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

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76 3 Ordinary Diﬀerential Equations

Theorem 3.18 Let the embedding i : D → M be completely continuous,

let the Cauchy problem (3.11)–(3.12) be regular and let f : D

→ R be a

continuous weakly proper function such that for any curve m

(t,m)

(s) there

exists a real constant C>0 for which the relation





(t,m)

)



− f



(t,m)

)





<C|s

− s

holds for every pair s

and s

in the domain of the curve m

(t,m)

(s).Thenall

solutions of (3.11)–(3.12) are well-deﬁned for s ∈ (−∞, ∞).

Proof. Let m

(t,m)

(s) be an arbitrary solution of the Cauchy problem (3.11)–

(3.12)thatexistsfors ∈ (t − ε



,t + ε). If both ε and ε



are inﬁnite, the

Theorem is proved. Suppose they are ﬁnite.

As in the proof of Theorem 3.8, from the inequality





(t,m)

)



− f



(t,m)

)





<C|s

− s

along the curve m

(t,m)

(s) it follows that the curve m

(t,m)

(s)on(t −ε



,t+ ε)

belongs to a bounded set in D. Since the embedding of D into B is completely

continuous, this set is relatively compact in B.So,thecurvem

(t,m)

(s), con-

tinuous and relatively compact on (t−ε



,t+ε), can be extended to [t−ε



,t+ε]

in B. But since the solutions of (3.11)–(3.12) are regular, the extension be-

longs to D. Thus the domain of m

(t,m)

(s) is both open and closed and so it

coincides with R. 

Theorem 3.19 Let the Cauchy problem (3.11)–(3.12) be locally well-posed

and all solutions of (3.11)–(3.12) exist for s ∈ (−∞, ∞). Then there exists

a continuous weakly proper function f : D

→ R such that for any curve

(t,m)

(s) there exists a real constant C>0 for which the relation





(t,m)

)



− f



(t,m)

)





<C|s

− s

holds for every pair s

and s

Proof. This proof is a simple modiﬁcation of that for Theorem 3.11.Thespace

is replaced by D

. Existence, uniqueness and continuity of solutions in

now follow from the local well-posedness of the Cauchy problem according

to Deﬁnition 3.16. The remaining arguments are the same as in the proof of

Theorem 3.11. 

Theorem 3.20 Let the embedding i : D → B be completely continuous and

let the Cauchy problem (3.11)–(3.12) be locally well-posed and regular. Then

all solutions of (3.11)–(3.12) are well-deﬁned for all s ∈ (−∞, ∞) if and only

if there exists a continuous weakly proper function f : D

→ R such that for

any curve m

(t,m)

(s) there exists a real constant C>0 for which the relation

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 77





(t,m)

)



− f



(t,m)

)





<C|s

− s

holds for every pair s

and s

in the domain of the curve m

(t,m)

(s).

Theorem 3.20 is a simple corollary to Theorems 3.18 and 3.19.

3.1.2.3 The case of manifolds

Let M be a smooth Banach manifold with model Banach space B.Forthe

sake of convenience we consider charts on M as triples (U, V, ϕ), where V is

an open ball in the model space B, U is an open set in M and ϕ : V → U is

a homeomorphism. Recall that the structure of a smooth manifold is given

by the maximal atlas for whose charts (U

,ϕ

)and(U

,ϕ

), such

that U

∩U

= U

αβ

= ∅, the changes of coordinates ϕ

−1

◦ϕ

: ϕ

−1

αβ

) →

−1

αβ

)aresmooth.

In addition to the maximal atlas, which deﬁnes the smooth structure, we

shall also consider atlases (called continuous) that deﬁne the structure of a

topological manifold on M with the same model space B. This means that the

changes of coordinates between charts of such an atlas are only continuous.

The charts of a continuous atlas will be denoted by (

, ˜ϕ

In this section we also assume an additional property of the manifold:

Condition 3.21 There exists a continuous atlas such that the images of all

bounded sets in B under its coordinate changes are bounded.

Sometimes we shall also assume that M satisﬁes the following:

Condition 3.22 M has a continuous (as in Condition 3.21) countable lo-

cally ﬁnite atlas such that for any its charts (

, ˜ϕ

) the set

⊂ B is

bounded with respect to the norm of the model space B.

