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

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36 2 Connections

other hand, by construction the vectors

∂

∂ϑ

,...,

∂

∂ϑ

are tangent to the ﬁber

, i.e., they belong to the tangent space T

Deﬁnition 2.1. The tangent space T

to the ﬁber of a bundle Θ at m is

called the vertical subspace in the tangent space T

(m,ϑ)

Θ to the total space

of the bundle and is denoted by V

(m,ϑ)

. The vectors of V

(m,ϑ)

are said to be

vertical.

For every vector Y

(m,ϑ)

at a point (m, ϑ) ∈ Θ we can ﬁnd its coordinate

decomposition with respect to the basis mentioned above: Y = Y

∂

∂q

∂

∂ϑ

, i =1,...,n, j =1,...,d. By introducing vectors Y

= Y

∂

∂q

∈ T

and Y

∂

∂ϑ

∈ T

the vectors Y

(m,ϑ)

∈ T

(m,V )

Θ are represented as

quadruples (m, ϑ, Y

). This notation is compatible with that of Convention

1.3 for tangent vectors as points of the tangent bundle: here (m, ϑ) is a point

of the manifold Θ and (Y

) is a tangent vector to Θ.

Let us ﬁnd the formula of transformation of Y

and Y

under standard

changes of coordinates of the form (ϕ

βα

(m)) (see Deﬁnition 1.32)onthe

total space Θ. Recall that, since Θ is a vector bundle, g

βα

(m) is a linear

operator in R

and so it is equal to its derivative. On the other hand the

derivative in m ∈ M of the linear operator g

βα

(m), depending on m,is

a bilinear operator. Denote it by g



βα

(m)(·, ·). The ﬁrst argument of this

operator is a vector from the ﬁber and the second argument is a vector

tangent to the base M. In particular, the derivative g

βα

(m)inm at the

point (m, ϑ) ∈ Θ takes the form g



βα

(m)(ϑ, ·). Since ϕ

βα

does not depend

on the points of ﬁber, the derivative of ϕ

βα

in R

equals zero. Taking this

into account it is easy to see that the derivative of the change of coordinates

(ϕ

βα

(m)) at the point (m, ϑ) ∈ Θ is represented in the form

(ϕ

βα

(m))



βα



βα

(m)(ϑ, ·) g

βα

(m)



This means that under the above-mentioned changes of coordinates the col-

umn (Y

) transforms by the formula

)

,ϑ

)



βα



βα

)(ϑ

, ·) g

βα

)







βα



βα

)(ϑ

)+g

βα

)(Y

)



. (2.1)

In terms of quadruples formula (2.1) takes the form

,ϑ

) (2.2)

=(ϕ

βα

)ϑ

,ϕ



βα



βα

)(ϑ

)+g

βα

)(Y

)).

2.1 The Structure of a Tangent Bundle to a Vector Bundle 37

By the deﬁnition of the projection π we have π(q

,...,q

,ϑ

,...,ϑ

,...,q

). Hence, the Jacobi matrix (presentation of the diﬀerential d

(m,ϑ)

in the given coordinate system) takes the form (I 0) where I and 0 are the

unit n × n matrix and zero k ×n matrix, respectively. As a consequence we

obtain the formula for Tπ : TΘ → TM in the form

Tπ(m, ϑ, Y

)=(m, Y

) (2.3)

(by deﬁnition the tangent mapping Tπ acts as π on the points (m, ϑ)and

as d

(m,ϑ)

π on the vectors (Y

)). Recall that by construction Y

on the

left-hand side of (2.3) belongs to T

(m,ϑ)

Θ while Y

on the right-hand side

of (2.3) belongs to T

M but both vectors have the same coordinates with

respect to

∂

∂q

,...,

∂

∂q

(the ﬁrst “half”of the basis in T

(m,ϑ)

Θ and the entire

basis in T

M, respectively, are isomorphic to each other, see above). We do

not distinguish between these two vectors or the frames in this notation.

Formula (2.3)meansthat

Tπ



∂

∂q

∂

∂ϑ



= Y

∂

∂q

. (2.4)

Remark 2.2. As on every manifold, there is a natural projection of TΘ onto

Θ. Denote it by π

: TΘ → Θ. In coordinates it is represented in the form

(m, ϑ, Y

)=(m, ϑ). (2.5)

We emphasize the diﬀerence between (2.3)and(2.5).

