Curvatures for horizontal lift of a Riemannian metric

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. LX, 2006 SECTIO A 17–21

ANNA GĄSIOR

Curvatures for horizontal lift of a Riemannian metric

Abstract. In this paper we give formulas for coefficients of linear connection and basic curvatures of the bundle of volume forms with the horizontal lift of metric.

1. Introduction. Kurek in [4] defined the horizontal lift of a linear con- nection on a manifold M to the bundle of linear frames. Mikulski in [7] de- duced natural linear operators transforming vector fields from n-dimensional manifold M to the vector bundle T^(r),aM and in [6] the author defined the natural vector bundle T^(0,0),aM associated to the principal bundle LM of linear frames on the smooth manifold M . On the other hand, Mozgawa and Miernowski in [5] presented a systematic approach of basic types of tensor with respect to a symmetric linear connection to the horizontal lift to the bundle of volume forms. In the present paper we are going to determine coefficients of a linear connection and its basic curvatures of the bundle of volume forms with the horizontal lift of a Riemannian metric.

Let M be an orientable n-dimensional manifold equipped with Riemann- ian metric g = (g_ij) and let V be the bundle of volume forms over M .

Throughout the paper we assume that m, n, . . . = 1, 2, 3, . . . and indices α, β, . . . = 0, 1, 2, . . . . The Einstein summation convention will be used with

2000 Mathematics Subject Classification. 53B20.

Key words and phrases. Horizontal lift, Ricci tensor, Christoffel symbol, Ricci curva- ture, scalar curvature.

respect to these system of indices. We also denote that Γ^r_r[a|n] = 1

2



Γ^r_ra|n− Γ^r_rn|a , Γ^r_r(a|n) = 1

2



Γ^r_ra|n+ Γ^r_rn|a

 , Γⁿ_[ikΓ^t_nt|j] = 1

2



Γⁿ_ikΓ^t_nt|j− Γⁿ_jkΓ^t_nt|i

 and we will use the following convention g_jk|i= ^∂g_∂x^jki (see [3]).

Let M be equipped with the Levi–Civita connection Γ^k_ij. Then Γ^k_ij are given by the following expressions (see [2])

Γ^m_ij = 1

2g^km g_jk|i+ g_ki|j− g_ij|k
, where g^ij
is the inverse matrix to (g_ij).

Moreover, the differentiability conditions for the existence of such a vol- ume forms in local coordinates are given by

Lemma 1.1 ([9], [1]). The Christoffel symbols for the metric g satisfy the following equality

Γ^p_kp|n= Γ^p_np|k.

In the sequel of the paper we make use of the following theorem.

Theorem 1.2 ([5]). Let g be a Riemannian metric on M . Then γ = g^H is a Riemannian metric on V and

γ = 1 Γ^k_ik Γ^k_ik g_ij+ Γ^k_ikΓ^t_jt



γ⁻¹ = (γ^ij) =1 + g^ijΓ_ik^kΓ^t_jt −g^ijΓ^k_ik

−g^ijΓ^k_ik g^ij

 .

2. Curvatures of the lifted metric. In this chapter we prove some basic facts on the properties of the curvatures of the lifted metric to the bundle of volume forms on M .

Theorem 2.1. The Levi–Civita connection eΓ^δ_αβ for the horizontal lift metric γ are given by the following formulas:

(a) eΓ^s_0n= 0, (b) eΓ⁰₀₀= 0, (c) eΓ⁰_n0= 0, (d) eΓ^s_mn= Γ^s_mn,

(e) eΓ⁰_mn= Γ^t_tm|n− Γ^j_mnΓ^t_jt.

Proof. Ad. (a). From the formula for a Levi–Civita connection we have eΓ^s_0n= 1

2 γ_nα|0+ γ_0α|n− γ_on|α
γ^sα

= 1

2 γ_n0|0+ γ_00|n− γ_0n|0
γ^s0+1

2 γ_na|0+ γ_0a|n− γ_0n|a
γ^sa

= 1

2 γ_0a|n− γ_0n|a
γ^sa= 1 2



Γ^p_kp|n− Γ^p_np|k g^sa.

Moreover, from Lemma 1.1 we have that Γ^p_kp|n= Γ^p_pn|k. Hence eΓ^s_0n= 0.

Ad. (b), (c). These statements follow from definition of Levi–Civita connection, Theorem 1.2 and simple calculations.

Ad. (d). Similarly as above, we have Γe^s_mn= 1

2 γ_mα|n+ γ_nα|m− γ_mn|α
γ^αs

= 1

2 γ_m0|n+ γ_n0|m− γ_mn|0
γ^0s+1

2 γ_ma|n+ γ_na|m− γ_mn|a
γ^as

= −Γ^t_t(m|n)g^jsΓ^t_jt+ Γ_mn^s + g^asΓ^r_arΓ^t_t(m|n)+ g^as



Γ^t_mtΓ^r_r[a|n]+ Γ^t_ntΓ^r_r[a|m]



= Γ^s_mn,

since from Lemma 1.1 we have Γ^r_r[a|m]= 0 and Γ^r_r[a|n] = 0.

Ad. (e). From the formula for a Levi–Civita connection and Theorem 1.2 we have

Γe⁰_mn = 1

2 γ_mα|n+ γ_nα|m− γ_mn|α
γ^α0

= 1

2 γ_m0|n+ γ_n0|m− γ_mn|0
γ⁰⁰+1

2 γ_ma|n+ γ_na|m− γ_mn|a
γ^a0

= Γ^t_t(m|n) 1 + g^ijΓ^r_irΓ^s_js
− Γ^j_mnΓ^s_js

− g^ajΓ^s_js



Γ^t_mtΓ^r_r[a|n]+ Γ^t_ntΓ^r_r[a|m]+ Γ^r_arΓ^t_t(m|n)



= Γ^t_t(m|n) 1 + g^ijΓ^r_irΓ^s_js
− Γ^j_mnΓ^s_js− g^ajΓ^s_jsΓ^r_arΓ^t_t(m|n)

= Γ^t_tm|n− Γ^j_mnΓ^s_js,

since Γ^t_t(m|n) = Γ^t_tm|n. 

