Lifting Dual Connections with the Riemann Extension

Lifting dual connections to the cotangent bundle

S. Puechmorel

1 ENAC, Université de Toulouse, stephane.puechmorel@enac.fr

* Correspondence: stephane.puechmorel@enac.fr; Tel.: +33-5-62259503 Version July 31, 2020 submitted to Mathematics

Abstract: Le

(

M,g

)

be a Riemannian manifold equipped with a pair of dual connections

1

(∇

,

∇

^∗

)

.Such a structure is known as a statistical manifold since it was defined in the context

2

of information geometry. This paper aims at defining the complete lift of such a structure to the

3

cotangent bundleT^∗Musing the Riemannian extension of the Levi-Civita connection ofM.In the first

4

section, common tensors associated with pairs of dual connections, emphasizing the cyclic symmetry

5

property of the so-called skewness tensor. In a second section, the complete lift of this tensor is

6

obtained, allowing the definition of dual connections on TT^∗M with respect to the Riemannian

7

extension.

8

1. Introduction

9

Information geometry was originally dealing with parameter spaces of families of probability

10

densities viewed as differentiable manifolds [1,2]. More specifically, letEbe a measure space and let

11

S

= {

p_θ,θ

∈

M

}

be a parameterized family of densities onEsatisfying:

12

1. Mis a topological manifold (in most of the case it is simply an open subset ofRⁿ).

13

2. The topology ofSinduced by theL¹norm is compatible with the topology ofM.

14

3. It exists a probability measureµonEsuch that for anyθ

∈

M,p_θ <<µ.

15

4. θ

7→ (

x

∈

E

→

p_θ

(

x

))

is smooth uniformly inx.

16

5. ∂_θE_µ

[

logp

(

x,θ

)] =

E_µ

[

∂_θlogp

(

x,θ

)]

.

17

6. The moments up to order 3 ofx

7→

∂_θlogp

(

x;θ

)

exist and are smooth.

18

7. The matrixFwith entriesF_ij

(

θ

) =

Ep_θ

h

∂_θilogp

(

x,θ

)

∂_θjlogp

(

x,θ

)

ⁱis positive definite.

19

The last assumption allows to endowMwith the structure of a Riemannian manifold with metric:

g_θ

∂_θi,∂_θj

=

Fij

(

θ

)

(1)

Parameterized families of the so-called exponential type, whose densities can be written as:

p

(

x;θ

) =

exp

(−h

θ,T

(

x

)i −

ψ

(

θ

) +

h

(

x

))

play a special role in statistics and have a well behaved Riemannian structure. WhenT

(

x

) =

x, the family is said to be natural and is defined entirely byψ.In such a case, the Fisher information matrix takes the form:

F_ij

(

θ

) = −

Ep_θ

"

∂²

∂θ_i∂θ_j

#

so that the Riemannian metric is Hessian. The structure of such manifolds has been thoroughly studied in [3]. Finally, from considerations arising in statistical estimation, a pair of dual connections

∇

,

∇

^∗ with respect to the Fisher metric can be constructed [4]. They possess vanishing torsion and are related by the skewness tensor:

g

(∇

_XY,Z

) −

g

(∇

^∗_XY,Z

) =

T

(

X,Y,Z

)

Submitted to Mathematics, pages 1 – 13 www.mdpi.com/journal/mathematics

with:

T_ijk

=

Ep_θ

h

∂_θilogp

(

x,θ

)

∂_θjlogp

(

x,θ

)

∂_θklogp

(

x,θ

)

ⁱ

As a generalization, a smooth Riemannian manifold

(

M,g

)

equipped with a pair

(∇

,

∇

^∗

)

of

20

torsionless dual connections is called a statistical manifold. It can be defined equivalently by

(

M,g,T

)

21

whereTis a fully symmetric

(

0, 3

)

-tensor. It turns out [5] that any statistical manifold can be embedded

22

as a statistical model, i.e. one related to a parameterized family of densities.

23

For a Riemannian manifold

(

M,g

)

, lifting geometric objects to the tangent bundleTM(resp.

24

cotangent bundleT^∗M) is a classical problem [6–8] that relies most of the time on the whithey sum

25

TTM

=

HTM

⊕

VTM(resp.TT^∗M

=

HT^∗M

⊕

VT^∗M) withVTMthe vertical bundled obtain from

26

the kernel of the canonical projectiondπ:TTM

→

TM(resp. dπ: TT^∗M

→

T^∗MandHTM the

27

horizontal subspace arising from a fixed affine connection

∇

. In the tangent bundle, [8] introduces a

28

lift based on horizontal and vertical lifts of vector fields and relies on a quasi-complex structure on

29

TM. ForT^∗M, the preferred method involves complete lifts [9] and Riemann extensions [10], which

30

are pseudo-riemannian metrics of neutral signature defined on the cotangent bundle and associated in

31

a canonical way to affine connections with vanishing torsion. The complete lift of the connection is

32

defined to be the Levi-Civita one with respect to its Riemann extension. Complete and vertical lifts of

33

different kind of tensors are also presented in [6]. Finally, horizontal lifts of connections are presented

34

in [7].

