y

the

vector ﬁeld on TM that at (m, X) ∈ TM and t ∈ R is the vertical lift of

Y (t, m, X)atthepoint(m, X) (see Deﬁnition 2.34). Evidently we obtain

Proposition 3.67 If a curve m(t) satisﬁes (3.23), it is a solution of the

second order diﬀerential equation Y

+ Z.

Without proof we present the following classical result of ﬁnite-dimensional

Riemannian geometry (for a proof, see e.g., [26, 161]).

Theorem 3.68 (Hopf-Rinow theorem) For a ﬁnite-dimensional Riemann-

ian manifold M the following four statements are equivalent:

(i) M is a complete Riemannian manifold (see Deﬁnition 1.49), i.e., it is

complete as a metric space with respect to the Riemannian distance ρ;

(ii) every set that is bounded with respect to the Riemannian distance ρ is

relatively compact;

(iii) the Levi-Civit´a connection is complete in the sense of Deﬁnition 2.40;

(iv) the geodesics of the Levi-Civit´a connection, starting at some speciﬁed

point, are well-deﬁned for all t ∈ (−∞, ∞).

From equivalent statements (i)–(iv) it follows that:

(v) for every pair of points m

∈ M there exists a geodesic of the Levi-

Civit´a connection that joins them and whose length is equal to ρ(m

Remark 3.69 (Hamiltonian systems). The additional vector bundle

structure on the tangent bundle TM yielded above the special class of vec-

tor ﬁelds on TM, second order diﬀerential equations. Analogously, the ad-

ditional structure on the cotangent bundle yields a single special object

called the canonical 1-form, the diﬀerential form θ whose value on a vec-

tor Y ∈ T

(m,α)

∗

M at a point (m, α) ∈ T

∗

M is given by the formula

(m,α)

(Y )=α

(TπY), where π : T

∗

M → M is the natural projection.

By routine calculation one can easily show that at α

= p

the canon-

ical 1-form obtains the coordinate presentation θ

(m,α)

= p

that coincides

with the coordinate expression of α at m ∈ M.Forθ the coordinates corre-

spond to the covectors from the ﬁrst half of the basis in T

∗

(m,α)

∗

M while for

α the entire basis in T

∗

M is involved.

The canonical 2-form on T

∗

M is Ω =dθ. Its coordinate presentation is

Ω =dp

∧dq

. We obtain directly from the deﬁnition that Ω is exact and so

it is closed. In addition one can easily prove that Ω is not degenerate, i.e.,

for every 1-form β at every point (m, α) ∈ T

∗

M there exists a unique vector

∈ T

(m,α)

∗

M such that for every vector Y ∈ T

(m,α)

∗

M the equality

β(Y )=Ω(Y,X

) holds.

Let H : T

∗

M → R be a smooth function and consider its diﬀerential

dH (i.e. a covector ﬁeld). The vector ﬁeld X

such that for every vector

ﬁeld Y the equality dH(Y )=Ω(Y,X

) holds is called the screw gradient of

H or Hamiltonian vector ﬁeld with Hamiltonian H. For the screw gradient

= X

∂

∂q

∂

∂p

the coordinates take the form X

∂H

∂p

= −

∂H

∂q

, and