d−Connections Compatible with Homogeneous Metric on the Cotangent Bundle

Liviu Popescu

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract In this paper is studied the cotangent bundle T̃^{∗}M = T^{∗}M\{0} with a 0homogeneous lift G^{∗} . The connection compatible with the homogeneous metric is determined.

Mathematics Subject classification : 53C15, 53C55, 53C60 Key words: nonlinear connection, adapted basis, homogeneous lift, metrical dconnections.

1 Introduction

Let (T^{∗}M, π^{∗}, M) be the cotangent bundle, where M is a C^{∞}-differentiable, real ndimensional manifold. If (U, φ) is a local chart on M and (x^{i}) are the coordinates of a point p ∈ M, p ∈ φ^{−1}(x) ∈ U, then a point u ∈ π^{∗−1}(U), π^{∗}(u) = p has the coordinates (x^{i}, p_{i}), (i = \overline{1, n}). The natural basis of the module X (T^{∗}M) is given by

(∂_{i} = \frac{∂}{∂x^{i}} , ∂^{r} = \frac{∂}{∂p_{r}}). Given a nonlinear connection N on T^{∗}M ([1]) there exist a

single system of functions N_{ia}(x, p) such that δ_{k} = ∂_{k} + N_{ka}(x, p)∂^{a}, (a = \overline{1, n}) and (δ_{k}, ∂^{a}) is a local basis of X (T^{∗}M), which is called the adapted basis to N. We have the dual basis (dx^{i}, δp_{a} = dp_{a}−N_{ka}(x, p)dx^{k}). For X ∈ X (T^{∗}M) is obtained a unique decompostion X = hX + vX, hX ∈ H, vX ∈ V, (V is the vertical distribution) and for ω ∈ X^{∗}(T^{∗}M) we have ω = hω+vω, where (hω)(X) = ω(hX), (vω)(X) = ω(vX). In the adapted basis (δ_{k}, ∂^{a}) we have X = X^{i}δ_{i} + X_{a}∂^{a} and ω = ω_{i}dx^{i} + ω^{a}δp_{a}. The homogeneous lift of the Riemannian and Finslerian metrics on the tangent bundle have been studied by Acad. Radu Miron ([3], [4]), while the properties of homogeneous structures on cotangent bundle were studied by P. Stavre and the author ([5], [6], [7]). More specific, details on the homogeneous lift of a Cartan metric on cotangent bundle and on integrability conditions of homogeneous almost complex structures are given in [6], the properties of the homogeneous lift of a Riemann metric on cotangent bundle are studied in [7], and the homogeneous almost product structure case is developed in [5].

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 43-47.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

2 Existence of metrical d-connections

Let (M, g_{ij}(x)) be a Riemannian space and (T^{∗}M, π^{∗}, M) its cotangent bundle. We introduce g^{rs}(x) with g_{ik}(x)g^{ks}(x) = δ_{i}^{s}. We consider

N^{c}_{kr} (x, p) =^{def} p_{s}γ_{rk}^{s}(x), (1)

where γ_{rk}^{s}(x) are the Christoffel symbols of g. Evidently {N^{c}_{kr} (x, p)} are the coefficients of a nonlinear connection on T̃^{∗}M = T^{∗}M \ {0} which is 1- homogeneous on

the fibres. Using N^{c}_{kr} we consider δ_{k} = ∂_{k}+ N^{c}_{kr} (x, p)∂^{r}; δp_{k} = dp_{k}− N^{c}_{ik} (x, p)dx^{i}.

We have

G^{∗}= h G^{∗} +v G^{∗}, G^{∗}= g_{ij}(x)dx^{i} ⊗ dx^{j} + g^{rs}(x)δp_{r} ⊗ δp_{s}. (2)

If we define the homothety h^{∗}_{t}: (x, p) → (x, tp), ∀t ∈ R, then

(G^{∗} ∘ h^{∗}_{t}) (x, p) = g_{ij}(x)dx^{i} ⊗ dx^{j} + t^{2}g^{rs}(x)δp_{r} ⊗ δp_{s} ≠G^{∗} (x, p). (3)

Proposition 1 G^{∗} is globally defined Riemannian metric on T̃^{∗}M and is not homogeneous on the fibres of T^{∗}M.

We consider the function H(x, p) = g^{rs}(x)p_{r}p_{s}. (4)

Obviously H is 2-homogeneous on the fibres of cotangent bundle T̃^{∗}M.

If G^{∗} is defined by

G^{∗}= g_{ij}(x)dx^{i} ⊗ dx^{j} + \frac{a^{2}}{H} g^{rs}(x)δp_{r} ⊗ δp_{s} , (5)

where a > 0 is a constant, then we get:

Proposition 2 The following properties hold:

1^{∘} The pair (T̃^{∗}M, G^{∗}) is a Riemannian space depending only on the metric g.

2^{∘} G^{∗} is 0-homogeneous on the fibres of T̃^{∗}M.