35

In this paper, the complete lift of dual connections is defined and yields a pair of dual connections

36

which have vanishing torsion if the original connections have. The strategy adopted is to lift the

37

skweness tensor, here defined in a more general setting as a

(

0, 3

)

-tensor with cyclic symmetry. The

38

procedure described in [6] is adapted to this case, effectively allowing to get a skewness tensor on

39

TT^∗M.

40

2. Statistical structures

41

In information geometry, dual connections are the basic objects defining the so-called statistical

42

manifold structure [4]. In the sequel,Mis a smoothn-dimensional manifold endowed with a Riemann

43

metricg.

44

Definition 1. Let

∇

,

∇

^∗be affine connections onTM. They are said to be dual if for any tripleX,Y,Z of vector fields:

Z

(

g

(

X,Y

)) =

g

(∇

_ZX,Y

) +

g

(

X,

∇

^∗_ZY

)

(2) The torsion of a connection

∇

is the tensorTdefined as: T

(

X,Y

) = ∇

_XY

− ∇

_YX

− [

X,Y

]

. The

45

next well known proposition relates the torsion tensors of dual connections.

46

Proposition 1. Let

∇

,

∇

^∗be dual connections. Let T (resp. T^∗) be the torsion tensor of

∇

(resp.

∇

^∗

)

. Then,

47

T

=

T^∗.

48

Proof. For any triple

(

X,Y,Z

)

of vector fields:

g

(

T^∗

(

X,Y

)

,Z

) =

g

(∇

^∗_XY,Z

) −

g

(∇

^∗_YX,Z

) −

g

([

X,Y

]

,Z

)

=

Xg

(

Y,Z

) −

g

(∇

_XZ,Y

) −

Yg

(

X,Z

) +

g

(∇

_YZ,X

) −

g

([

X,Y

]

,Z

)

=

g

(

Z,

∇

_XY

) −

g

(

Z,

∇

_YX

) −

g

([

X,Y

]

,Z

)

=

g

(

T

(

X,Y

)

,Z

)

49

As a particular, but important case, if the torsion ofTvanishes, so does the torsion ofT^∗.

50

Proposition 2. Let

∇

,

∇

^∗be dual connections. Then

∇

g

= −∇

^∗g

51

Proof. For any triple

(

X,Y,Z

)

of vector fields:

(∇

_Zg

)(

X,Y

) =

Z

(

g

(

X,Y

)) −

g

(∇

_ZX,Y

) −

g

(

X,

∇

_ZY

)

and

(∇

^∗_Zg

)(

X,Y

) =

Z

(

g

(

X,Y

)) −

g

(∇

^∗_ZX,Y

) −

g

(

X,

∇

^∗_ZY

)

Using the relations:

Z

(

g

(

X,y

) −

g

(∇

^∗_ZX,Y

) =

g

(

X,

∇

_ZY

)

and:

Z

(

g

(

X,y

) −

g

(

X,

∇

^∗_ZY

) =

g

(∇

_ZX,Y

)

the claim follows.

52

Definition 2. Let

∇

¹,

∇

²be affine connections onTM. Their mutual torsion is the tensor:

D∇₁,∇₂

(

X,Y

) = ∇

¹_XY

− ∇

²_YX

− [

X,Y

]

Remark1. The divergence tensor is defined for dual connections

∇

,

∇

^∗asD

(

X,Y

) = ∇

_XY

− ∇

^∗_XY,

53

which is related toD∇,∇^∗by the relationD∇,∇^∗

=

T

(

X,Y

) +

D

(

X,Y

)

. For torsion-less connections,

54

the two notions agree, i.e.D∇,∇^∗

=

D.

55

In the case of dual connections with vanishing torsion, the commutation defect of the divergence

56

is related to the mutual curvature of the connections.

57

Definition 3. Let

(∇

¹,

∇

²

)

be a pair of connections. Their mutual curvature is the tensor

(

1, 3

)

-tensor:

R_∇1∇²

(

X,Y,Z

) = ∇

¹_X

∇

_Y²Z

− ∇

_Y¹

∇

²_XZ

− ∇

¹_[X,Y]Z (3) As in the case of the curvature, it is often useful to introduce the

(

0, 4

)

-tensor:

R_∇1∇²

(

X,Y,Z,U

) =

g

(

R_∇1∇²

(

X,Y,Z

)

_,U

)

The curvature and the mutual curvature of dual connections enjoy symmetry properties.