Deﬁnition 3.23. AsetΘ ⊂ M is called relatively weakly compact if there

exists a ﬁnite collection of charts (

, ˜ϕ

) of a continuous atlas on M such

that Θ ⊂



and for every i the set ˜ϕ

−1

(Θ



) ⊂

is bounded with

respect to the norm of the model space B that contains

Remark 3.24. Unlike the case of linear Banach spaces, the weak topology

is (generally speaking) ill-deﬁned on Banach manifolds while the slightly

stronger topology of weak convergence is well-deﬁned (for details, see [195]).

If the model space B of M is a reﬂexive Banach space, then under some nat-

ural conditions the relatively weakly compact set described in Deﬁnition 3.23

is relatively compact with respect to the topology of weak convergence on M

(see [195]). If M itself is a reﬂexive Banach space, then any relatively weakly

compact set (described in Deﬁnition 3.23) is relatively weakly compact (i.e.,

with respect to weak convergence) by the well-known properties of reﬂexive

Banach spaces (cf. Remark 3.7).

78 3 Ordinary Diﬀerential Equations

Deﬁnition 3.25. A function f : N → R on a Banach manifold N is called

weakly proper if for any relatively compact set in R its pre-image is relatively

weakly compact in N as in Deﬁnition 3.23.

Deﬁnition 3.26. A vector ﬁeld X(t, m)onM, t ∈ R, is called completely

continuous if its restriction to any chart (U, V, ϕ), as a mapping from R × V

into the model space B, is completely continuous in the ordinary sense.

For X(t, m) we shall consider the Cauchy problem (3.5)–(3.6)onM.We

retain the Deﬁnition 3.1 of a complete vector ﬁeld as well as the Deﬁnition

3.4 of local well-posedness of solutions of the Cauchy problem.

As above, we consider the extended phase space M

= R × M and the

vector ﬁeld X

(t,m)

=(1,X(t, m)) on it. By formula (3.7) we also introduce

the curves m

(t,m)

(s) that satisfy the Cauchy problem (3.8)–(3.9)onM

Theorem 3.27 Let X(t, m) be a completely continuous vector ﬁeld on a

Banach manifold M. Let there exist a continuous weakly proper function

f : M

→ R on M

such that for any curve m

(t,m)

(s), as introduced by

(3.7), there exists a real constant C>0 for which the relation (3.10) holds

for every pair s

and s

in the domain of the curve m

(t,m)

(s).ThenX(t, m)

is complete.

The proof of Theorem 3.27 is a simple modiﬁcation of that for Theorem 3.8

and is left to the reader.

We cannot prove a necessary condition of completeness in the same way

as in Theorem 3.11 since a weakly proper function such as r

(m)maynot

exist on an arbitrary inﬁnite-dimensional manifold and so it is a problem to

determine whether at least one weakly proper function on a Banach manifold

exists. We say that a Banach manifold M admits a continuous weakly proper

function if at least one such function is well-deﬁned on M . The following

statements describe suﬃcient conditions for a Banach manifold M to admit

a weakly proper function of a sort diﬀerent from r(m) on a Banach space.

Theorem 3.28 If M satisﬁes Condition 3.22, it admits a continuous weakly

proper function.

Proof. Let {(

, ˜ϕ

)} be an atlas as in the hypothesis. In particular, the

covering {

} is locally ﬁnite and so there exists a continuous partition of

unity θ

(m), i =1,...,∞ corresponding to {

} (see, e.g., [172]).

Deﬁne the functions ψ

: M → R by the formulae

(m)=



i if m ∈

0ifm/∈

and construct the function ψ(m)onM by the formula:

ψ(m)=

∞



i=1

(m)ψ

(m).

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 79

By construction ψ(m) is continuous and weakly proper. 

The construction of ψ above is a modiﬁcation of the construction in [134]

(cf. the proof of Theorem 3.3).

Corollary 3.29

(i) If M is a paracompact Lindel¨of topological space (see, e.g., [168]), it

admits a continuous weakly proper function.

(ii) If M is paracompact and separable, it admits a continuous weakly

proper function.

