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

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136 6 Essentials from Stochastic Analysis in Linear Spaces

An important role in the investigation of stochastic ﬂows is played by the

so-called inﬁnitesimal generators of ﬂows (for short we shall often refer to

inﬁnitesimal generators simply as generators). Let the equation describing

the ﬂow be given in Itˆoform:(6.16). We introduce the following notation:

,...,q

) denote coordinates in R

and A

∗

denotes the matrix of the op-

erator adjoint to A. Take a relatively compact open neighborhood U ⊂ R

of x ∈ R

and let τ be the Markov time that the process ξ

t,x

(s)ﬁrsthitsthe

boundary of U.Letf : R

→ R be a real-valued function that has continuous

bounded derivatives up to the second order.

Theorem 6.33 (see, e.g., [76]) The equality

lim

Δt→+0

Δt

(E(f (ξ

t,x

((t + Δt) ∧ τ))) − f(x)) = A(t, x)f (6.28)

holds where (t + Δt) ∧ τ =min((t + Δt),τ), Δt → +0 means that Δt → 0

and Δt > 0, the diﬀerential operator A(t, x) is given by the relation

A(t, x)f =



i,j

(t, x)

∂

∂q

+ a(t, x)f, (6.29)

the matrix (σ

)=A ◦ A

∗

is the diﬀusion coeﬃcient of the equation, and

a(t, x)f denotes the derivative of f in the direction of the vector ﬁeld a(t, x).

Proof. It is known that



(t+Δt)∧τ

A(s, ξ

t,x

(s))dw(s) is a martingale. Since

for Δt = 0 this integral equals zero, E(



(t+Δt)∧τ

A(s, ξ

t,x

(s))dw(s)) = 0 (see

Section 6.1.4). Taking into account this equality, one can derive from the Itˆo

formula that

E(f (ξ

t,x

(t + Δt) ∧ τ))) − f(x)

= E

⎛

⎜

⎝

(t+Δt)∧τ





(a(s, ξ

t,x

(s)))ds

(t+Δt)∧τ



tr f



(A(s, ξ

t,x

(s)),A(s, ξ

t,x

(s))ds

⎞

⎟

⎠

Direct calculations in coordinates show that f



(a) is the derivative a(t, x)f of

the function f in the direction of the vector ﬁeld a(t, x) and that tr f



(A, A)=



i,j

∂

∂q

. So, the latter equality can be presented in the form:

E(f (ξ

t,x

((t + Δt) ∧ τ))) − f(x)=E





(t+Δt)∧τ

(Af)(s, ξ

t,x

(s))ds



6.3 Stochastic Flows and their Generators 137

Since for s ∈ [t, (t + Δt) ∧τ ) the process ξ

t,x

(s) belongs to the compact set

U and since f is smooth, the values of f and of its derivatives are bounded.

Hence we can apply Lebesgue’s theorem on limits under integrals (expecta-

tions) and so there exists a random value s



∈ [t, (t + Δt) ∧ τ ] such that:



(t+Δt)∧τ

(Af)(s, ξ

t,x

(s)))ds =(Af )(s



,ξ

t,x



))((t + Δt) ∧ τ − t).

Evidently (t + Δt) ∧ τ − t =((t + Δt) −t) ∧ (τ − t)=Δt ∧(τ −t)and

lim

Δt→+0

Δt

E((A f )(s



,ξ



))((t + Δt) ∧ τ − t))

= E lim

Δt→+0

((Af)(s



,ξ



))

Δt ∧(τ −t)

Δt

. (6.30)

Since at time t the process under consideration has value x, i.e., t is not the

ﬁrst time that the sample path hits the boundary of the compact set U,we

see that a.s. τ − t>0. In addition the expression τ − t does not depend on

Δt. Hence lim

Δt→+0

τ−t

Δt

=+∞. From the last expression it follows that

lim

Δt→+0

Δt

(Δt ∧(τ − t)) = 1 ∧ lim

Δt→+0

τ − t

Δt

=1.