3^{∘} The distribution N and V are ortogonal with respect to G^{∗}

G^{∗} (hX, vY) = 0, ∀X, Y ∈ X (T^{∗}M).

G^{∗}= g_{ij}(x)dx^{i} ⊗ dx^{j} + h^{rs}(x, p)δp_{r} ⊗ δp_{s}, (6)

where

h^{rs}(x, p) = \frac{a^{2}}{H} g^{rs}(x) (7)

From [1] we have:

Definition 1 A linear connection D on T^{∗}M is called metrical d−linear connection with respect to G^{∗} if D G^{∗} = 0 and D preserves by parallelism the horizontal distribution N.

d−Connections Compatible with Homogeneous Metric on the Cotangent Bundle 45

We will prove the existence of metrical d−linear connections. In the adapted frame we have:

\begin{matrix}D_{δ_{k}}δ_{j} =F_{jk}^{i}^{h} δ_{i} + F̃_{j(r)k}∂^{r}, D_{δ_{k}}∂^{r} = − ˜F̃_{k}^{i(r)}δ_{i}− F_{(j)k}^{(r)}^{v} ∂^{j}, \\ D_{∂^{k}}δ_{j} =C_{j}^{i(k)}^{h} δ_{i} + C̃_{j(r)}^{(k)}∂^{r}, D_{∂^{k}}∂^{r} = − ˜C̃^{(r)i(k)}δ_{i}− C_{(j)}^{(r)(k)}^{v} ∂^{j}, \end{matrix} (8)

where F_{jk}^{i}^{h}, F̃_{j(r)k}, ˜ F̃_{k}^{i(r)}, F_{(j)k}^{(r)}^{v}, C_{j}^{i(k)}^{h}, C̃_{j(r)}^{(k)}, ˜ C̃^{(r)i(k)}, C_{(j)}^{(r))k)}^{v}) are the coefficients of D.

Theorem 1 There exists a metrical d− linear connection D on T̃^{∗}M with respect to

G^{∗}, which depends only on the metric tensor g; its components are

\{ \begin{matrix}F̃_{j(r)k} = ˜ F̃_{k}^{i(r)} = C̃_{j(r)}^{(k)} = ˜ C̃^{(r)i(k)} =C_{j}^{i(k)}^{h}= 0, \\ F_{jk}^{i}^{h}=F_{(j)k}^{(i)}^{v}= γ_{jk}^{i}(x), \\ C_{(j)}^{(r)(k)}^{v}= \frac{1}{H} (δ_{j}^{k}p^{r} + δ_{j}^{r}p^{k} − g^{rk}p_{j}), \end{matrix} (9)

where g^{rm}p_{m} = p^{r}.

Proof. In the general case of a vector bundle we have a canonical metrical connection given by [2],

\{ \begin{matrix}F_{jk}^{i}^{h}= \frac{1}{2}g^{is}(δ_{j}g_{si} + δ_{k}g_{js} − δ_{s}g_{jk}), \\ F_{(j)k}^{(r)}^{v}= ∂^{r}N_{jk} + \frac{1}{2}h^{rs}h_{js||k}, \\ C_{j}^{i(k)}^{h}= \frac{1}{2}g_{js}g^{is}||^{k} = \frac{1}{2}g_{js}∂^{k}g^{is}, \\ C_{(j)}^{(r)(k)}^{v}= −\frac{1}{2}h_{js}(∂^{r}h^{ks} + ∂^{k}h^{rs} − ∂^{s}h^{rk}), \end{matrix}

where_{,,|´|”} and ,,||” are the h−, and v− covariant derivative with respect to the Berwald connection (B_{jk}^{r} = ∂^{r}N_{jk}, 0).

But g = g(x), so δ_{j}g_{si} = ∂_{j}g_{si} and ∂^{k}g^{is} = 0 ⇒ F_{jk}^{i}^{h}= γ_{jk}^{i}(x) and C_{j}^{i(k)}^{h}= 0.

From h^{rs}(x, p) = \frac{a^{2}}{H} g^{rs}(x) it follows h_{rs}(x, p) = \frac{H}{a^{2}} g_{rs}(x). But

∂^{r}h^{ks}(x, p) = ∂^{r} (\frac{a^{2}}{g^{rm}p_{m}p_{r}} g^{ks}(x)) = −\frac{2a^{2}}{H^{2}} g^{ks}(x)g^{rm}p_{m},

C_{(j)}^{(r)(k)}^{v}= −\frac{1}{2}h_{js}(∂^{r}h^{ks} + ∂^{k}h^{rs} − ∂^{s}h^{rk}) =

= −\frac{1}{2} \frac{H}{a^{2}} g_{js}(x) (−\frac{2a^{2}}{H^{2}} g^{ks}g^{rm}p_{m} − \frac{2a^{2}}{H^{2}} g^{rs}g^{km}p_{m} + \frac{2a^{2}}{H^{2}} g^{rk}g^{sm}p_{m})

C_{(j)}^{(r)(k)}^{v}= \frac{1}{H} (δ_{j}^{k}p^{r} + δ_{j}^{r}p^{k} − g^{rk}p_{j}) , g^{rm}p_{m} = p^{r}.