Let Y be a cross-section of the bundle Θ.OverthechartU

we have

the decomposition Y = Y

(see above). Recall that we consider the cross-

section Y as a mapping Y : M → Θ such that πY = id (see Deﬁnition 1.38).

Consider also its tangent mapping TY : TM → TΘ.SinceY has the form

Y (q

,...,q

)=(q

,...,q

),...,Y

,...,q

)),

its Jacobi matrix takes the form

Y =





∂Y

∂q





where I is the unit n ×n matrix and



∂Y

∂q



is the n ×d Jacobi matrix of Y .

Thus for (m, X) ∈ TM we obtain

TY(m, X)=



m, Y, X,



∂Y

∂q



(2.6)

(recall that TY acts as Y on points m and as d

X on X).

38 2 Connections

On the vector bundle Θ the so-called action of the real line is given as

follows. For every a ∈ R deﬁned a : Θ → Θ (we denote the number and the

corresponding mapping by the same symbol a)by:

a(m, ϑ)=(m, aϑ), (2.7)

where (m, ϑ) ∈ Θ, i.e. the action consists of multiplying all vectors from all

ﬁbers of Θ by a.Thusa(q

,...,q

,ϑ

,...,ϑ

)=(q

,...,q

,aϑ

,...,aϑ

)

and evidently d

(m,ϑ)

a =



I 0

0 aI



, where I and 0 are the unit and zero

matrices, respectively, of corresponding dimensions. Hence

Ta(m, ϑ, Y

)=(m, aϑ, Y

,aY

). (2.8)

Below we shall often use the constructions described in this section on

tangent and cotangent bundles. For these cases we have to deﬁne the previous

formulae and notation more precisely.

Since the ﬁbers of a tangent bundle are tangent spaces, they already have

the standard frames

∂

∂q

,...,

∂

∂q

. For the case of a tangent bundle we most

often use such frames and the trivialization generated by them in a tangent

bundle over charts (the construction of this trivialization is described in Sec-

tion 1.1). Sometimes we shall also use alternative trivializations but those

cases will be mentioned explicitly.

In this case the notation ˙q

for coordinates in ﬁbers is compatible with the

interpretation of a tangent vector as a velocity of some curve. We replace ϑ

by this notation. Thus the frames in tangent spaces to TM have the form

∂

∂q

,...,

∂

∂q

∂

∂ ˙q

,...,

∂

∂ ˙q

For an analogous trivialization in a cotangent bundle we use the basis

,..., dq

and coordinates in ﬁbers with respect to those frames are de-

noted by p

(here we take into account the interpretation of cotangent vectors

as momenta). Respectively, the frame in a cotangent space to T

∗

M takes the

form

∂

∂q

,...,

∂

∂q

∂

∂p

,...,

∂

∂p

The frame in a cotangent space to TM is denoted by dq

,...,dq

d˙q

,...,d˙q

and in a cotangent space to T

∗

M by dq

,...,dq

, dp

,...,dp

We consider in detail the case of tangent bundles. The constructions on

cotangent bundles are analogous.

Deﬁnition 2.3. The tangent bundle to a tangent bundle TM is called the

second tangent bundle to the manifold M and is denoted by TTM or T

The vectors of the second tangent bundle, i.e., tangent vectors to TM,are

described as quadruples of the form (m, X, Y

)whereX and Y

belong to

M while Y

is a vector tangent to T

The Jacobi matrix of the natural projection π : TM → M has the form

(I,0) where I is the unit matrix and 0 is the zero matrix, both n × n.Thus

Tπ



∂

∂q

∂

∂ ˙q



= Y

∂

∂q

2.1 The Structure of a Tangent Bundle to a Vector Bundle 39

The action of the real line on TM is deﬁned as a particular case of the

general deﬁnition. For a ∈ R the Jacobi matrix of the corresponding mapping

a : TM → TM has the same form as above.

