Horizontal lift of symmetric connections to the bundle of volume forms V

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXIV, NO. 1, 2010 SECTIO A 45–61

ANNA GĄSIOR

Horizontal lift of symmetric connections to the bundle of volume forms V

Abstract. In this paper we present the horizontal lift of a symmetric affine connection with respect to another affine connection to the bundle of volume formsV and give formulas for its curvature tensor, Ricci tensor and the scalar curvature. Next, we give some properties of the horizontally lifted vector fields and certain infinitesimal transformations. At the end, we consider some substructures of aF (3, 1)-structure on V.

1. Introduction. Throughout the paper we assume that i, k, . . . = 1, 2, 3, . . . , n and α, β, . . . = 0, 1, 2, . . . , n. Moreover, the Einstein summation con- vention will be used with respect to these systems of indices.

Let M be an orientable n-dimensional manifold,V be the bundle of vol- ume forms over M and let π : V → M be a projection of the bundle. We consider two local charts (U, xⁱ) and (U^, x^i) on M , U ∩ U^ = ∅ and the volume form ω∈ V. Assume that form ω is given by

ω = v(x)dxⁱ∧ . . . ∧ dxⁿ,

in the local chart (U, xⁱ), where v > 0 is a smooth function and ω = v^(x^)dx^i ∧ . . . ∧ dx^n

in the chart (U, x^i). Let functions x^i = x^i(x) be orientation-preserving transition functions on manifold M . Then the transition functions onV are

2000 Mathematics Subject Classification. 53B05.

Key words and phrases. Horizontal lift,π-conjugate connection, Killing field, infinites- imal transformation,F (3, 1)-structure, F K, F AK, F NK, F QK, F H-structure.

given by the following formulas

v^ = ¯I · v, x^i = x^i(x), where ¯I = det

∂x^i

∂x^j



is the Jacobian of the map x^i = x^i(x). Following Dhooghe ([3]), we introduce a new coordinate system (x⁰, x¹, . . . , xⁿ) on V, where x⁰ = ln v. Then the transition functions in the terms of these coordinate system are

x^0 = x⁰(x) + ln ¯I(x) x^i = x^i(x).

Let ¯J (x) = ln ¯I(x) and J (x^) = ln I(x^). Since I · ¯I = 1, we have

∂J

∂x^i = −∂ ¯J

∂x^j

∂x^j

∂x^i and

∂ ¯J

∂xⁱ = −∂J

∂x^j

∂x^j

∂xⁱ .

Then the Jacobi matrix of the transition functions on V has the following

form ⎡

⎢⎣1 ∂ ¯J

∂xⁱ 0 ∂x^j

∂xⁱ

⎤

⎥⎦ .

For the further purposes we quote some theorems describing the properties of geometrical objects on the bundle of volume forms.

Theorem 1.1 ([3]). Let ∇ = (Γ^k_ij) be a symmetric connection and v = v^{i ∂}_∂xi

be a vector field on a manifold M . Then

¯

v = −vⁱΓ^k_ik ∂

∂x⁰ + vⁱ ∂

∂xⁱ

is globally defined vector field on V, which is called the horizontal lift of v.

Theorem 1.2 ([9]). Let ∇ = (Γ^k_ij) be a symmetric connection and g = (gij) be a tensor of type (0, 2) on a manifold M . Then

¯ g =

1 Γ^k_ik Γ^k_ik g_ij + Γ^k_ikΓ^t_jt



is globally defined (0, 2)-tensor on V, which is called the horizontal lift of g.

Theorem 1.3 ([9]). Let ∇ = (Γ^k_ij) be a symmetric connection and g be a Riemannian metric on a manifold M . Then ¯g is a Riemannian metric onV and

(¯g)⁻¹=

1 + g^ijΓ^k_ikΓ^t_jt −g^ijΓ^k_ik

−g^ijΓ^k_ik g^ij 

.