Proof. (i) For each m ∈ M choose a chart (U

,ϕ

) such that m ∈ U

Then take an open bounded ball V

⊂ V

centered at ϕ

−1

m and deﬁne

= ϕ

.SinceM is paracompact, there exists a locally ﬁnite reﬁnement

of the covering {U

} (see [172]). Since M is a Lindel¨of space, we can select a

countable subcovering {U

} of that reﬁnement. By construction each U

is a

subset of some U

and so there exists an open ball V

= ϕ

−1

⊂ V

. Denote

by ϕ

the restriction of ϕ

to V

. Then the atlas {(U

,ϕ

)} satisﬁes the

conditions of Theorem 3.28.

(ii) Let Ξ be a countable everywhere dense subset of M.Foreachm ∈ Ξ

construct a chart (U

,ϕ

) and a locally ﬁnite reﬁnement of the covering

} in the same manner as in the proof of (i). Since every point of Ξ is

contained only in a ﬁnite number of sets from that reﬁnement, the collection

of sets in the reﬁnement is countable. The rest of the proof is the same as

in (i). 

Theorem 3.30 Let M admit a continuous weakly proper function and let

a vector ﬁeld X(t, m) on M be such that the Cauchy problem (3.5)–(3.6) is

locally well-posed. If X(t, m) is complete, there exists a continuous weakly

proper function f : M

→ R on M

such that for any curve m

(t,m)

(s),as

deﬁned by (3.7), there exists a real constant C>0 for which the relation

(3.10) holds for every pair s

and s

Proof. Let X(t, m) be complete, i.e., the solutions m

(t,m)

(s) of the problem

(3.5)–(3.6) exist on the entire line. Let ψ be a weakly proper function on M

that exists by the hypothesis.

Denote by g

: M → M the ﬂow of the vector ﬁeld X, i.e., g

(m)=

(0,m)

(s). Since X(t, m) is complete and the Cauchy problem is locally well-

posed for it, g

is well-deﬁned and forms a continuous family of homeomor-

phisms of M. From the completeness of X(t, m) it also follows that for every

pair (t, m) ∈ M

the value m

(t,m)

(0) of the solution m

(t,m)

(s) is well-deﬁned

and continuously depends on (t, m).

Now construct a continuous atlas on M

as follows: for a chart (U

,ϕ

)

from the smooth atlas on M and for an interval (s

) ⊂ R deﬁne

α,(s

)



s∈(s

)

(s, g

)),

α,(s

)

=(s

) ×V

80 3 Ordinary Diﬀerential Equations

and

˜ϕ

α,(s

)

(s, x)=(s, g

(ϕ

(x))

for x ∈ V

. Construct also the continuous function Φ : M

→ R by assigning

the value Φ(t, m)=ψ(m

t,m

(0)) to the point (t, m) ∈ M

Consider the continuous function f : M

→ R, f (t, m)=Φ(t, m)+t.

Taking into account the construction of the continuous atlas



α,(s

)

α,(s

)

, ˜ϕ

α,(s

)



one can easily see that f is weakly proper.

By its construction, Φ takes constant values along the curves m

(t,m)

(s)on

. Indeed, for m

(t,m)

(s)=(s, m

(t,m)

(s)) the equality m

(s,m

(t,m)

(s))

(0) =

(t,m)

(0) holds for all s. Hence,





(t,m)

)



− f



(t,m)

)







(t,m)

(0)



+ s

− ψ



(t,m)

(0)



− s



= |s

− s

Thus, the constant C in the conclusion of the theorem may take any value

greater than or equal to 1. 

Theorem 3.31 Let M be a Banach manifold that admits a continuous

weakly proper function. A completely continuous vector ﬁeld X(t, m) on M

such that the Cauchy problem (3.5)–(3.6) is locally well-posed is complete if

and only if there exists a continuous weakly proper function f : M

→ R such

that for any curve m

(t,m)

(s), as deﬁned by (3.7), there exists a real constant

C>0 for which the relation





(t,m)

)



− f



(t,m)

)





<C|s

− s

holds for every pair s

and s

from the domain of the curve m

(t,m)

(s).

Theorem 3.31 follows from Theorems 3.27 and 3.30.

As in the case of linear Banach spaces the following statement holds since

for locally Lipschitz continuous vector ﬁelds the Cauchy problems are locally

well-posed.