The transformation rule for quadruples describing vectors of the second

tangent bundle under changes of coordinates also has to be speciﬁed. First of

all on TM the transformation g

βα

(m) in ﬁbers takes the form g

βα

(m)=ϕ



βα

Hence g



βα

(m)(·, ·)=ϕ



βα

(·, ·)where



denotes the second derivative of the

change of coordinates ϕ

βα

. Since the ﬁber of TM at m is the tangent space

M, both arguments in ϕ



βα

(·, ·) have the same nature: they are vectors

tangent to M. This is why we replace the symbol ϑ in the notation for an

element in the ﬁber of Θ by the symbol X of a tangent vector to M.

Thus formula (2.1) is transformed into

)



βα



βα

, ·) ϕ



βα







βα



βα

)(X

)+ϕ



βα

)(Y

)



. (2.9)

So, by formula (2.9) the transformation of quadruples as vectors tangent to

TTM under the change of coordinates ϕ

βα

on M has the form

)

=(ϕ

βα

,ϕ



βα

,ϕ



βα

,ϕ



βα

)+ϕ



βα

)). (2.10)

We return to the general case.

Recall that by Deﬁnition 2.1 the space T

is called the vertical subspace

in T

(m,ϑ)

Θ and is denoted by V

(m,ϑ)

. The vectors belonging to V

(m,ϑ)

are said

to be vertical.

As a direct consequence of the construction we obtain the following:

Proposition 2.4 The space V

(m,ϑ)

does not depend on the choice of the

chart U

, its coordinate system (q

,...,q

) in a neighborhood of m ∈ M,or

on the choice of the trivialization of π

−1

Indeed, the ﬁber Θ

and hence the tangent space T

= V

(m,ϑ)

are

determined without use of any coordinate system. The system (q

,...,q

)is

involved only in representing V

(m,ϑ)

as the linear span of

∂

∂ϑ

,...,

∂

∂ϑ

. Notice

that these vectors do depend on the trivialization of π

−1

Recall that by formula (1.2) we introduced the linear isomorphism p of a

tangent space to a vector space onto the vector space. Thus here p : V

(m,ϑ)

→

is well-deﬁned and takes the coordinate representation



∂

∂ϑ



= e

. (2.11)

40 2 Connections

Denote by H

(m,ϑ)

the linear span of the vectors

∂

∂q

,...,

∂

∂q

in T

(m,ϑ)

Θ.

By construction, H

(m,ϑ)

is the tangent space to the submanifold U

× ϑ in

−1

with respect to the given trivialization of π

−1

. Immediately from

the deﬁnition we get T

(m,ϑ)

Θ = H

(m,ϑ)

⊕ V

(m,ϑ)

where ⊕ is the direct sum.

Notice that for the vector Y

(m,ϑ)

=(m, ϑ, Y

) ∈ T

(m,ϑ)

Θ by deﬁnition

∈ H

(m,ϑ)

and Y

∈ V

(m,ϑ)

so that Y

(m,ϑ)

= Y

⊕ Y

Proposition 2.5 The subspace H

(m,ϑ)

depends on the choice of trivialization

of π

−1

and hence on the chart U

Proof. Indeed, consider another chart U

with non-empty intersection U

αβ

with U

. Let a trivialization of π

−1

be given so that the standard basis in

generates another ﬁeld of bases e



,...,e



diﬀerent from the ﬁeld e

,...,e

generated by the above trivialization of π

−1

. The layers in U

×R

of the

vectors U

× ϑ and in U

× R

of the vectors U

× ϑ for a speciﬁed vector

ϑ ∈ R

are diﬀerent since the former is generated by those ϑ



’s from Θ



∈ U

whose coordinates with respect to e

,...,e

are the same as the

coordinates of ϑ and the latter by those whose coordinates with respect to



...,e



are the same as those of ϑ. Since the layers going through (m, ϑ)

are diﬀerent, their tangent spaces at (m, ϑ) are also diﬀerent. 

Remark 2.6. From the deﬁnitions it immediately follows that the quadruple

for a vector from H

(m,ϑ)

takes the form (m, ϑ, Y

, 0) and, for a vector from

(m,ϑ)

,theform(m, ϑ, 0,Y

Proposition 2.7 Tπ sends any H

(m,ϑ)

isomorphically onto T

M and V

(m,ϑ)

is the kernel of Tπ at any T

(m,ϑ)

Θ.

Indeed, a vector from H

(m,ϑ)

takes the form (m, ϑ, Y

, 0) and from V

(m,ϑ)

the form (m, ϑ, 0,Y

) (see Remark 2.6). So, by (2.3) Tπ(m, ϑ, Y

, 0) = (m, Y

)

and Tπ(m, ϑ, 0,Y

)=(m, 0).