2. The horizontal lift of a symmetric connection. Curvatures of a horizontally lifted connection. At the beginning, we present a theo- rem on a horizontal lift of a symmetric connection with respect to another connection to the bundle of volume formsV. Next, we give formulas for its curvature tensor, the Ricci tensor and the scalar curvature. In the sequel we will use the following conventions

Γ^k_ij|m= ∂Γ^k_ij

∂x^m, Γ^k_[ik|j]= 1

2



Γ^k_ik|j− Γ^k_jk|i , Γ^k_[ik|jm]= 1

2



Γ^k_ik|jm− Γ^k_mk|ij , g_jk|i= ∂g_jk

∂xⁱ .

Theorem 2.1. Let ∇ = (Γ^k_ij) be a symmetric connection and ∇₁ = (Φ^k_ij) be a connection on M . Then an operator ¯∇₁ whose nonzero coefficients are given by

∇¯_1i ∂

∂x^j = ¯∇_1j ∂

∂xⁱ =



Γ^t_it|j− Φ^r_ijΓ^t_rt

 ∂

∂x⁰ + Φ^k_ij ∂

∂x^k

is a linear connection on V, which will be called the horizontal lift of the connection∇1 with respect to the connection ∇.

Proof. We are going to check that the coefficients ( ¯Φ^γ_αβ) of the connection

∇¯₁ satisfy the transformation rule of the connection. This transformation rule for the zero coefficients of the connection ¯∇1 follows from simple cal- culations. For the nonzero coefficients we have

Φ¯^k_i^j^ = ∂x^α

∂x^i

∂x^β

∂x^j

∂x^k

∂x^γΦ¯^γ_αβ+∂x^k

∂x^α

∂²x^α

∂x^i∂x^j

= ∂xⁱ

∂x^i

∂x^j

∂x^j

∂x^k

∂x^kΦ¯^k_ij+∂x^k

∂x^k

∂²x^k

∂x^i∂x^j = Φ^k_i^j^. In the next part of the proof we use the following equality

∂²x^0

∂xⁱ∂x^j = ∂

∂xⁱ ∂J

∂x^j

= ∂

∂x^j ∂J

∂xⁱ

. We receive from above identity

∂²x^d

∂x^a∂x^r

∂²x^r

∂x^j∂x^d

∂x^a

∂xⁱ = ∂²x^d

∂x^a∂x^r

∂²x^r

∂xⁱ∂x^d

∂x^a

∂x^j .