58

Proposition 3. Let

(∇

,

∇

^∗

)

be a pair of dual connections. Then, for any vector fields X,Y,Z,U;

(R

(

X,Y,Z,U

) =

R^∗

(

X,Y,U,Z

)

R∇^∗∇

(

X,Y,Z,U

) =

R∇∇^∗

(

X,Y,U,Z

)

⁽⁴⁾ Proof. The proof of the first property is found in, e.g. [4]. For the second, the definition ofR∇∇^∗is written as:

R∇^∗∇

(

X,Y,Z,U

) =

g

(∇

^∗_X

∇

_YZ,U

) −

g

(∇

^∗_Y

∇

_XZ,U

) −

g

(∇

^∗_[X,Y]Z,U

)

Using the duality property:

R∇^∗∇

(

X,Y,Z,U

) =

X

(

g

(∇

_YZ,U

)) −

g

(∇

_YZ,

∇

_XU

)

−

Y

(

g

(∇

_XZ,U

)) +

g

(∇

_XZ,

∇

_YU

)

−

g

(∇

^∗_[X,Y

]

Z,U

)

Using duality once again:

R∇^∗∇

(

X,Y,Z,U

) =

XY

(

g

(

Z,U

)) −

Xg

(

Z,

∇

_Y^∗U

) −

Y

(

g

(

Z,

∇

^∗_XU

)) +

g

(

Z,

∇

^∗_Y

∇

_XU

)

−

YX

(

g

(

_Z,U

)) +

_Y

(

g

(

_Z,

∇

^∗_XU

)) +

X

(

g

(

Z,

∇

^∗_XU

)) −

g

(

Z,

∇

^∗_X

∇

_YU

)

− [

X,Y

]

g

(

Z,U

) +

g

Z,

∇

_[X,Y]U

= −

R^∗_∇∇∗

(

Y,X,U,Z

) =

R^∗_∇∇∗

(

X,Y,U,Z

)

59

In the case of dual connections without torsion, the definition ofD

(

X,Y

)

simplifies to

∇

_XY

−

60

∇

^∗X_Y. Letting DX : Y

→

D

(

X,y

)

, the next proposition relates the commutation defect to the

61

curvatures.

62

Proposition 4. For any vector fields X,Y,Z:

DXDYZ

−

DYDXZ

=

R

(

X,Y,Z

) +

R^∗

(

X,Y,Z

) −

R∇∇^∗

(

X,Y,Z

) −

R∇^∗∇

(

X,Y,Z

)

Proof. By simple computation:

DXDYZ

−

DYDXZ

= (∇

_X

− ∇

^∗_X

) (∇

_YZ

− ∇

_Y^∗Z

) − (∇

_Y

− ∇

_Y^∗

) (∇

_XZ

− ∇

^∗_XZ

)

= ∇

_X

∇

_YZ

− ∇

_X

∇

^∗YZ

− ∇

^∗_X

∇

_YZ

+ ∇

^∗_X

∇

^∗_YZ

− ∇

_Y

∇

_XZ

+ ∇

_Y

∇

^∗_XZ

+ ∇

^∗_Y

∇

_XZ

− ∇

^∗_Y

∇

^∗_XZ and the claims follows by identification of the terms.

63

Proposition 5. Let

∇

,

∇

^∗be dual affine connections on TM. Then, for any triple X,Y,Z of vector fields:

g

(∇

_XY,Z

) =

g

∇

^lc_XY,Z

+

¹

2

[

g

(

D∇,∇^∗

(

Z,X

)

,Y

) −

g

(

D∇,∇^∗

(

Y,Z

)

,X

) +

g

(

D∇,∇^∗

(

X,Y

)

,Z

)]

(5) where

∇

^lcis the Levi-Civita connection.

64

Proof. Since the two connections are dual:

X

(

g

(

Y,Z

)) =

g

(∇

_XY,Z

) +

g

(

Y,

∇

^∗_XZ

)

Using the definition ofD∇,∇^∗it comes:

X

(

g

(

Y,Z

)) =

g

(∇

_XY,Z

) +

g

(

Y,

∇

_ZX

) −

g

(

D∇,∇^∗

(

Z,X

)

,Y

) −

g

([

Z,X

]

,Y

)

Using then an alternating sum over the cyclic permutations of

(

X,Y,Z

)

and the Koszul formula:

2g

∇

^lc_XY,Z

=

X

(

g

(

Y,Z

)) −

Z

(

g

(

X,Y

)) +

Y

(

g

(

Z,X

)) +

g

(

Y,

[

Z,X

] −

g

(

X,

[

Y,Z

]) +

g

(

Z,

[

X,Y

])

yields the result.

65

Remark2. Prop.5is the analogue of the Kozsul formula for dual connections. It is a defining property

66

givenD∇∇^∗.