Since s



∈ [t, (t+Δt)∧τ], by Lebesgue’s theorem s



→ t as Δt → +0. Passing

to the limit in (6.30) completes the proof. 

Deﬁnition 6.34. The operator A deﬁned by (6.29) is called the inﬁnitesimal

generator (or simply generator) of the ﬂow ξ.

If a and A satisfy some regularity conditions, the generator A determines

the diﬀusion process uniquely: any two processes having the same generator

and having the same initial value induce the same measure on the path space

(see Section 6.1.1). This means the process is weakly unique, see Section 6.2.3.

It is easy to see that the generator of the Wiener process is

Δ where Δ

is the Laplace operator in R

We should point out the following classical result of A.V. Skorokhod:

Theorem 6.35 (see, e.g., [81]) Let the coeﬃcients of equation (6.16) be con-

tinuous and bounded together with their ﬁrst and second derivatives in x on

the product R × R

. Suppose that the function f : R

→ R has continuous

partial derivatives up to the second order and that it, and its ﬁrst and sec-

ond derivatives, is bounded on R

. Choose an arbitrary interval [0,l] ⊂ R.

Then the function v(t, x)=E(ξ

t,x

(l)) on [0,l] × R

is C

-smooth in t and

-smooth in x and it satisﬁes the equation

∂v

∂t

+ Av =0.

Thus, solutions of stochastic diﬀerential equations play for parabolic equa-

tions a role similar to that of characteristics for ﬁrst order partial diﬀerential

equations.

138 6 Essentials from Stochastic Analysis in Linear Spaces

By analogy with the case of ordinary diﬀerential equations (see Section 1.1

and Section 3.1), the mapping ξ

t,(·)

(s)isarandom diﬀeomorphism (random

homeomorphism – in the case of continuous coeﬃcients of the equation) of

onto its image. If the image coincides with R

, ξ(t, s) is said to be a

stochastic ﬂow of diﬀeomorphisms (homeomorphisms, respectively).

Note that unlike the case of ordinary ﬂows (see Section 1.1), without ad-

ditional constructions the evolution (or group) property makes no sense for

stochastic ﬂows since for such properties the ﬂow must be deﬁned for all

t, s ∈ (−∞, ∞). Thus, we need to deﬁne a solution ξ

t,x

(s)fors ≤ t and prove

the solution’s existence on the entire real line.

If ξ(t, s) is a ﬂow of diﬀeomorphisms, there exists a backward ﬂow

ξ(s, t)

consisting of diﬀeomorphisms inverse to those in the ﬂow ξ(t, s).

Theorem 6.36 [167] If a ﬂow of diﬀeomorphisms ξ(t, s) is given by equation

(6.16), the backward ﬂow

ξ(s, t) is given by the Itˆo equation of the form

ξ(t)=[−a(t, ξ(t)) + tr A



(A(t, ξ(t))]dt − A(t, ξ(t))dw(t). (6.31)

If both forward and backward ﬂows are strictly complete, it is not hard

to see that having deﬁned for s ≤ t a value of ξ(t, s)equalto

ξ(s, t), we

obtain a family of random diﬀeomorphisms for which the evolution property

,ξ

t,x

)

)=ξ

t,x

+ s

) holds for all t, s

∈ (−∞, +∞).Note that

equation (6.31) is well-deﬁned even if ξ(t, s) is not a ﬂow of diﬀeomorphisms.

Deﬁnition 6.37. If the ﬂow ξ(t, s) is not a ﬂow of diﬀeomorphisms, the ﬂow

generated by equation (6.31) is called the backward ﬂow to ξ(t, s).

Deﬁnition 6.38. The generator

A of the backward ﬂow generated by (6.31)

is called the backward generator of the ﬂow ξ(t, s).

One can easily see that

A is given by the relation

A(t, x)f =



i,j

(t, x)

∂

∂q

− a(t, x)f +trA



(A(t, x))f, (6.32)

where tr A



(A(t, x))f is the derivative of f along the vector ﬁeld tr A



(A(t, x)).