For the coefficients ( ¯Φ⁰_i^j^) we have Φ¯⁰_i^j^ = ∂x^α

∂x^i

∂x^β

∂x^j

∂x^0

∂x^γΦ¯^γ_αβ+∂x^0

∂x^α

∂²x^α

∂x^i∂x^j

= ∂x^a

∂x^i

∂x^b

∂x^jΦ¯⁰_ab+ ∂x^a

∂x^i

∂x^b

∂x^j

∂J

∂x^pΦ¯^p_ab+ ∂²x⁰

∂x^i∂x^j + ∂J

∂x^p

∂²x^p

∂x^i∂x^j

= ∂x^a

∂x^i

∂x^b

∂x^j



Γ^t_at|b− Φ^r_abΓ^t_rt

 + ∂x^a

∂x^i

∂x^b

∂x^j

∂J

∂x^pΦ^p_ab + ∂²x⁰

∂x^i∂x^j + ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j

= ∂x^a

∂x^i

∂x^b

∂x^j

∂

∂x^b

∂x^c

∂x^aΓ^t_c^t^ + ∂ ¯J

∂x^a



− ∂x^a

∂x^i

∂x^b

∂x^j



∂x^d

∂x^a

∂x^e

∂x^b

∂x^r

∂x^fΦ^f_d^e^ + ∂x^r

∂x^e

∂²x^e

∂x^a∂x^b



·



∂x^c

∂x^rΓ^t_c^t^+ ∂ ¯J

∂x^r



+ ∂x^a

∂x^i

∂x^b

∂x^j

∂ ¯J

∂x^p

∂x^d

∂x^a

∂x^e

∂x^b

∂x^p

∂x^fΦ^f_d^e^+ ∂x^p

∂x^e

∂²x^e

∂x^a∂x^b



+ ∂²x⁰

∂xⁱ∂x^j + ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j

= ∂²x^c

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^jΓ^t_c^t^+ Γ_i^tt^|j+ ∂²J¯

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j − Φ^c_i^j^Γ^t_c^t^

+ ∂J

∂x^fΦ^f_i^j^− ∂²x^c

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^jΓ^t_c^t^+ ∂J

∂x^e

∂²x^e

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j

− ∂J

∂x^fΦ^f_i^j^− ∂J

∂x^e

∂²x^e

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j + ∂²J

∂x^i∂x^j + ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j

= Γ^t_i^t^|j^ − Φ^c_i^j^Γ^t_c^t^+ ∂²J¯

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j + ∂²J

∂x^i∂x^j + ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j. Now, we have

∂²J¯

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j = ∂²x^e

∂x^p∂x^i

∂²x^p

∂x^e∂x^b

∂x^b

∂x^j + ∂³x^p

∂x^e∂x^b∂x^a

∂x^e

∂x^p

∂x^a

∂x^i

∂x^b

∂x^j and

∂³x^p

∂x^e∂x^a∂x^b

∂x^e

∂x^p

∂x^a

∂x^i

∂x^b

∂x^j = −2 ∂²x^p

∂x^e∂x^b

∂²x^e

∂x^p∂x^i

∂x^b

∂x^j − ∂³x^e

∂x^p∂x^i∂x^j

∂x^p

∂x^e

− ∂J

∂x^r

∂²x^r

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j,

∂²J

∂x^i∂x^j = ∂²x^e

∂x^a∂x^b

∂x^b

∂x^i

∂²x^a

∂x^e∂x^j + ∂³x^a

∂x^e∂x^j∂x^i

∂x^e

∂x^a.

Moreover, 0 = ∂

∂x^b

∂x^r

∂x^a

∂x^a

∂x^i



= ∂²x^r

∂x^a∂x^b

∂x^a

∂x^i +∂x^r

∂x^a

∂²x^a

∂x^i∂x^p

∂x^p

∂x^b and hence

−∂J

∂x^r

∂²x^r

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j = ∂J

∂x^r

∂x^r

∂x^a

∂x²x^a

∂x^i∂x^p

∂x^p

∂x^b

∂x^b

∂x^j

− ∂J

∂x^r

∂²x^r

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j

= −∂ ¯J

∂x^a

∂²x^a

∂x^i∂x^j. Using the above formulas, we get

Φ¯⁰_i^j^ = Γ_i^t^t^|j^ − Φ_i^c^j^Γ^t_c^t^+ ∂²x^e

∂x^p∂x^i

∂²x^p

∂x^e∂x^b

∂x^b

∂x^j − 2 ∂²x^p

∂x^e∂x^b

∂²x^e

∂x^p∂x^i

∂x^b

∂x^j

− ∂³x^e

∂x^p∂x^i∂x^j

∂x^p

∂x^e − ∂J

∂x^r

∂²x^r

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j + ∂²x^e

∂x^a∂x^b

∂x^b

∂x^i

∂²x^a

∂x^e∂x^j + ∂³x^a

∂x^e∂x^i∂x^j

∂x^e

∂x^a + ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j

= Γ_i^tt^|j^ − Φ_i^c^j^Γ^t_c^t^− ∂J

∂x^r

∂²x^r

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j + ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j

= Γ_i^tt^|j^ − Φ_i^c^j^Γ^t_c^t^− ∂J

∂x^r

∂x²x^r

∂x^a∂x^b

∂x^a

∂x^i

∂x^b

∂x^j + ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j

= Γ_i^tt^|j^ − Φ_i^c_j^Γ^t_c^t^− ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j + ∂ ¯J

∂x^p

∂²x^p

∂x^i∂x^j

= Γ^t_i^t^|j^ − Φ_i^c^j^Γ^t_c^t^.  Now, we give formulas for coefficients of a curvature tensor, a Ricci tensor and a scalar curvature for the horizontally lifted connection ¯∇₁. Let R be the curvature tensor of a connection ∇ on a manifold M. The coefficients (R^s_ijk) of R are expressed in terms of the connection ∇ = (Γ^k_ij) by the formula ([1])

R^s_ikj = Γ^l_ikΓ^s_jl− Γ_jk^l Γ^s_il+ Γ^s_ik|j− Γ^s_jk|i.