67

Notation 1. The

(

0, 3

)

-tensor:

U∇₁,∇₂

(

X,Y,Z

) =

g D∇₁,∇₂

(

Z,X

)

,Y

−

g D∇₁,∇₂

(

Y,Z

)

,X

+

g D∇₁,∇₂

(

X,Y

)

,Z

(6)

is the skewness tensor associated the connections

∇

₁,

∇

₂. When no confusion is possible in the case of

68

dual connections, the subscripts will be dropped so thatU

(

X,Y,Z

)

stands forU∇,∇^∗

(

X,Y,Z

)

69

Remark3. The formula of prop.5can be rewritten to give the expression of

∇

^∗: g

(∇

^∗_XY,Z

) =

g

∇

^lc_XY,Z

−

¹

2U

(

Y,X,Z

)

Proposition 6. For any triple

(

X,Y,Z

)

:

U

(

X,Y,Z

) =

U

(

Y,X,Z

) +

2g

(

T

(

X,Y

)

,Z

)

(7) where T is the torsion of

∇

.

70

Proof. Using the definition:

∇

_XY

= ∇

_YX

+ [

X,Y

] +

T

(

X,Y

)

and the fact that the Levi-Civita has vanishing torsion:

g

(∇

_XY,Z

) =

g

∇

^lc_XY,Z

+

¹

2U

(

X,Y,Z

)

thus:

g

(∇

_YX,Z

) =

g

∇

^lc_YX,Z

−

g

(

T

(

X,Y

)

,Z

) +

¹

2U

(

X,Y,Z

)

=

g

∇

^lc_YX,Z

+

¹

2U

(

Y,X,Z

)

and so:

U

(

X,Y,Z

) =

U

(

Y,X,Z

) +

2g

(

T

(

X,Y

)

,Z

)

71

Proposition 7. The tensor U has the cyclic symmetry propery, that is for any triple

(

X,Y,Z

)

of vector fields:

U

(

X,Y,Z

) =

U

(

Z,X,Y

)

(8)

Proof. Using the symmetry of the Riemann metric, the same derivation as in prop.5but applied to the termsX

(

g

(

Z,Y

)

,Y

(

g

(

X,Z

)

,Z

(

g

(

Y,X

)

yields:

2g

∇

^lc_XZ,Y

=

2g

(∇

_XZ,Y

)

−

g

(

Z,D

(

Y,X

)) +

g

(

X,D

(

Z,Y

)) −

g

(

Y,D

(

X,Z

))

(9)

By identification it comes:

U

(

X,Z,Y

) =

U

(

Y,X,Z

)

(10)

72

Proposition 8. Let U be a tensor with cyclic symmetry, then the connections defined by:

g

(∇

_XY,Z

) =

g

∇

^lc_XY,Z

+

¹

2U

(

X,Y,Z

)

g

(∇

^∗_XY,Z

) =

g

∇

^lc_XY,Z

−

¹

2U

(

Y,X,Z

)

(11)

are dual.

73

Proof. For any triple

(

X,Y,Z

)

of vector fields:

X

(

g

(

Y,Z

)) =

g

∇

^lc_XY,Z

+

g

Y,

∇

^lc_XZ Under the assumption of eq.10, it comes:

X

(

g

(

Y,Z

)) =

g

(∇

_XY,Z

) +

¹

2U

(

X,Y,Z

) +

g

((

Y,

∇

^∗_XZ

) −

¹

2U

(

Z,X,Y

)

and sinceUhas cyclic symmetry:

X

(

g

(

Y,Z

)) =

g

(∇

_XY,Z

) +

g

((

Y,

∇

^∗_XZ

)

74

Proposition 9. Let

∇

₁,

∇

₂be a pair of affine connections. For any triple

(

X,Y,Z

)

of vector fields:

g Y,D∇₁,∇₂

(

Z,X

)

=

¹ 2

U∇₁,∇₂

(

X,Y,Z

) +

U∇₁,∇₂

(

Z,X,Y

)

(12) Proof. Direct computation from the definition ofU.

75

Remark4. Prop.9shows that the mutual torsion of a pair of dual connections is uniquely defined by a

76

cyclic symmetric tensor. Conversely, for a pair

∇

₁,

∇

₂of connections, the cyclic symmetry defect of

77

the tensorU∇₁,∇₂, namelyA

(

X,Y,Z

) =

U∇₁,∇₂

(

X,Y,Z

) −

U∇₁,∇₂

(

Z,X,Y

)

is the obstruction of being

78

dual. Please note also that the torsion for a pair of dual connections can be seen as the obstruction for

79

the tensorUto be totally symmetric.

80

Remark 5. A statistical manifold may be defined as a quadruple

(

M,g,

∇

,U

)

with M a a smooth

81

manifold,g a Riemannian metric,

∇

an affine connection andUa tensor with cyclic symmetry. It

82

slightly more general than the usual definition sinceUis not required to be totally symmetric, thus

83

allowing connections with torsion.