If a ﬂow is given by an equation in Stratonovich form (6.18), it is not hard

to derive from formula (6.25) that in terms of the coeﬃcients of equation

(6.18) the generators A and

A have the following presentations:

A(t, x)=



i,j

(t, x)

∂

∂q

+(a(t, x)+

tr A



(A(t, x)), (6.33)

A(t, x)=



i,j

(t, x)

∂

∂q

+(−a(t, x)+

tr A



(A(t, x)). (6.34)

We refer the reader to, e.g., [1, 167] for a more detailed discussion of

stochastic ﬂows.

Chapter 7

Stochastic Analysis on Manifolds

The purpose of this chapter is to describe and investigate the main features

of stochastic analysis on smooth manifolds. Our principal focus shall be on

stochastic diﬀerential equations. A monographic presentation of various al-

ternative aspects of and approaches to stochastic analysis on manifolds can

be found in [23, 66, 69, 147, 179, 180, 190, 205].

7.1 Stochastic Diﬀerential Equations in Stratonovich

Form on a Manifold

7.1.1 General construction

The theory of stochastic diﬀerential equations in Stratonovich form can be

generalized to the case of manifolds in a very simple way. The reason for

this is that by formula (6.14) a solution of an equation in Stratonovich form

(6.18) under the coordinate change ϕ

βα

transforms into

dϕ

βα

(ξ(t)) = ϕ



βα

[a(t, ξ(t)) + A(t, ξ(t)) ◦ dw(t)] (7.1)

which coincides with formula (1.1) for the transformation of a tangent vector.

Equations in Itˆo form are transformed by formula (7.6), i.e., they are cross-

sections of a special bundle, and so they require special constructions for their

description. This is why Stratonovich’s approach to stochastic diﬀerential

equations on manifolds is used so extensively in the literature.

In this section we describe an approach to stochastic diﬀerential equations

on manifolds which is based on that of Stratonovich. Itˆo’s approach will be

described in later sections.

Let M be a smooth ﬁnite dimensional manifold with dimension n.

Deﬁnition 7.1. The pair (a(t, m),A(t, m)), where a(t, m) is a vector ﬁeld

on M and A(t, m) is a ﬁeld of linear operators A(t, m):R

→ T

M sending

Y.E. Gliklikh, Global and Stochastic Analysis with Applications

to Mathematical Physics, Theoretical and Mathematical Physics,

DOI 10.1007/978-0-85729-163-9

7, © Springer-Verlag London Limited 2011

139

140 7 Stochastic Analysis on Manifolds

a certain Euclidean space R

to the tangent spaces to M , is called an Itˆo

vector ﬁeld.

For Itˆo vector ﬁelds we often use the notation (a, A). The value at the

point m is denoted by (a

)or(a(m),A(m)) (in the non-autonomous

case the notation (a(t, m),A(t, m)) is also used).

Since the space R

is speciﬁed (i.e., it is not subjected to coordinate

changes), one can easily see that under a change of coordinates ϕ

βα

in a

chart on M the coordinate descriptions A

(t, m)andA

(t, m) are connected

by formula A

(t, m)=ϕ



βα

(t, m), i.e., under changes of coordinates both

components of the pair (a(t, m),A(t, m)) are transformed by formula (1.1).

This is the reason for using the term “Itˆo vector ﬁeld”.

Lemma 7.2 Let A(t, m) be as above. Then AA

∗

is a symmetric (2, 0)-tensor

ﬁeld on M.