Theorem 2.2. Let ( ¯R^γ_αβ) be the coefficients of the curvature tensor ¯R of the horizontal lift of the connection ∇1 = (Φ^k_ij) with respect to the symmetric connection ∇ = (Γ^k_ij). Then nonzero coefficients of the tensor ¯R are given by the following formulas

R¯^s_ikj = R^s_ikj,

R¯_ikj⁰ = −Γ^t_rtR^r_ikj+ 2Φ^d_ikΓ^t_[jt|d]+ 2Φ^d_jkΓ^t_[dt|i]+ 2Γ^t_[it|kj], where R = (R^s_ijk) is the curvature tensor of the connection ∇₁ on M .

Proof. From the definition of the curvature tensor we have R¯_βγδ^τ = ¯Φ^α_βγΦ¯^τ_δα− ¯Φ^α_δγΦ¯^τ_βα+ ¯Φ^τ_βγ|δ− ¯Φ^τ_δγ|β. The statements

R¯^β₀₀₀= ¯R^β_i00= ¯R^β_0k0= ¯R^β_00j= ¯R^β_0kj = ¯R^β_i0j = ¯R^β_ik0= 0 and

R¯^s_ikj = R^s_ikj

follows from the definition of the curvature tensor, Theorem 2.1 and simple calculations. Next, we have

R¯⁰_ikj = ¯Φ^α_ikΦ¯⁰_jα− ¯Φ^α_jkΦ¯⁰_iα+ ¯Φ⁰_ik|j− ¯Φ⁰_jk|i

= Φ^d_ik



Γ^t_jt|d− Φ^r_jdΓ^t_rt

− Φ^d_jk

Γ^t_it|d− Φ^r_idΓ^t_rt

 + ∂

∂x^j



Γ^t_it|k− Φ^r_ikΓ^t_rt



− ∂

∂xⁱ



Γ^t_jt|k− Φ^d_jkΓ^r_rt



= −Γ^t_rt



Φ^d_ikΦ^r_jd− Φ^d_jkΦ^r_id+ Φ^r_ik|j− Φ^d_jk|i + Φ^d_ik



Γ^t_jt|d− Γ^t_dt|j + Φ^d_jk



Γ^t_dt|i− Γ^t_it|d

+ Γ^t_it|kj− Γ^t_jt|ki

= −Γ^t_rtR^r_ikj+ 2Φ^d_ikΓ_[jt|d]^t + 2Φ^d_jkΓ^t_[dt|i]+ 2Γ^t_[it|kj].  We recall that the linear connection ∇ is locally volume preserving at p ∈ M, if there exists a neighbourhood U of p in M and a volume form ω∈ U such that ∇ω = 0 ([4]). The integrability conditions for existence of such a local volume forms in local coordinates (U, xⁱ) on a neighbourhood of p∈ M are given by

Γ^k_ik|j = Γ^k_jk|i,

where (Γ^k_ij) are the coefficients of the connection ∇ ([13]). The connection is called globally volume preserving, if such a volume form exists on M . More- over, it is well known that the Riemannian connection on the Riemannian manifold M is locally volume preserving.

Corollary 2.1. Let ∇ be a Riemannian connection on a Riemannian man- ifold (M, g) and ∇₁ be a connection on the manifold M . Then the nonzero coefficients of the curvature tensor ¯R = R¯^δ_αβγ

of the horizontally lifted connection ¯∇1 with respect to the connection ∇ = (Γ^k_ij) are given by

R¯^s_ikj = R^s_ikj, R¯⁰_ikj = −Γ^t_rtR^r_ikj,

where R = (R^s_ijk) is the curvature tensor of the connection ∇₁ on M . For a Ricci tensor and a scalar curvature of the horizontally lifted con- nection we have the following theorems.

Theorem 2.3. Let ∇ be a symmetric connection, ∇₁ be a connection on the manifold M and ( ¯R_αβ) be the coefficients of a Ricci tensor ¯R of the horizontally lifted connection ¯∇₁ with respect to connection ∇. Then the nonzero coefficients of ¯R are given by the formulas

R¯_ik= Rik,

where (Rik) are the coefficients of Ricci tensor R of the connection ∇1 on the manifold M .