Corollary 2.1. If n = m then Christoffel symbols for lifted metric γ are given by the following formulas:

eΓ^s_mm = Γ^s_mm,

eΓ⁰_mm = Γ^p_mp|m− Γ^j_mmΓ^t_jt.

Let R be the curvature tensor of a Riemmanian manifold M and let R^s_ijk be coefficients of R. Then R^s_ijk is expressed in terms of the coefficients Γ^k_ij

of the Riemannian connections by the formula (see [2]) R_ikj^s = Γ^l_ikΓ^s_jl− Γ_jk^l Γ^s_il+ Γ^s_ik|j− Γ^s_jk|i.

Theorem 2.2. Let eR^σ_αβ be the coefficients of the curvature tensor eR of the lifted metric γ. Then eR^σ_αβ are related to the curvature tensor R^s_ijk of the matric g by the following formulas:

(a) eR^s_0j0= 0, (b) eR^s_ij0 = 0, (c) eR^s_0jk= 0,

(d) eR⁰_ijk= −Rⁿ_ijkΓ^t_nt+ 2Γⁿ_[jkΓ^t_nt|i], (e) eR^s_ijk= R^s_ijk.

Proof. Ad. (a). From the definition one get

Re^s_0j0= eΓ^l₀₀eΓ^s_lj− eΓ^l_j0Γe^s_l0+ eΓ^s_00|j− eΓ^s_j0|0. From Theorem 2.1 we have eΓ^s_l0= 0. Hence eR_0j0^s = 0.

Ad. (b), (c), (e). These statements follow from definition, Theorem 2.1 and simple calculations.

Ad. (d). From Lemma 1.1 we have Γ^t_tn|k = Γ^t_tk|n. Hence Γ^t_t(l|j) = Γ^t_tl|j. Next, we have

Re⁰_ijk= eΓ^l_ikΓe⁰_lj− eΓ^l_jkΓe⁰_li+ eΓ⁰_ik|j− eΓ⁰_jk|i

= Γ^l_ik

Γ^t_tl|j− Γⁿ_ljΓ^t_nt

− Γ^l_jk

Γ^t_tl|i− Γⁿ_liΓ^t_nt +

Γ^t_ti|k− Γⁿ_ikΓ^t_nt

|j−

Γ^t_tj|k− Γⁿ_jkΓ^t_nt

|i

= Γ^t_nt



Γ^l_jkΓⁿ_li− Γ^l_ikΓⁿ_lj+ Γⁿ_jk|i− Γⁿ_ik|j

+ Γⁿ_jkΓ^t_nt|i− Γⁿ_ikΓ^t_nt|j

= −Rⁿ_ijkΓ^t_nt+ 2Γⁿ_[jkΓ^t_nt|i].

 Let R_ik= R^j_ijk be the coefficients of the Ricci tensor ([2], [8]). Then from Theorem 2.2 we get directly

Theorem 2.3. Let eRαβ are the coefficients of a Ricci tensor of the lifted metric γ. Then

(a) eR₀₀= 0, (b) eRi0= 0,

(c) eR_ik= R_ik,

where R_ij are the coefficients of Ricci tensor of g on the Riemannian man- ifold M .

Theorem 2.4. Let eK be a scalar curvature of the lifted metric γ. Then K =e n − 1

n + 1K,

where K is a scalar curvature of g on a Riemmanian manifold M .

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

K =e 1

n(n + 1)Re_ikγ^ik= 1

n(n + 1)R_ikg^ik= n − 1 n + 1K.

 References

[1] Dhooghe, P. F., Van Vlierden, A., Projective geometry on the bundle of volume forms, J. Geom. 62 (1998), 66–83.

[2] do Carmo, M. P., Riemannian Geometry, Birkh¨auser, Boston, 1992.

[3] Gołąb, S., Tensor Calculus, Elsevier Scientific Publ. Co., Amsterdam–London–New York, 1974.

[4] Kurek, J., On a horizontal lift a linear connection to the bundle of linear frames, Ann.

Univ. Mariae Curie-Skłodowska Sect. A 41 (1987), 31–38.

[5] Miernowski, A., Mozgawa, W., Horizontal lift to the bundle of volume forms, Ann.

Univ. Mariae Curie-Skłodowska Sect. A 57 (2003), 69–75.

[6] Mikulski, W., The natural affinors on generalized higher order tangent bundles, Rend.

Mat. Appl. (7) 21 (2001), 339–349.

[7] Mikulski, W., The natural operators lifting vector fields to generalized higher order tangent bundles, Arch. Math. (Brno) 36 (2000), 207–212.

[8] Sokolnikoff, I. S., Tensor Analysis. Theory and Applications to Geometry and Mechan- ics of Continua, 2nd ed., Wiley, New York–London–Sydney, 1964.

[9] Schouten, J. A., Ricci-Calculus, 2nd ed., Springer-Verlag, Berlin–G¨ottingen–Heidel- berg, 1954.

Anna Gąsior

Institute of Mathematics M. Curie-Skłodowska University pl. Marii Curie-Skłodowskiej 1 20-031 Lublin, Poland

e-mail: agasior@golem.umcs.lublin.pl Received May 4, 2005