Proof. If the matrix of the operator A(t, m) is calculated with respect to the

standard basis in R

and the basis in T

M is generated by the coordinate

system of some chart, the transposed matrix A

∗

(t, m) is the matrix of the

conjugate operator calculated with respect to the dual basis in the cotangent

space T

∗

M and the standard basis in R

(we identify R

with its conjugate

space in terms of the standard inner product). Note that A(t, m)A

∗

(t, m)

sends T

∗

M to T

M, i.e., a pair of cotangent vectors is transformed into a

pair comprising a tangent vector and a cotangent vector whose “coupling”

yields a real number that linearly depends on both cotangent vectors from

the initial pair. Thus A(t, m)A

∗

(t, m) is a bilinear form on cotangent vectors,

i.e., it is a (2, 0)-tensor ﬁeld. The fact that the matrix of AA

∗

is symmetric

in any coordinate system is obvious. 

Let w(t) be a Wiener process in R

and (a(t, m),A(t, m)) be an Itˆo vector

ﬁeld on M. The expression

dξ(t)=a(t, ξ(t))dt + A(t, ξ(t)) ◦ dw(t) (7.2)

is called a stochastic diﬀerential equation in Stratonovich form on M, given

by Itˆo vector ﬁeld (a(t, m),A(t, m)). This means that in every chart on M

the solution ξ(t) satisﬁes equation (6.17). As said above, (7.2) has the correct

transformation rule under changes of coordinates, i.e., it is well-deﬁned.

Remark 7.3. Let M be embedded into a Euclidean space R

as a subman-

ifold. Since the Itˆo vector ﬁeld (a, A)onM is transformed as a vector under

changes of coordinates, the phrase “an Itˆo vector ﬁeld is tangent to the sub-

manifold M in R

” is well-deﬁned. This means that a(t, m) ∈ T

M ⊂ T

and A(t, m):R

→ T

M ⊂ T

at every point m ∈ M; notice that these

relations remain true under changes of coordinates. For Itˆo equations the

property of being tangent to M is ill-deﬁned.

7.1 Stochastic Diﬀerential Equations in Stratonovich Form on a Manifold 141

We consider some examples of stochastic diﬀerential equations in Stratono-

vich form on manifolds.

Example 7.4. Embed the manifold M by Whitney’s Theorem (Theorem 1.2)

into a Euclidean space R

for some suﬃciently large k. Denote by P

: R

→

M the operator of orthogonal projection. Let a(t, m) be a vector ﬁeld

and B(t, m)bea(1, 1)-tensor ﬁeld on M . Deﬁne the ﬁeld of linear operators

A(t, m):R

→ T

M by the formula A(t, m)=B(t, m) ◦P

. It is clear that

(a(t, m), A(t, m)) is an Itˆo vector ﬁeld on M that generates the equation

dξ(t)=a(t, ξ(t))dt + A(t, ξ(t)) ◦ dw(t). (7.3)

Example 7.5. Let M be a Riemannian manifold with dim M = n.Inthiscase

we set k = n. Consider the total space OM of the bundle of orthonormal

frames over M and the Levi-Civit´a connection H on OM (see Section 2.7). Let

a(t, m) be a vector ﬁeld and B(t, m)bea(1, 1)-tensor ﬁeld on M. Consider

their horizontal lifts a

(t, b)andB

(t, b)ontoOM. Recall that a

(t, b)=

Tπ

−1

(a(t, πb)

and B

(t, b) is deﬁned analogously. Consider also the basic

vector ﬁeld E(w(t)) on OM (see Deﬁnition 2.68), constructed from the vector

w(t)inR

. The pair (a

(t, b),B

(t, b) ◦E

)isanItˆo vector ﬁeld on OM and

together with w(t)fromR

it determines the equation

dξ(t)=a

(t, ξ(t))dt + B

(t, ξ(t)) ◦ E

ξ(t)

(◦dw(t)). (7.4)

on OM.

Example 7.6. We present an example of a stochastic diﬀerential equation with

random coeﬃcients. Let the following objects be given: a non-decreasing fam-

ily of σ-subalgebras B

of the σ-algebra F (t ∈ [0,l], l>0) to which a Wiener

process w(t)inR

is adapted; a stochastic process a(t)inR

and a stochas-

tic process A(t) with values in the space of linear mappings from R

to R

that are non-anticipating with respect to B

and having a.s. continuous sam-

ple trajectories; the ﬁeld of linear operators E

: R

→ T

M,smoothin

m ∈ M. The pair (E

a(t),E

A(t)) on M is a random Itˆo vector ﬁeld and

it generates the stochastic diﬀerential equation

dξ(t)=E

ξ(t)

a(t)dt + E

ξ(t)

A(t) ◦ dw(t)).