2.2 Connections on Vector Bundles

Connection and connector

Deﬁnition 2.8. Let Θ be a vector bundle and suppose that in every tangent

space T

(m,ϑ)

Θ a subspace H

(m,ϑ)

, complementary to V

(m,ϑ)

(i.e. T

(m,ϑ)

Θ =

(m,ϑ)

⊕V

(m,ϑ)

at any (m, ϑ) ∈ Θ), is speciﬁed such that the total family of

subspaces H = {H

(m,ϑ)

|(m, ϑ) ∈ Θ} satisﬁes the following two properties:

(i) the space H

(m,ϑ)

depends smoothly on (m, ϑ) ∈ Θ (in the sense de-

scribed below);

(ii) the family H is invariant with respect to the action of the real line on

Θ, i.e., TaH

(m,ϑ)

= H

(m,aϑ)

for every a ∈ R and (m, ϑ) ∈ Θ.

Then H is said to be a connection on Θ.

The subspaces H

(m,ϑ)

of a connection H are called horizontal,asarethe

vectors of T

(m,ϑ)

Θ belonging to H

(m,ϑ)

2.2 Connections on Vector Bundles 41

The precise meaning of the statement that H

(m,ϑ)

is smooth in (m, ϑ)isas

follows. In a neighborhood of any point (m, ϑ) ∈ Θ there are n smooth linearly

independent vector ﬁelds such that, for any (m



,ϑ



) in the neighborhood, the

subspace H



,ϑ



)

is the linear span of vectors of those ﬁelds at (m



,ϑ



Proposition 2.9 The family H

(m,ϑ)

introduced in Section 2.1 is a connection

on π

−1

Proof. The presentation T

(m,ϑ)

Θ = H

(m,ϑ)

⊕ V

(m,ϑ)

was derived in Section

2.1 from the deﬁnition of H

(m,ϑ)

. Also by deﬁnition H

(m,ϑ)

is the linear span

of smooth linearly independent vectors

∂

∂q

,...,

∂

∂q

At any point (m, ϑ) ∈ Θ the space H

(m,ϑ)

is the set of all vectors whose

quadruple presentation takes the form (m, ϑ, Y

, 0) (see Remark 2.6). By for-

mula (2.8)weseethatTa is a one-to-one mapping sending (m, ϑ, Y

, 0) to

the quadruple (m, aϑ, Y

, 0). Thus TaH

(m,ϑ)

= H

(m,aϑ)

. 

Deﬁnition 2.10. The family of subspaces H

(m,ϑ)

is called the Euclidean con-

nection of a given trivialization of π

−1

Indeed, H

(m,ϑ)

depends on the trivialization (see Proposition 2.5).

There exist connections {H

(m,ϑ)

} that may not be presented as the Eu-

clidean connection of a trivialization. Of course, at a given point (m, ϑ)fora

subspace H

(m,ϑ)

, complimentary to V

(m,ϑ)

, one can ﬁnd a trivialization such

that H

(m,ϑ)

coincides with the Euclidean connection of a trivialization at

(m, ϑ), but in general this cannot be achieved for subspaces at all points in

a given neighborhood. In order not to exclude general connections, we have

not limited ourselves to Euclidean connections.

Proposition 2.11 Tπ : H

(m,ϑ)

→ T

M is a linear isomorphism.

42 2 Connections

Proof. Recall that Tπ : T

(m,ϑ)

Θ → T

M is surjective and (by Proposi-

tion 2.7) V

(m,ϑ)

is the kernel of Tπ. Thus, the Proposition follows from the

general result of linear algebra that a surjective linear operator is one-to-one

on a complement to the kernel. 

The above proof is also valid in the analogous case of H

(m,ϑ)

in Proposi-

tion 2.7. However, we used a coordinate proof there for simplicity.

Combining Propositions 2.7 and 2.11 we see that Tπ is connected with the

decomposition T

(m,ϑ)

Θ = H

(m,ϑ)

⊕ V

(m,ϑ)

as follows:

Lemma 2.12

(i) Tπ : H

(m,ϑ)

→ T

M is an isomorphism.

(ii) V

(m,ϑ)

=kerTπ.