84

3. Dual connections lifts

85

LetUbe a coordinate neighborhood in M and letπ:T^∗M

→

Mbe the canonical projection.

86

φ⁻¹

(

U

)

is a coordinate neighborhood inT^∗Mwith coordinates denoted as

(

x¹, . . . ,xⁿ,p1, . . . ,pn

)

.

87

The lift of connections on the cotangent bundle has been studied in [6,7] using the Riemann extension defined in [10]. Another kind of lift is introduced in [11] along with a metric onT^∗M Let

(

M,g

)

be a smooth Riemannian manifold and let

∇

be an affine connection. The kernel of dπ: TT^∗M

→

T^∗M defines an integrable distribution, called the vertical distribution, hereafter denoted byVT^∗M. It is spanned by the vectors:

ej+n

=

δ^j

=

^∂

∂p_j, j

=

1 . . .n (13)

Complementary to it, there is an horizontal distribution spanned by the vectors:

ej

=

∂_j

+

Γ^k_jipkδⁱ, j

=

1 . . .n (14) with:

∂_j

=

^∂

∂x^j

These basis vectors are conveniently put into a matrix form, following the convention of [11]:

L

=

^{Id 0} Γ Id

!

(15) whereΓis the matrix with entries:

Γji

=

Γ^k_jip_k (16)

Definition 4. The Riemannian extension of a torsion-free affine connection

∇

onTMis the symmetric

(

0, 2

)

-tensor with component matrix:

∇

^R

= −

_2Γ Id Id 0

!

whereΓis the matrix defined in16.

88

Proposition 10. Let

∇

be a torsion-free affine connection on M and let

(

e_j

)

_1,...,2n be its adapted frame in TT^∗M. With respect to it, the component matrix of the Riemannian extension is:

0 Id Id 0

!

Proof. In the adapted frame, the expression of the component matrix of the Riemannian extension is:

L^t

−

2Γ Id Id 0

! L

which is equal to:

−

2Γ

+

Γ

+

Γ^t Id

Id 0

!

using the assumption that

∇

is torsion-free,Γ^t

=

_Γand the claim follows.

89

Definition 5. The Levi-Civita connection with respect to Riemannian extension, denoted by

∇

^c, is

90

called the complete lift of the connection

∇

. .

91

Proposition 11. The Christoffel symbols of the complete lift

∇

^care given by:

cΓ^k_ji

=

_Γ^k_ji,^cΓ^k+n_ji

=

p_lR^l_kij,^cΓ^k+n_j(i+n)

= −

_Γⁱ_jk,i,j,k

=

1, . . . ,n

When

∇ = ∇

^lc, the torsion-free assumption is automatically satisfied, so that in an adapted frame

92

the Riemannian extension reduces to the one of prop.10

93

Proposition 12. Let

(∇

,

∇

^∗

)

be a pair of dual affine connections on TM. Then, with respect to the Riemannian extension

∇

^Rof

∇

^lc, the following relations hold:

L^t

∇

^RL^∗

=

L^∗t

∇

^RL

=

^{0 Id} Id 0

!

(17)

L^t

∇

^RL

=

1

2 D˜

+

D^˜t Id

Id 0

!

(18)

L^∗t

∇

^RL^∗

= −

¹₂ D^˜

+

D^˜t Id

Id 0

!

(19)

whereD is the matrix with entries:˜

D˜_ji

=

p_kD^k_ji

and L (resp. L^∗) is the component matrix of the adapted frame to

∇

(resp.

∇

^∗).

94

Proof. In the case of dual connections, eq.12yields:

g

(

D

(

X,Y

)

_,Z

) =

U

(

X,Y,Z

)

and so:

∇ = ∇

^lc

+

¹

2D (20)

∇

^∗

= ∇

^lc

−

¹

2D^t (21)

whereD^t

(

X,Y

) =

D

(

Y,X

)

. From20(resp.21), it comes:

Γ

=

_Γ^lc

+

¹

2D˜ (22)

Γ^∗

=

_Γ^lc

−

¹

2D˜^t (23)

(24) When then have:

∇

^RL

= −

Γ^lc

+

^D₂^˜ Id

Id 0

!

and:

L^∗t

∇

^RL

= −

^D₂^˜

+

^D₂^˜ Id

Id 0

!

=

^{0 Id}

Id 0

!

The other equations are proved the same way.

95

The above relations show that the horizontal subspaces of

∇

and

∇

^∗are related by the Riemannian extension in a very simple way. LetX,Ybe a vector inTx,pT^∗Mwith decompositionX

=

XV

+

XH

(resp.Y

=

Y_V^∗

+

Y_H^∗) according to the horizontal subspace of

∇

(resp.