The notions of strong and weak solutions continue to be meaningful for

stochastic diﬀerential equations on manifolds.

Theorem 7.7 Let F : M → N be a smooth mappings of manifolds and ξ(t)

be a solution of equation (7.2) on M.ThenF (ξ(t)) satisﬁes the equation

dF (ξ(t)) = TFa(t, ξ(t))dt + TFA(t, ξ(t)) ◦ dw(t),

on N where TF is the tangent mapping of F.

Theorem 7.7 follows from formula (6.14).

142 7 Stochastic Analysis on Manifolds

Local existence of solution theorems for equation (7.2) with smooth enough

coeﬃcients (a, A) can be proved via the following argument. Specify an initial

condition ξ(0) = m

∈ M and consider a relatively compact chart U

 m

Since A(t, m) is smooth enough, in this chart it is possible to pass from

equation (6.17) (whose solution in the chart is ξ(t) by deﬁnition) to equation

(6.25). If the coeﬃcients (a, A) are smooth enough, applying the existence

theorems for linear spaces from Section 6.2.3 (applicable in every chart) one

can easily see that the latter equation has a strong solution with given initial

condition. This solution exists on a random time interval [0,τ

)whereτ

the time that a sample trajectory ξ

(t) ﬁrst hits the boundary of the domain

(the stopping time, see Section 6.1.3).

We describe two tricks that from the very beginning have been applied

in this theory for proving the global existence of solutions (alternative tech-

niques will be considered later). The ﬁrst trick embeds M into a Euclidean

space, as in Example 7.4 (but it is applicable in a much broader setting).

Embed M into a Euclidean space R

of suﬃciently large dimension. Con-

sider the normal bundle N(M )ofM in R

with ﬁber N

. Take a tubular

neighborhood Θ of M. Recall that Θ is retracted onto M by the ﬁbers of N

(see [172]). Denote by ˆr : Θ → M the corresponding retraction.

The neighborhood Θ is represented as a direct product M × W where W

is an open ball in R

N−n

that at every point m ∈ M we can identify with a

ball of the normal space N

to M. From the presentation of Θ as M ×W it

follows that the tangent space T

(m,x)

Θ = T

M ×T

.ItisobviousthatT ˆr

(where ˆr : Θ → M is the retraction introduced above) sends the subspaces

into zero vectors in T

M. Thus on the subspaces T

M the mapping

T ˆr is a linear isomorphism. Introduce vectors a(t, (m, x)) = T ˆr

−1

a(t, m)and

linear operators A(t, (m, x)) = T ˆr

−1

A(t, m)atx ∈ Θ. By Remark 7.3 it is

clear that (a(t, x),A(t, x)) is a well-deﬁned Itˆo vector ﬁeld on Θ.

Let O be a neighborhood of M such that

O⊂Θ,where

O is the closure of

O.Letϕ(y):R

→ R be a smooth function that meets the conditions: 0 ≤

ϕ ≤ 1, ϕ(y)=1forx ∈

O and ϕ(y)=0forx ∈ Θ (Urysohn’s function). Con-

sider the Itˆo vector ﬁeld (¯a(t, x),

A(t, x)) on R

(¯a(t, x),

A(t, x)) of the form

¯a(t, x)=



ϕ(x)a(t, x) x ∈ Θ,

0 x/∈ Θ,

A(t, x)=



ϕ(x)A(t, x) x ∈ Θ,

0 x/∈ Θ.

On R

this vector ﬁeld has the same smoothness as (a(t, m),A(t, m)) on M.

Specify an arbitrary initial condition ξ(0) = m

∈ M.