Theorem 2.4. Let g be a Riemannian metric on the manifold M and let

¯

g be a horizontally lifted Riemannian metric on V. If ¯K is a scalar curva- ture of the horizontally lifted connection ¯∇1 with respect to the symmetric connection∇, then

K =¯ n− 1 n + 1K,

where K is a scalar curvature of the connection ∇ on M.

Proof. From the definition of the scalar curvature, Theorem 2.3 and for- mula for (¯g^αβ) we have

K =¯ 1

n(n + 1)R¯_αβ¯g^αβ = 1

n(n + 1)R_ikg^ik= n− 1

n + 1K. 

Let π be a non-singular tensor field of type (1, 1) on the manifold M . Rompała in [12] described some properties of π-conjugate connection with respect to a given connection. Now, we prove that if we have a π-conjugate connection ∇₂ with respect to a connection ∇₁ on the manifold M , then the horizontally lifted connection ¯∇2 is ¯π-conjugate with respect to the horizontally lifted connection ¯∇₁ on V, where ¯π is the horizontal lift of π.

Definition 2.1. Let ∇ = (Γ^k_ij) be the linear connection and let π be a non- singular tensor field of the type (0, 2) on M . The connection ∇^∗ = (Gⁱ_ks) which is given by

Gⁱ_ks= π^pi∇kπ_ps+ Γⁱ_ks

is said to be a π-conjugate connection with respect to the connection ∇.

For the horizontally lifted connections ¯∇1 and ¯∇2 with respect to the connection∇ and the horizontally lifted tensor field ¯π of type (0, 2) we have the following theorem.

Theorem 2.5. Let ∇₂ be a π-conjugate connection with respect to a connec- tion ∇1 on manifold M . Let ¯∇1 and ¯∇2 be horizontally lifted connections with respect to a connection ∇ on V. Then ¯∇₂ is a ¯π-conjugate connection with respect to a horizontally lifted connection ¯∇₁, where ¯π is horizontal lift of π with respect to a connection ∇.

Proof. We are going to show the nonzero coefficients ( ¯G^γ_αβ) of the connec- tion ¯∇₂ are given by the formulas

G¯^k_ij = G^k_ij,

G¯⁰_ij = Γ^t_it|j− G^r_ijΓ^t_rt.

From Definition 2.2 and definition of the covariant derivative of a tensor field of type (0, 2) we have

G¯^τ_γ= ¯π^βτ(¯π_β|γ − ¯Φ^α_βγπ¯_α− ¯Φ^α_γ¯π_βα) + ¯Φ^τ_γ,

where ( ¯Φ^γ_αβ) are the coefficients of the connection ¯∇1 and ¯π = (¯π_αβ). For the nonzero coefficients we get

¯

π^β0(¯π_βs|k− ¯Φ^α_βkπ¯_αs− ¯Φ^α_skπ¯_βα) + ¯Φ⁰_ks

= −π^dbΓ^v_bv

π_ds|k− Φ^r_dkπ_rs− Φ^r_skπ_dr

+ Γ^t_kt|s− Φ^r_ksΓ^u_ru

= Γ^t_kt|s− Γ^v_bvG^b_ks= ¯G⁰_ks,

¯

π^βi(¯π_βs|k− ¯Φ^α_βkπ¯_αs− ¯Φ^α_skπ¯_βα) + ¯Φⁱ_ks

= ¯π⁰ⁱ(Γ^t_st|k− Γ^t_st|k− Φ_sk^r Γ^u_ru− Φ^r_skΓ^t_rt) + π^di



π_ds|k+ Γ^t_dt|kΓ^u_su+ Γ^t_dtΓ^u_su|k− Γ^t_dt|kΓ^u_su+ Φ^r_dkΓ^t_rtΓ^u_su

−Φ^r_dk(πrs+ Γ^t_rtΓ^u_su) − Γ^t_st|kΓ^u_du+ Φ_sk^r Γ^t_rtΓ^u_du− Φ_sk^r (π_dr+ Γ^t_dtΓ^u_ru)