∇

^∗), then:

∇

^R

(

Y,X

) = h

Y_V^∗,X_H

i + h

X_V,Y_H^∗

i

with

h·

,

·i

the euclidean inner product.

96

Another interesting fact is that with respect to the adapted frames of

∇

(resp.

∇

), the Riemannian

97

extension becomes a modified Riemannian extension in the sense of [12]. To a given modified

98

Riemannian extension, it is thus possible to associate a pair of dual connections with a given torsion

99

(this last restriction comes from the fact that only the symmetric part of the tensor D enters the

100

expression).

101

Since duality is related to metric, it is not so obvious how to lift a pair of mutually dual connections

102

in a canonical way since the complete lifts of

∇

and

∇

^∗involve different Riemannian extensions.

103

The preferred approach will be thus to lift the mutual torsionDto a

(

0, 3

)

-tensor, what can be done

104

extending the approach of [6], and to exploit the fact that it has the cyclic symmetry property.

105

In the sequel, the symmetric (resp. anti-symmetric) part with respect to the contravariant indices of the

(

1, 2

)

-tensorDwill be denoted by^sD(resp.^aD), i.e.:

sD_ij^k

=

¹ 2

D^k_ij

+

D^k_ji

aD^k_ij

=

¹ 2

D^k_ij

−

D^k_ji

Proposition 13. The expression:

σ

=

¹

2p_k^aD^k_ijdxⁱ

∧

dx^j defines a2-form on TT^∗M. Its exterior derivative dσis given by:

dσ

=

¹ 2p_l∂^aD^l_ij

∂x^k dx^k

∧

dxⁱ

∧

dx^j

+

¹ 2

aD^k_ijdp_k

∧

dxⁱ

∧

dx^j Rearranging the terms, the formdσcan be rewritten as:

6dσ

=

p_l ∂^aD^l_ij

∂x^k

+

^∂

aD^l_ki

∂x^j

+

^∂

aD^l_jk

∂xⁱ

!

dxⁱ

∧

dx^j

∧

dx^k

+

^aD^k_ijdp_k

∧

dxⁱ

∧

dx^j

+

^aDⁱ_jkdx^k

∧

dp_i

∧

dx^j

+

^aD_ki^j dx^k

∧

dxⁱ

∧

dp_j

(25)

It turns out that the above tensor has cyclic symmetry since it is

(

0, 3

)

and skew-symmetric. This

106

can made more explicit by first noticing that the first line in the right hand side has obviously this

107

property. In the second line, considering as an example the first term ^aD^k_ijdpk

∧

dxⁱ

∧

dx^j, a cyclic

108

permutation of the arguments yields^aD_ij^kdx^j

∧

dp^k

∧

dxⁱ. Now, the indices changej

→

k,k

→

i,i

→

j

109

gives^aDⁱ_jkdx^k

∧

dpi

∧

dx^j, which is exactly the original second term. The remaining terms can be

110

worked the same way.

111

Considering now the symmetric part ofD, a similar procedure can applied to obtain a fully symmetric

(

0, 3

)

-tensor. Let us denote by

the symmetric tensor product, that is:

x

y

= (

x

⊗

y

+

y

⊗

x

)

/2 . From^sD, a symmetric tensor onTT^∗Mcan be defined as:

θ

=

¹

2p_k^sD^k_ijdxⁱ

dx^j

Following the construction of13and the formula of [13], a fully symmetric lift can be defined.

112

Definition 6. The symmetric lift of^sDis the

(

0, 3

)

-tensor with components:

1

6 p_l ∂^sD_ij^l

∂x^k

+

^∂

sD^l_ki

∂x^j

+

^∂

sD^l_jk

∂xⁱ

!

dxⁱ

dx^j

dx^k

+

^sD_ij^kdp_k

dxⁱ

dx^j

+

^sDⁱ_jkdx^k

dp_i

dx^j

+

^sD_ki^j dx^k

dxⁱ

dp_j

(26)

Gathering things together, both the symmetric and the anti-symmetric part ofDcan be lifted to a

113

cyclic symmetric

(

0, 3

)

-tensor. In the sequel, the notation of [6] is adopted: Latin lettersi,j, . . . refer to

114

xcomponents, overlined lettersi,j, . . . refers topcomponents and capital letters can be used for both.

115

As an example,dxⁱ

=

dpi,δⁱ

=

∂_i.