Theorem 7.8 Let the above-mentioned Itˆo vector ﬁeld (¯a(t, x),

A(t, x)) on

be smooth and C

-uniformly bounded. Then equation (7.2) on M with

initial condition ξ(0) = m

∈ M has a unique strong solution on M that

exists for all t ∈ [0, +∞).

7.1 Stochastic Diﬀerential Equations in Stratonovich Form on a Manifold 143

Proof. Consider the following equation in Stratonovich form in R

ξ(t)=¯a(t,

ξ(t))dt +

A(t,

ξ(t)) ◦ dw(t).

Pass by formula (6.25) to the corresponding equation in Itˆo form. Since

by hypothesis the coeﬃcients (¯a(t, x),

A(t, x)) and their ﬁrst derivatives are

smooth and uniformly bounded (from this it in particular follows that ¯a(t, x),



(

A(t, x)) and

A(t, x) are uniformly bounded), the corresponding equa-

tion in Itˆo form that we have obtained has a unique strong solution that

exists for t ∈ [0, +∞) (see Section 6.2.3). Since by construction the Itˆo vector

ﬁeld (¯a(t, x),

A(t, x)) at the points x ∈ M coincides with (a(t, x),A(t, x)),

(i.e., it is tangent to M), it is easy to show that the solution

ξ(t) with initial

conditions

ξ(0) = m

a.s. lies in M. 

7.1.2 Riemannian uniform atlases

The second traditional trick that allows one to prove the existence of global

in time solutions of stochastic diﬀerential equations is based on the existence

of a certain covering, uniform in the following sense:

(i) for every point there exists a chart that together with the point contains

a ball of some speciﬁed radius, this radius being independent of the

point, the chart, etc.;

(ii) on every ball as described in (i), the coeﬃcients of the equation are

bounded by a constant and that constant is independent of the ball,

the point, the chart etc.

The technique of proving the existence of a solution for t ∈ [0, +∞)isas

follows. Applying theorems of existence of strong solutions in vector spaces

(see Section 6.2.3), we can prove the existence and uniqueness of a solution

in the chart with center at m

on a random interval [0,τ

]whereτ

(ω)isthe

Markov time of the ﬁrst hit of a sample trajectory ξ(t, ω) on the boundary of

the above-mentioned ball with speciﬁed radius and center at m

in this chart

(if ξ(t, ω) does not hit the boundary at all, τ(ω)=∞). Then in an analogous

manner we start a solution from ξ(τ

,ω) given for [τ

,τ

]whereτ

(ω)isthe

Markov time of the ﬁrst hit to the boundary of the ball centered at ξ(τ

,ω)in

the corresponding charts, etc. Some estimates are proved for the probability

of hitting the boundary of the ball for small t, based on boundedness of

the equation coeﬃcients. Then it is derived from those estimates that a.s.

sup

= ∞.

Such a trick was used for the ﬁrst time by Itˆo for equations in Itˆoformin

[149] (this is perhaps the ﬁrst paper in the literature devoted to stochastic

diﬀerential equations on manifolds). For equations in Stratonovich form we

refer the reader to the paper by Clark [40](seealso[66]). The same trick was

used in [23]forItˆo equations in Belopolskaya-Daletskii form (see below). In

144 7 Stochastic Analysis on Manifolds

[66] the global existence of solutions for the equation from Example 7.6 was

proved by this method.

Usually the radii of balls are estimated in terms of Euclidean distance in

charts. This is why it is not clear whether such a covering exists on a given

manifold for a given equation.

Following [95], here we present a modiﬁcation of such conditions in which

the radii of balls are measured with respect to a certain Riemannian metric

on the manifold, i.e., we introduce a special class of Riemannian metrics. We

prove an important result that such metrics exist on every ﬁnite-dimensional

manifold.

Let M be a connected ﬁnite-dimensional Riemannian manifold with Rie-

mannian metric ·, · and let ρ be the Riemannian distance function on M.