Corollary 3.32 Let M be a Banach manifold that admits a continuous

weakly proper function. Both completely continuous and locally Lipschitz con-

tinuous vector ﬁelds X(t, m) are complete if and only if there exists a con-

tinuous weakly proper function f : M

→ R on M

such that for any curve

(t,m)

(s), as deﬁned by (3.7), there exists a real constant C>0 for which

the relation





(t,m)

)



− f



(t,m)

)





<C|s

− s

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 81

holds for every pair s

and s

in the domain of the curve m

(t,m)

(s).

An analog of Theorem 3.15 takes the following form:

Theorem 3.33 Let a Banach manifold M have a smooth countable locally

ﬁnite atlas such that for any its charts (U

,ϕ

) the set V

⊂ B is bounded

with respect to the norm of the model space B.LetM also admit a smooth

partition of unity. A smooth completely continuous vector ﬁeld X(t, m) on

M is complete if and only if there exists a smooth weakly proper function

f : M

→ R and a real constant C>0 such that for the derivative X

f of

f in the direction of X

the inequality |X

f| <C holds on M

A new point in the proof of Theorem 3.33 is that having taken a smooth

partition of unity in the construction of the function ψ in the proof of Theo-

rem 3.28, one obtains a smooth ψ. Since the ﬂow of a smooth X

is smooth,

the function f constructed from ψ in the proof of Theorem 3.30 is also smooth.

A direct calculation shows that |X

f| = 1. The rest of proof is analogous to

the previous ones.

Remark 3.34. All ﬁnite dimensional manifolds are paracompact and sepa-

rable and they all admit smooth partitions of unity. This means that they all

admit smooth proper functions (all ﬁnite dimensional weakly proper func-

tions are proper with respect to the strong topology). Notice also that all

continuous ﬁnite-dimensional vector ﬁelds are completely continuous. Thus,

it follows from Theorem 3.33 that any smooth vector ﬁeld X(t, m)onany

ﬁnite-dimensional manifold M is complete if and only if there exists a smooth

proper function f : M

→ R such that |X

f| <Con M

for some constant

C>0. This is the assertion of Theorem 3.3.

Let D be a Banach manifold that is embedded into M by a continuous

map so that the image of D is everywhere dense in M.Letforeverym ∈ D

and t ∈ R a vector X(t, m) ∈ T

M be given. Consider the Cauchy problem

(t,m)

(s)=X



s, m

(t,m)

(s)



(3.13)

(t,m)

(t)=m ∈ D. (3.14)

We use the same deﬁnitions of local well-posedness and regularity as in the

case of linear spaces (see Deﬁnitions 3.16 and 3.17). As above, we introduce

the manifold D

= R × D, the vector ﬁeld X

(t, m)=(1,X(t, m)) and the

curves m

(t,m)

(s)=(s, m

(t,m)

(s)).

By a combination of arguments used above in the proofs for cases of man-

ifolds and for right-hand sides deﬁned on an everywhere dense subset in a

Banach space, we obtain the following:

Theorem 3.35 Let D be a Banach manifold that admits a weakly proper

function and let the embedding i : D → M be completely continuous. Let

the Cauchy problem (3.13)–(3.14) be locally well-posed and regular. Then all

82 3 Ordinary Diﬀerential Equations

solutions of (3.13)–(3.14) are well-deﬁned for all s ∈ (−∞, ∞) if and only if

there exists a continuous weakly proper function f : D

→ R such that for

any curve m

(t,m)

(s) there exists a real constant C>0 for which the relation





(t,m)

)



− f



(t,m)

)





<C|s

− s

holds for every pair s

and s

in the domain of the curve m

(t,m)

(s).

3.1.3 A necessary and suﬃcient condition for

completeness of a vector ﬁeld of two-sided type

An alternative type of suﬃciency condition for the completeness of ﬂows may

be formulated in terms of the so-called two-sided estimates, i.e., estimates of

the norm of the right-hand side under some additional condition. In R

have, for example, the condition of sub-linear growth X(t, x) <C(1 +

x) and the famous Wintner theorem [144] (formulated below). Under the

hypotheses of some of these theorems, one can deﬁne a new Riemannian

metric on the phase space in such a way that the right-hand side of the

equation is uniformly bounded by a constant with respect to this metric.

Thus, in these cases, the extendability of solutions (the completeness of a

vector ﬁeld) follows from the fact that a solution has bounded length on every

ﬁnite interval with respect to a complete Riemannian metric and, therefore,

is relatively compact.

It turns out that the requirement that the vector ﬁeld should be bounded

with respect to a complete Riemannian metric can be modiﬁed in such a way

that it becomes necessary and suﬃcient.