116

Definition 7. The cyclic symmetric complete lift of the

(

1, 2

)

-tensorD, denotedU^c, is the

(

0, 3

)

-tensor with componentsu^c_ABCdx^A

⊗

dx^B

⊗

dx^C:















u^c_ijk

=

p_{l ∂Dl}

ij

∂x^k

+

^∂Dlki

∂x^j

+

^∂D

ljk

∂xⁱ

u^c_ijk

=

Dⁱ_jk,u^c_ijk

=

D^j_ki,u^c_ijk

=

D^k_ij u^c_ijA

=

u^c_iAj

=

u^c_Aij

=

0

FromU^c, the complete lift ofDcan be defined as the

(

1, 2

)

-tensorD^csuch that for any triple of vector fields:

∇

^R

(

X,D^c

(

_Y,Z

)) =

_U^c

(

_X,Y,Z

)

₍₂₇₎ Given the matrix form of the Riemannian extension:

∇

^R

= −

_2Γ Id Id 0

!

its inverse is readily obtained as:

∆

=

^{0 Id} Id 2Γ

!

The components ofD^cin coordinates can be obtained by composing the matrix A, yielding:

D^{c C}_AB

=

∆^CDuDAB



























D^{c i}_jk

=

_Dⁱ_jk D^{c i}_jk

=

p_l

_∂Dl

ij

∂x^k

+

^∂Dlki

∂x^j

+

^∂D

ljk

∂xⁱ

+

_2Γⁱ_lD^l_jk D^{c i}_jk

=

D_ki^j

D^{c i}_jk

=

D_ij^k

D^{c i}_jk

=

D^{c i}_jk

=

D^{c i}_jk

=

D^{c i}_jk

=

0

(28)

with the notationΓⁱ_l

=

Γ_il. Please note that the above relations are different from the one given in [6]

117

for the complete lift of a skew-symmetric

(

1, 2

)

-tensor since here the Riemann extension is used in

118

place of the canonicale

(

1, 1

)

-tensor and only the cyclic symmetry is assumed. This last fact can be

119

noticed in the third and forth lines of eq. (28).

120

The next definitions are recalled for the sake of completeness.

121

Definition 8. Letω

=

ω_idxⁱbe a degree 1 differential form. Its vertical lift toTT^∗Mis the vector field:

ω^V

=

ω_iδⁱ

Vector fields admit both a vertical and a complete lift. Only the later will be used here.

122

Definition 9. LetX

=

Xⁱ∂_ibe a vector field onM. Its complete lift toTT^∗Mis the vector field:

X^c

=

Xⁱ∂_i

−

p_l∂X^l

∂x^kδ^k Finally

(

1, 1

)

-tensors can be lifted in a quite obvious way:

123

Definition 10. LetFbe a

(

1, 1

)

-tensor field. Its vertical lift toTT^∗Mis the vector field:

F^V

=

_plFk^lδ^k

The action ofD^con vertical and complete lift can now be obtained.

124

Proposition 14. Let X be a vector field andω,θbe1-forms. Then:

( D^c

(

ω^V,θ^V

) =

₀

D^c

(

ω^V,X^c

) = (

ωDX

)

^V, D^c

(

X^c,ω^V

) =

ωD^XV (29)

where DX(resp. D^X) is the

(

1, 1

)

-tensor defined by: DX

(

Y

) =

D

(

X,Y

)

(resp. D^X

(

Y

) =

D

(

Y,X

)

).

125

Proof. Letω

=

ω_idxⁱ,θ

=

θ_jdx^j. ThenD

(

ω^v,θ^V

) =

ωⁱθ^jD^{c A}_ij

=

0. LetXbe vector field andX^cits complete lift. By linearity:

D^{c A}

(

ω^V,X^c

) =

ωⁱX^jD^{c A}_ij

−

pl

∂X^l x^k D^{c A}_ik

SinceD_ik^{c A}

=

0, the second term in the right hand side vanishes. For the fist one, onlyD^{c k}_ij

=

Dⁱ_jkis non-zero, so that:

D^c

(

ω^V,X^c

) =

ω_iX^jDⁱ_jkδ^k

The tensorD_Xhas expressionD_X

(

Y

) =

D_ij^kXⁱY^j∂_k, so thatωD_Xis the formωD_X

=

ω_kXⁱD_ij^kdx^j, whose

126

vertical lift isω_kXⁱD^k_ij∂^k.

127

Please note while the expression obtained is similar to the one of [6], the sign is opposite.

128

The case of the action on two complete lifts is a little bit more complicated. First of all, given two vector fieldsX

=

Xⁱ∂_i,Y

=

Y^j∂_j, a simple computation yields:

D^c

(

X^c,Y^c

) =

XⁱY^jD_ij^k∂_k

+

XⁱY^jp_l ∂D^l_ij

∂x^k

+

^∂D

l ki

∂x^j

+

^∂D

l jk

∂xⁱ

! δ^k

+

2plΓklD_ij^lXⁱY^jδ^k

−

Xⁱpl

∂Y^l

∂x^jD_ki^j δ^k

−

Y^jpl

∂x^l

∂xⁱDⁱ_jkδ^k

(30)

After rewriting, eq. (30)becomes:

D^c

(

X^c,Y^c

) =

XⁱY^jD^k_ij∂_k

+

Xⁱp_l Y^j∂D_ki^l

∂x^j

−

^∂Y

l

∂x^jD_ki^j

! δ^k

+

Y^jpl Xⁱ∂D^l_jk

∂xⁱ

−

^∂X

l

∂xⁱ Dⁱ_jk

!