Our next step is to construct a map that is one-to-one on V

(m,ϑ)

and

whose kernel is H

(m,ϑ)

. Recall that the operator p that establishes a linear

isomorphism between the vector space Θ

and the tangent space T

it acts by formula (2.11). Hence, for a vector from V

(m,ϑ)

with the quadruple

(m, ϑ, 0,Y

) (see Remark 2.6)whereY

∂

∂ ˙q

,wehave

p(m, ϑ, 0,Y

)=(m, pY

. (2.12)

The decomposition T

(m,ϑ)

Θ = H

(m,ϑ)

⊕ V

(m,ϑ)

yields the decomposition

(m,ϑ)

= HY ⊕ VY for every Y

(m,ϑ)

∈ T

(m,ϑ)

Θ,whereHY ∈ H

(m,ϑ)

and

VY ∈ V

(m,ϑ)

. The symbols H and V may be considered as projections H :

(m,ϑ)

Θ → H

(m,ϑ)

and V : T

(m,ϑ)

Θ → V

(m,ϑ)

in the above decomposition.

Deﬁnition 2.13. The map K = pV : T

(m,ϑ)

Θ → Θ is called the connector

of the connection H.

2.2 Connections on Vector Bundles 43

Thus K(Y

(m,ϑ)

)=p(VY

(m,ϑ)

). Evidently K on V

(m,ϑ)

coincides with p and

so K maps V

(m,ϑ)

onto Θ

isomorphically. On the other hand, V(H

(m,ϑ)

0 ∈ V

(m,ϑ)

and so K(H

(m,ϑ)

)=0∈ Θ

. We summarize these properties in

the following lemma:

Lemma 2.14

(i) K : V

(m,ϑ)

→ Θ

is a linear isomorphism.

(ii) H

(m,ϑ)

=kerK.

Compare Lemmas 2.12 and 2.14. Notice the diﬀerence: we know that

(m,ϑ)

and Tπ exist on each vector bundle Θ while H

(m,ϑ)

and K must

be given “by hand”.

In order to work with H

(m,ϑ)

and K we need to describe them by means

of coordinates. The best way to do that is to compare H

(m,ϑ)

with H

(m,ϑ)

a certain trivialization over a chart U

since the coordinate presentation of

(m,ϑ)

is known.

Consider a vector Y

∈ T

M.SinceTπ sends both H

(m,ϑ)

and H

(m,ϑ)

onto T

M one-to-one, each of the spaces contains a unique vector whose im-

age under Tπ is Y

. The vector in H

(m,ϑ)

is usually denoted by the same

symbol Y

. Denote the vector in H

(m,ϑ)

by HY . Consider the diﬀerence

(ϑ, Y

)=Y

− HY ∈ T

(m,ϑ)

Θ. By construction we have TπΓ

(ϑ, Y

Tπ(Y

) − Tπ(HY )=Y

− Y

=0∈ T

M. Hence, Γ

(ϑ, Y

) ∈ V

(m,ϑ)

since V

(m,ϑ)

is the kernel of Tπ. Thus we can apply p to Γ

(ϑ, Y

)and

obtain the vector pΓ

(ϑ, Y

) ∈ Θ

. We have constructed the operator

pΓ

(·, ·):Θ

× T

M → Θ

Deﬁnition 2.15. The operator pΓ

(·, ·) is called the local connector (or local

connection coeﬃcient) of the connection H.

44 2 Connections

The word “local” means that the operator is constructed and calculated

in a certain chart U

on M with respect to a certain trivialization of π

−1

Theorem 2.16 The operator pΓ

(·, ·) is linear in the second argument.

Indeed, pΓ

(ϑ, Y

)=Tπ

−1

)

(m,ϑ)

− Tπ

−1

)

(m,ϑ)

.Sincetheop-

eration Tπ

−1

and the operation of taking the diﬀerence are both linear,

pΓ

(ϑ, Y

) is linear in Y

Theorem 2.17 The operator pΓ

(·, ·) is linear in the ﬁrst argument.

To prove Theorem 2.17 we need the following:

Lemma 2.18 Let B : E → E be a map in the vector space E, smooth and

homogeneous with degree 1.ThenB is a linear operator.