F_{(j)k}^{(r)}^{v}= ∂^{r} N^{c}_{jk} +\frac{1}{2}h^{rs}h_{js||k} = ∂^{r} N^{c}_{jk} +\frac{1}{2}h^{rs}[δ_{k}h_{js} − ∂^{m}(N^{c}_{sk})h_{jm} − ∂^{m}(N^{c}_{jk})h_{sm}].

Since N_{jk} = γ_{jk}^{r}p_{r} and

F_{(j)k}^{(r)}^{v}= γ_{jk}^{r} + \frac{1}{2} \frac{a^{2}}{H} g^{rs}(x)[\frac{H}{a^{2}} ∂_{k}g_{js} + \frac{1}{a^{2}} g_{js}∂_{k}g^{ml}p_{m}p_{l} + \frac{2}{a^{2}} γ_{kl}^{m}p_{m}g_{js}g^{lm}p_{m}−

−\frac{H}{a^{2}} γ_{sk}^{m}g_{jm} − \frac{H}{a^{2}} γ_{jk}^{m}g_{sm}],

we obtain

F_{(j)k}^{(r)}^{v}= \frac{1}{2}γ_{jk}^{r} + \frac{1}{2}g^{rs}∂_{k}g_{js} + \frac{1}{2H} ∂_{k}g^{ml}p_{m}p_{l}δ_{j}^{r} + \frac{1}{2H} g^{ms}g^{li}p_{m}p_{i}∂_{k}g_{ls}δ_{j}^{r}+

+\frac{1}{2H} g^{sm}g^{li}p_{m}p_{i}∂_{l}g_{ks}δ_{j}^{r} − \frac{1}{2H} g^{sm}g^{li}p_{m}p_{i}∂_{s}g_{kl}δ_{j}^{r}−

−\frac{1}{4}g^{rs}∂_{s}g_{kj} − \frac{1}{4}g^{rs}∂_{k}g_{sj} + \frac{1}{4}g^{rs}∂_{j}g_{sk}.

But g^{ms}∂_{k}(g_{ls}) = −∂_{k}(g^{ms})g_{ls}, and consequently

F_{(j)k}^{(r)}^{v}= \frac{1}{2}γ_{jk}^{r} + \frac{1}{4}g^{rs}(∂_{k}g_{js} + ∂_{j}g_{sk} − ∂_{s}g_{kj}) + \frac{1}{2H} ∂_{k}g^{ml}p_{m}p_{l}δ_{j}^{r} − \frac{1}{2H} ∂_{k}g^{ms}p_{m}p_{s}δ_{j}^{r}−

−\frac{1}{2H} g^{li}g_{ks}p_{m}p_{i}∂_{l}g^{sm}δ_{j}^{r} + \frac{1}{2H} g^{lm}g_{ks}p_{m}p_{i}∂_{l}g^{si}δ_{j}^{r} = \frac{1}{2}γ_{jk}^{r} + \frac{1}{2}γ_{jk}^{r} = γ_{jk}^{r},

which ends the proof.

□

References

[1] R. Miron, Hamilton geometry, Seminarul de mecanică, No. 3, Univ. Timişoara, 1987.

[2] R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces. Theory and Applications, Kluwer Academic Publishers, no. 59 (1994).

[3] R. Miron, The homogeneous lift of a Riemannian metric, An. Şt. Univ. “A. I. Cuza” Iasi, (to appear).

[4] R. Miron, D. Hrimiuc, H. Shimada, S. Sabău, The geometry of Hamilton and Lagrange Spaces, Kluwer Academic Publishers, 2000.

[5] L. Popescu, On the homogeneous lift G in the cotangent bundle (II), Differential Geometry - Dynamical Systems 2 (2000), 1, 32-35.

[6] P. Stavre, L. Popescu, The homogeneous lift to cotangent bundle of a Cartan metric, Proceedings of National Conference of Finsler and Lagrange Geometry and Applications, Univ. of Bacau, Studies and Scientific Research, ser. Mathematics, no. 10, 2000, 273-285.

[7] P. Stavre, L. Popescu, The homogeneous lift G on the cotangent bundle, Novi-Sad Journal of Mathematics 32 (2002), 2, 1-7.

[8] P. Stavre, On the integrability of the structures (T^{∗}M, (G, F^{∗})), Algebras, Groups and Geometries, Hadronic Press, SUA, 16 (1999), 107-114.

University of Craiova Faculty of Economic Sciences Department of Applied Mathematics 13 ”Al.I. Cuza” Street, 1100, Craiova, Romania email: liviunew@yahoo.com