Let M be a ﬁnite-dimensional smooth manifold and X(t, m) be a vector

ﬁeld which is jointly smooth in t and m. Denote the extended phase space

R×M by M

. Clearly, T

(t,m)

= R×T

M. As in the previous subsections,

deﬁne a vector ﬁeld X

on M

setting X

(m,t)



1,X(m, t)



Theorem 3.36 AﬁeldX on M is complete if and only if there exists a

complete Riemannian metric on M

with respect to which X

is uniformly

bounded.

Proof. Clearly, the completeness of X is equivalent to the completeness of

the vector ﬁeld X

Assume that there exists a complete Riemannian metric on M

with re-

spect to which the ﬁeld X

is bounded. Then every integral curve of X

has ﬁnite length on every ﬁnite interval. Since the metric is complete, the

last assertion implies the relative compactness of the integral curve on every

ﬁnite interval. As above, we deduce that the domain of every integral curve

is both open and closed in R. This yields the completeness of the ﬁeld.

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 83

Let us prove the “only if” assertion. Let X be complete, then so is X

Since, by hypothesis, the ﬁeld X is smooth, the ﬁeld X

is also smooth.

Consider an arbitrary smooth proper real-valued function ψ on the manifold

M (see Deﬁnition 3.2). The function g, satisfying the aforesaid conditions,

can be constructed in the same way as ψ in (3.4).

Pick an inner product depending smoothly on (m, t) on each tangent space

(m,t)

({t}×M) to the submanifold {t}×M of the manifold M

. For example,

one can take a Riemannian metric on M and extend it in a natural way. Now

we can construct a Riemannian metric ·, ·

on M

by regarding the vectors

of the ﬁeld X

as being of unit length and orthogonal to the subspaces

(m,t)

(M ×{t}).

Denote by Φ

the diﬀeomorphism of the manifold M ×{0} to the manifold

M ×{t} along the trajectories of the ﬁeld X

. The function ψ can be regarded

as given on M ×{0}. Since the integral curves of the ﬁeld X

are globally

extendable, the function f : M

→ R given by the formula

f(m, t)=ψ



−1

(m, t)



+ t

is, obviously, smooth and proper. Clearly, X

f =1,whereX

f is the deriva-

tive of the function f in the direction of the ﬁeld X

Let us now choose an arbitrary smooth function ϕ: M

→ R such that

ϕ(m, t) > max exp(Yf)

where Y ∈ T

(m,t)



M ×{t}



and Y 

= 1. Such a function can be deﬁned

as follows. For a relatively compact neighborhood of each point (m



) ∈

, there exists a constant greater than sup max exp(Yf)

,where,asabove,

Y ∈ T

(m,t)



M ×{t}



and Y 

= 1, and the supremum is taken over all

points (m, t) from the neighborhood. Then, using the paracompactness of

and, as a consequence, the existence of a smooth partition of unity, we

glue together the function ϕ so that it is deﬁned on the whole of M

At every point (m, t) ∈ M

, deﬁne the inner product on T

(m,t)

by the

formula

Y,Z

= ϕ

(m, t)p

Y,p

Z

+ p

Y · p

where Y, Z ∈ T

(m,t)

and p

, p

are (in the metric , 

) orthogonal

projections of T

(m,t)

onto T

(m,t)



M ×{t}



and X

, respectively. Clearly,

X



=1.

Lemma 3.37 The Riemannian metric ·, ·

is complete on M

Proof. [of the lemma] By the Hopf-Rinow Theorem (Theorem 3.68)itis

enough to prove that every geodesic is extendable to the whole real axis.

It suﬃces to consider the geodesics with unit velocity vector norm. The other

geodesics can be obtained from these by linear changes of time.

Let c(s) be a geodesic with unit velocity vector norm, i.e., ˙c(s)

=1for

all s. One can easily see that

f(c(s)) = ˙c(s)f =(p

˙c(s))f +(p

˙c(s))f.

84 3 Ordinary Diﬀerential Equations

Since ˙c(s)

= 1 and since p

˙c(s)andp

˙c(s) are orthogonal to each other

with respect to the metric ·, ·

, p

˙c(s)

≤ 1andp

˙c(s)

≤ 1. Hence,



f(c(s))



≤



˙c(s)

p

˙c(s)



˙c(s)

p

˙c(s)



ϕ(c(s))

˙c(s)

p

˙c(s)



+ |X

f| < 2

by construction of the functions ϕ and f.