 + Φⁱ_ks

= π^di(π_ds|k− Φ^r_dkπ_rs− Φ^r_skπ_dr) + Φⁱ_ks= Gⁱ_ks= ¯Gⁱ_ks.  3. Some properties of a horizontally lifted vector field. Dhooghe in [3] described the horizontal lift of a vector field to the bundle of volume forms V. In this chapter we give some properties of such horizontally lifted vector fields. Let us consider when a horizontally lifted vector field is a Killing field onV. We have

Theorem 3.1. Let (M, g) be a Riemannian manifold. If X is a vector field and∇ is a symmetric, locally volume preserving connection on M, then the horizontally lifted vector field ¯X is a Killing field on (V, ¯g) if and only if X is a Killing field on M .

Proof. Let ∇ = (Γ^k_ij). Then X is a Killing field on the manifold (M, g) if and only if LXg = 0, where L is a Lie derivative of the Riemannian metric g ([7]). Let X be the Killing field on the manifold (M, g). From the assumption that the ∇ is the symmetric, locally volume preserving connection and from the formula of the Lie derivative of the metric ¯g with respect to ¯X we have

LX¯g¯₀₀= (−Γ^k_jkX^k)_|0+ (−Γ^k_jkX^k)_|0 = 0, LX¯¯g_0b= LX¯g¯_b0= X^e(Γ^t_bt|e− Γ^t_et|b) = 0,

LX¯¯g_bc= ¯X^α¯g_bc|α+ ¯g_αcX¯_|b^α+ ¯g_bαX¯_|c^α

= X^d

g_bc+ Γ^t_btΓ^u_cu

|c+ Γ^t_ct

−X^dΓ^u_du



|b

+

g_dc+ Γ^u_duΓ^t_ct

X_|b^d+ Γ^t_bt

−X^dΓ^u_du



|c+

g_bd+ Γ^u_buΓ^t_dt X_|c^d

= X^dg_bc|d+ X^dΓ^t_bt|dΓ^u_cu+ X^dΓ^t_btΓ^u_cu|d= LXg_bc= 0.

We have LX¯g = 0 so ¯¯ X is the Killing field on V.

Let ¯X be the Killing on the bundle of volume forms V. Then we have LX¯g = 0 and from the first part of the proof we get L¯ Xg = 0, so X is the

Killing field on the manifold (M, g). 

Yamauchi in [15] studied certain types of an infinitesimal transformations on tangent bundles. Now, we show that the horizontally lifted vector field X is an infinitesimal affine transformation of horizontally lifted connection¯ onV if and only if the vector field X is an infinitesimal affine transformation of a connection on the manifold M.

Theorem 3.2. Let ¯∇₁ be the horizontal lift of the connection ∇₁ with re- spect to the symmetric connection ∇ on M and let ¯X be the horizontal lift of the vector field X onV. Then ¯X is an infinitesimal affine transformation of the horizontally lifted connection ¯∇1 if and only if X is an infinitesimal affine transformation of the connection ∇₁ on M .

Proof. Let X be the infinitesimal affine transformation ([15]) of the con- nection∇₁= (Φ^k_ij) on M . Then we have

LXΦ^k_ij = 0.

For the horizontally lifted connection ¯∇1= ( ¯Φ^γ_αβ) and the horizontally lifted vector field ¯X we have

LX¯Φ¯^χ₀₀= LX¯Φ¯^χ_0l = LX¯Φ¯^χ_m0= 0, LX¯Φ¯^h_ml = LXΦ^h_ml = 0,

LX¯Φ¯⁰_ml = −Γ^t_dtLXΦ^d_ml = 0.

Thus, ¯X is the infinitesimal affine transformation of the connection ¯∇₁onV.

On the other hand, let ¯X be the infinitesimal affine transformation of the connection ¯∇₁ on V. Then from the first part of this proof we get

LXΦ^k_ij = 0

and X is the infinitesimal affine transformation of the connection∇₁ on the

manifold M . 

Now, we will examine a problem of a fibre-preserving infinitesimal trans- formation on the bundle of volume formsV. We have the following theorem.