δ^k

+

XⁱY^jpl

∂D_ij^l

∂x^k δ^k

+

2plΓklD^l_ijXⁱY^jδ^k

(31)

Let us consider, forX,Yfixed vector fields, the

(

1, 1

)

-tensor

∇

^lcD

(

X,Y

)

:

Z

7→ ∇

^lc_Z

(

D

(

X,Y

)) =

Z^k∂D^l_ijXⁱY^j

∂xk

+

_Γ^l_kmD^m_ijXⁱY^jZ^k

Its vertical lift is then:

∇

^lcD

(

X,Y

)

^V

=

pl

∂D^l_ijXⁱY^j

∂X^k δ^k

+

plΓ^l_kmD_ij^mXⁱY^jδ^k (32) On the other hand, the complete lift of the vector fieldD

(

X,Y

)

is:

(

D

(

X,Y

))

^C

=

D_ij^kXⁱY^j∂_k

−

p_l∂D^l_ijX_iY_j

∂x^k δ^k (33)

Combining ed. (32) and eq. (33) yields:

2p_lΓ_klD^l_ijXⁱY^jδ^k

+

XⁱY^jD_ij^k∂_k

=

2

∇

^lcD

(

X,Y

)

^V

+ (

D

(

X,Y

))

^C

−

p_l∂D^l_ijXiYj

∂x^k δ^k (34)

Putting the expression in eq. (31) yields:

D^c

(

X^c,Y^c

) =

2

∇

^lcD

(

X,Y

)

^V

+ (

D

(

X,Y

))

^C

+

Xⁱpl Y^j∂D^l_ki

∂x^j

−

^∂Y

l

∂x^jD_ki^j

! δ^k

+

Y^jp_l Xⁱ∂D^l_jk

∂xⁱ

−

^∂X

i

∂xⁱDⁱ_jk

!

δ^k

−

p_lD^l_ij∂X^l

∂x^kY^jδ^k

−

p_lD_ij^lXⁱ∂Y_j

∂x^kδ^k

(35)

LetKbe a

(

1, 1

)

tensorK. Its Lie derivative can be written [14, p. 32, prop. 35.]:

L

_XK

(

Y

) = [

X,K

(

Y

)] −

K

([

X,Y

])

It thus comes:

L

_YDX

(

Z

) = [

Y,DX

(

Y

)] −

DX

([

Y,Z

])

(36) Which can be written in coordinates:

L

_YDX

(

Z

)

^l

=

Y^j∂D^l_ikXⁱZ^k

∂x^j

−

D_ik^j Xⁱ∂Y^l

∂x^jZ^k

−

Y^j∂Z^k

∂x^jXⁱD^l_ik

+

Z^j∂Y^k

∂x^jXⁱD^l_ik

=

Xⁱ Y^jD^l_ik

∂x_j

−

D_ik^j ∂Y^l

∂x^j

!

Z^k

+

^∂Y

k

∂x^jXⁱD_ik^l Z^j

(37)

Plugging it into eq. (35) finally gives the reduced expression:

D^c

(

X^c,Y^c

) =

2

∇

^lcD

(

X,Y

)

^V

+ (

D

(

X,Y

))

^C

+ (L

_YD^X

+ L

_XD_Y

)

^V

+

2

((∇

⁰D

)(

X,Y

) − ∇

⁰

(

D

(

X,Y

))

^V

(38)

with

∇

⁰the trivial connection with 0 Christoffel symbols. The equation eq. (38) completely defines the

129

tensorD^c.

130

From the complete liftD^c, dual connections with respect to the Riemannian extension can be obtained:

(

∇

˜

= ∇

^c

+

¹₂D^c

∇

˜^∗

= ∇

^c

−

¹₂D^{c t} (39) The pair

( ∇

^˜, ˜

∇

^∗

)

defines the complete lift of the original statistical structure to the pseudo-Riemannian

131

manifold

(

T^∗M,

∇

^R

)

. When

∇

is without torsion, thenDis symmetric. Using eq. (38) and the fact that

132

in such a caseDX

=

D^Xshow thatD^cis itself symmetric, proving that ˜

∇

has vanishing torsion.

133

Conflicts of Interest:‘The author declare no conflict of interest

134

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135

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c

2020 by the authors. Submitted to Mathematics for possible open access publication

164

under the terms and conditions of the Creative Commons Attribution (CC BY) license

165

(http://creativecommons.org/licenses/by/4.0/).

166