ACTA UNIVERSITATIS APULENSIS No 11/2006 Proceedings of theInternational Conference on Theory and Application of
Mathematics and Informatics ICTAMI 2005 - Alba Iulia, Romania
N-linear connections and JN-linear connections on second order tangent bundle T2
Atanasiu Gheorghe
Abstract.On second order tangent bundle T2M we define an N-linear connection which have nine coefficients in comparation with the JN-linear con-nection which have three coefficients, only. To work with an N-linear connec-tion on T2M is an advantage for the physical applications to electrodinamics, elasticity, quantum field theories, etc.
2000 Mathematical Subject Classification: 53B05, 53B15, 53B40 Let M be a real C∞
- manifold with n dimensions and (T2M, π, M) its tangent bundle, [1] - [3]. The local coordinates on 3n-dimensional manifolds
T2M are denoted by xi, y(1)i, y(2)i
= x, y(1), y(2)=u,(i= 1,2, ..., n).
Let ∂x∂i,
∂ ∂y(1)i,
∂ ∂y(2)i
be the natural basis of the tangent space T T2M at
the point u ∈ T2M and let us consider the natural 2−tangent structure on
T2M, J :χ(T2M)→χ(T2M) given by J
∂ ∂xi
= ∂
∂y(1)i, J
∂ ∂y(1)i
= ∂
∂y(2)i, J
∂ ∂y(2)i
= 0. (1)
We denote withN a nonlinear connection onT2M with the local coefficients
Ni Ni
(1)j (2)j
(i, j = 1,2, ..., n) ,[3], [6]. Hence, the tangent space of T2M in
the point u∈T2M is given by the direct sum of the linear vector spaces:
TuT2M =N0(u)⊕N1(u)⊕V2(u),∀u∈T2M (2)
An adapted basis to the direct decomposition (2) is given byn δ δxi,
δ δy(1)i,
δ δy(2)i
o
, where
Gh. Atanasiu - N-linear connections and JN-linear connections on ... χ(T2M)is called an N-linear connection onT2M if it preserves by parallelism
the horizontal and vertical distributions N0, N1 and V2 on T2M.
An N-linear connection D on T2M is characterized by its coefficients, in
the adapted basis (3), in the form:
D δ The system of nine functions
DΓ (N) = are called the coefficientsof the N−linear connection D.
The torsion tensor T of an N− linear connection DΓ (N) is expressed, as usually, by T(X, Y) = DXY −DYX −[X, Y] and, in the adapted basis (3),
Gh. Atanasiu - N-linear connections and JN-linear connections on ... the adapted basis (3), have eighteencomponents: R
(0α)
Generally, an N-linear connection DΓ (N) on T2M is not compatible with the natural 2-tangent structureJ given by (1).
Definition.An N-linear connectionDΓ (N)on T2M is called JN−linear connection if it is absolut parallel with respect to J, i.e.:
DXJ = 0, ∀X ∈χ T2M
. (7)
Theorem 1 (Gh. Atanasiu [2], pg. 39, [3], pg.25)A JN− linear connec-tion on T2M is characterized by the coefficients JDΓ (N) given by(6), where
L
It results that a JN− linear connection on T2M has three essentially coefficients JDΓ (N) =
In the adapted basis (3), the torsion tensor T of aJN− linear connection on T2M have thirteen components Q
(21)
and the curvature ten-sorR of JDΓ (N) have six componentsRi
Of course, for the physical applications there exists an advantage to work with an N-linear connection (see [4],[5],[8]) in comparation with aJN−linear connection (see [1],[6],[7]).
Gh. Atanasiu - N-linear connections and JN-linear connections on ...
References
[1] Atanasiu,Gh., The Equations of Structure of an N-Linear Connection in the Bundle of Accelerations. Balkan J. Geom. and Its Appl., Vol.1, No.1, 1996, 11-19
[2] Atanasiu,Gh., New Aspect in the Differential Geometry of the Second Order. Sem. de Mecanic˘a, Univ. de Vest din Timi¸soara, No. 82, 2001, 1-81.
[3] Atanasiu,Gh.,Linear Connections in the Differential Geometry of Order Two. Lagrange and Hamilton Geometries and Their Applications, Radu Miron (Ed.), Handbooks. Treatises. Monographs, Fair Partners Publ., Bucure ¸sti, 49, 2004, 11-30.
[4] Atanasiu,Gh., Voicu, N., Einstein Equations in the Geometry of Sec-ond Order, Studia Mathematica, Univ. Babe¸s-Bolyai, Cluj-Napoca, 2005 (to appear).
[5] Atanasiu,Gh., Brinzei, N., Maxwell Equations on the Second Order Tan-gent Bundle, Central European J. of Math., (to appear).
[6] Miron, R. and Atanasiu,Gh.,Lagrange Geometry of Second Order. Math. Comput. Modelling, vol. 20, no.4, 1994,41-56.
[7] Miron, R. and Atanasiu,Gh., Geometrical Theory of Gravitational and Electromagnetic Fields in Higher Order Lagrange Spaces, Tsukuba J. of Math., vol. 20, 1996, 137-149.
[8] Voicu (Brinzei), N., Deviations of Geodesics in the Geometry of the Order Two(in Romannian), Ph. D. Thesis, Univ.Babe¸s-Bolyai, Cluj-Napoca, 2003, 1-134.
Gheorghe Atanasiu
Department of Algebra and Geometry ”Transilvania” University Brasov, Romˆania e-mail: [email protected]