Proof. (of Lemma 2.18) Recall that B is homogeneous with degree k if for

any vector X ∈ E and any λ ∈ R we have B(λX)=λ

B(X). From the

homogeneity it follows that B(0) = 0.

Since B is smooth, we can expand it by the Taylor formula in a neighbor-

hood of 0 ∈ E up to a certain degree greater than 1. Thus, since B(0) = 0,

B(X)=B



(X)+



(X, X)+... where B



is the ﬁrst derivative of B at

the origin (recall that B



is a linear operator), B



is the second derivative of

B at the origin (recall that B



is a bilinear operator), etc. On the right-hand

side only B



is homogeneous with degree 1; B



(X, X) is homogeneous with

degree 2 and the other summands have greater degrees of homogeneity. Thus

the left-hand side is homogeneous with degree 1 only if all summands on the

right hand side except B



are equal to zero. Hence B = B



and so it is a

linear operator. 

Proof. (of Theorem 2.17) Since by Deﬁnition 2.8(i) both H

(m,ϑ)

and H

(m,ϑ)

are smooth in ϑ,sotooispΓ

(ϑ, Y

). We shall show that pΓ

(ϑ, Y

)is

homogeneous with degree 1 in ϑ so that the statement of Theorem 2.17 will

follow from Lemma 2.18.

The vector HY = Tπ

−1

) ∈ H

(m,ϑ)

is presented as a quadruple in the

form (m, ϑ, Y

,Γ

(ϑ, Y

)). By Deﬁnition 2.8(ii) and by formula (2.8)(de-

scribing Ta) the vector Ta(HY )=(m, aϑ, Y

,aΓ

(ϑ, Y

)) belongs to H

(m,aϑ)

Using formula (2.3)wegetTπ((m, aϑ, Y

,aΓ

(ϑ, Y

)) = (m, Y

). Since Tπ

is one-to-one on H

(m,aϑ)

, it is the unique vector in H

(m,aϑ)

whose image

under Tπ is (m, Y

). But the vector HY = Tπ

−1

) ∈ H

(m,aϑ)

,whose

quadruple takes the form (m, aϑ, Y

,Γ

(aϑ, Y

)), also has this property:

Tπ(m, aϑ, Y

,Γ

(aϑ, Y

)) = (m, Y

). Hence,

(m, aϑ, Y

,aΓ

(ϑ, Y

)) = (m, aϑ, Y

,Γ

(aϑ, Y

))

and so, since p is linear, pΓ

(aϑ, Y

)=apΓ

(ϑ, Y

). 

2.2 Connections on Vector Bundles 45

So, pΓ

(·, ·) is linear in both arguments. It is useful to ﬁnd its values

on basis vectors. Consider pΓ

∂

∂q

). It is a vector from Θ

and so it

can be expanded in coordinates Γ

with respect to the basis e

,...,e

pΓ

∂

∂q

)=Γ

. The coordinates Γ

depend on m ∈ U

(as well as on

a trivialization), this means that they are real-valued functions of m ∈ U

Deﬁnition 2.19. The functions Γ

are called Christoﬀel symbols of the sec-

ond kind for the connection H.

Knowing Γ

, we can calculate the values pΓ

(X, Y ) for any X ∈ Θ

Y ∈ T

M. Indeed, let X = X

and Y = Y

∂

∂q

, then by linearity we get

pΓ (X, Y )=X

. (2.13)

Now let us turn back to the connector K. Recall that K(Y )=pVY for Y ∈

(m,ϑ)

Θ. Thus we need to describe pVY .ForY we have two decompositions:

Y = Y

and Y = HY +VY . Hence Y

= HY +VY and so VY −Y

− HY = Γ

(ϑ, Y

). Thus VY = Y

+ Γ

(ϑ, Y

) and consequently VY =

V(m, ϑ, Y

)=(m, ϑ, 0,Y

+ Γ

(ϑ, Y

)). Finally we obtain the formula for

K in the form:

K(m, ϑ, Y

)=pV(m, ϑ, Y

)=(m, pY

+ pΓ (ϑ, Y

)). (2.14)

Compare (2.14) with (2.3)and(2.5).

Let Y

∂

∂ϑ

and ϑ =˙q

.Using(2.12) we describe (2.14)incoordi-

nates as follows

K(m, ϑ, Y

)=(

+˙q

. (2.15)