Thus, the function f



c(s)



is bounded on any ﬁnite interval (a, b) and,

since f is proper, the set of points c(s)fors ∈ (a, b) is relatively compact.

This proves the unlimited extendability of the geodesic. 

As mentioned above, X



= 1, which completes the proof of Theo-

rem 3.36. 

Remark 3.38. If the ﬁeld X(t, m)isC

-smooth on M

, then the above

construction gives a C

-smooth complete Riemannian metric on M

, with

respect to which X

is bounded.

3.1.4 Some suﬃcient conditions

As already mentioned above, suﬃcient conditions for the completeness of a

vector ﬁeld are known and, when they hold, it is easy to ﬁnd a Riemannian

metric such that the vector ﬁeld is bounded. In this subsection, we construct

a complete Riemannian metric using a Wintner type hypothesis. Later, this

metric will be used to study complicated diﬀerential equations (stochastic,

with delay, etc).

Let us recall the classical Wintner theorem (see, e.g., [144]). Consider the

following diﬀerential equation on the Euclidean space R

˙x = f



t, x(t)



(3.15)

where f(x, t) is continuous in (t, x).

Theorem 3.39 (Wintner). Suppose that



f(x, t)



≤ ϕ(t) ·L



x



where the function ϕ(t) is positive and integrable on any ﬁnite interval [0,l],

and L:[0, ∞) → (0, ∞) is continuous and satisﬁes the condition



∞

L(u)

= ∞. (3.16)

Then all solutions of (3.36) are deﬁned on (−∞, ∞).

3.1 Global in Time Existence of Solutions of Ordinary Diﬀerential Equations 85

The Wintner theorem will be derived from the following result.

Theorem 3.40 Let M be a complete Riemannian manifold with Riemannian

metric ,  and L:[0, ∞) → (0, ∞) be a smooth function satisfying (3.16).

Choose a point m

∈ M and deﬁne a Riemannian metric ·, ·

∗

·, ·

∗



ρ(m

,m)



·, ·

(3.17)

at a point m ∈ M,whereρ is the Riemannian distance on M in the metric

·, ·.If·, · is complete, then so is ·, ·

∗

Lemma 3.41 The minimal geodesics of ·, · and ·, ·

∗

starting at m

coin-

cide to within a parametrization.

Proof. The lemma can be proved with a straightforward calculation: it can

be shown that the minimal geodesics (beginning at m

) of the ﬁrst metric

satisfy the geodesic equation of the second metric and the cut loci of both

metrics coincide.

Let us continue the proof of the theorem. It suﬃces to prove that the

metric ball U

, centered at m

, of any ﬁxed radius T in the metric ·, ·

∗

is compact. (By the Hopf-Rinow theorem, this implies completeness.) Sup-

pose the contrary, i.e., assume for some l that the ball U

is not compact.

Since ·, · is a complete metric, it follows by Lemma 3.41 that U

contains a

minimal geodesic u(s) of inﬁnite length in the metric ·, ·, beginning at m

Here s is the natural parameter (the length) in the metric ·, ·.Denotethe

reparametrization of u(s) with the length in ·, ·

∗

by u(t). By Lemma 3.41,

u(t) is a minimal geodesic of the metric ·, ·

∗

The deﬁnition of the metric ·, ·

∗

yields

= L (u(s)) = L(s)

and since u(s) lies in U



∞

L(s)

≤



dt = l.

This contradicts condition (3.16). 

Corollary 3.42 Let M be a complete Riemannian manifold, X(t, m) avec-

tor ﬁeld continuous in (t, m),andL :[0, ∞) → (0, ∞) a continuous function

satisfying (3.16). Suppose there exists a point m

∈ M such that at every

point m ∈ M the inequality



X(t, m)



<ϕ(t) · L



ρ(m

,m)



holds, where ρ

is the Riemannian distance on M and ϕ is a positive function integrable on

every ﬁnite interval. Then the ﬁeld X(t, m) is complete.

Proof. It is obvious that for any constant C>0, the function L + C satisﬁes

(3.16). There exists a smooth function Ψ(u) such that L<Ψ<L+ C for