Deﬁnition 7.9. An atlas on M is called a uniform Riemannian atlas if for

every point m ∈ M there exists a chart (V,ϕ), m ∈ U = ϕ(V), of this

atlas such that U contains the metric ball V

(r) centered at m and having

a speciﬁed radius r>0 independent of m and U,whereV

(r) is taken with

respect to the Riemannian distance ρ.

Note that, in general, the metric ball V

(r)={m



∈ M|ρ(m, m



) <r}

may not be homeomorphic to a ball in the model space and may have a

complicated topological structure.

Obviously, a Riemannian metric possessing a uniform Riemannian atlas is

complete.

Theorem 7.10 [95] For any Riemannian metric on M , there exists a Rie-

mannian metric conformal to it that possesses a uniform Riemannian atlas.

To prove Theorem 7.10, we reﬁne the methods developed in [140, 191]for

the investigation of convex neighborhoods and complete Riemannian metrics.

Theorem 7.10 is a generalization of the main result from [191]provingthat

any Riemannian metric is conformal to a complete Riemannian metric.

Pick a Riemannian metric ·, · on M, i.e., let ·, ·

be an inner product

in the tangent space T

M,andletρ be the Riemannian distance on M

corresponding to ·, ·.

It is known (see [161]) that for any point m ∈ M, there exists a number

a(m) > 0 such that the ρ-metric ball V



a(m)



lies in a normal coordinate

neighborhood (chart) of any point m



∈ V



a(m)



.Letr(m) be the least

upper bound of such a(m).

If r(m

∗

)=∞ for a point m

∗

∈ M, the proof is clear. Assume that r(m) <

∞ for all m ∈ M.

Lemma 7.11 For any two points m, m



∈ M, the following inequality holds:



r(m) − r(m



)



≤ ρ(m, m



). (7.5)

7.1 Stochastic Diﬀerential Equations in Stratonovich Form on a Manifold 145

Proof. First, consider the case where m



∈ V

(r(m)). Then V



(r(m) −

ρ(m, m



)) ⊂ V

(r(m)) and, by the deﬁnition of r(m



), we have r(m



) ≥

r(m) − ρ(m, m



). If r(m) ≥ r(m



), then (7.5) follows. On the other hand,

m ∈ V

(r(m



)) if r(m



) >r(m), and, therefore, r(m) ≥ r(m) − ρ(m, m



which proves (7.5). The case where m ∈ V





r(m



)



can be dealt with in

the same manner. In the remaining case, the inequalities r(m) ≤ ρ(m, m



)

and r(m



) ≤ ρ(m, m



) follow from m



∈ V

(r(m)) and, at the same time,

m ∈ V



(r(m



)) . Therefore, |r(m) −r(m



)| <ρ(m, m



), which completes the

proof of the lemma. 

Without loss of generality, we may assume that the function r(m)is

smooth. If r(m) is not smooth, it can be approximated by a smooth function

∗

(m) such that 0 <r

∗

(m) <r(m)andr

∗

(m) satisﬁes (7.5).

Let us introduce a new metric ·, ·

∗

on M by the formula

·, ·

∗

(m)

·, ·

Denote by ρ

∗

the Riemannian distance on M corresponding to ·, ·

∗

Lemma 7.12 If ρ(m, m



) ≥ r(m) then ρ

∗

(m, m



) ≥ 1/2.

Proof. Let γ(t) be an arbitrary piecewise smooth curve such that γ(a)=m

and γ(b)=m



. Denote its length in the metric ·, · by L, i.e.,

L =



˙γ(t) dt.

The length of γ in the metric ·, ·

∗

can be found by the formula

∗



˙γ(t)

r (γ(t))

dt.

Using the classical mean value theorem, we obtain

∗

r (γ(τ))



˙γ(t) dt =

r (γ(τ))

where τ ∈ [a, b]. Then

∗

r(γ(τ )) −r(m)+r(m)

and, by Lemma 7.11,

∗

r(m)+ρ



m, γ(τ)

