scholarly journals PARALLELISM OF DISTRIBUTIONS AND GEODESICS ON F(±a2; ±b2)-STRUCTURE LAGRANGIAN MANIFOLD

2021 ◽  
pp. 157
Author(s):  
Mohammad Nazrul Islam Khan ◽  
Lovejoy S. Das
Keyword(s):  
Tangent Space ◽  
Tensor Field ◽  
Vertical Structure ◽  
Product Structure ◽  
Lagrangian Manifold ◽  
Almost Product Structure ◽  
Vertical Tangent ◽  
B2 Structure

This paper deals with the Lagrange vertical structure on the vertical space TV (E) endowed with a non null (1,1) tensor field FV satisfying (Fv2-a2)(Fv2+a2)(Fv2 - b2)(Fv2 + b2) = 0. In this paper, the authors have proved that if an almost product structure P on the tangent space of a 2n-dimensional Lagrange manifold E is defined and the F(±a2; ±b2)-structure on the vertical tangent space TV (E) is given, then it is possible to define the similar structure on the horizontal subspace TH(E) and also on T(E). In the next section, we have proved some theorems and have obtained conditions under which the distribution L and M are r-parallel, r¯ anti half parallel when r = r¯ . The last section is devoted to proving theorems on geodesics on the Lagrange manifold

On a Structure Defined by a Tensor Field F of Type (1, 1) Satisfying F2 = 0

1973 ◽  
Vol 16 (3) ◽  
pp. 447-449
Author(s):  
C. S. Houh
Keyword(s):  
Tangent Space ◽  
Tensor Field ◽  
Class C ◽  
Almost Tangent Structure

Professor Eliopoulous studied almost tangent structure on manifolds M2n in [1], [2], [3]. An almost tangent structure F is a field of class C∞ of linear operations on M2n such that at each point x in M2n, Fx maps the complexified tangent space into itself and that Fx is of rank n everywhere and satisfies F2=0.

Note on Almost Product Manifolds and their Tangent Bundles

1966 ◽  
Vol 9 (05) ◽  
pp. 621-630
Author(s):  
Chorng Shi Houh
Keyword(s):  
Tangent Bundle ◽  
Tangent Vector ◽  
Product Structure ◽  
Dimensional Manifold ◽  
Almost Product Structure ◽  
Local Coordinates ◽  
Natural Frame ◽  
Class C ◽  
Tangent Bundles ◽  
Differentiability Class

Let Mn be an n-dimensional manifold of differentiability class C∞ with an almost product structure . Let have eigenvalue +1 of multiplicity p and eigenvalue -1 of multiplicity q where p+q = n and p≧1, q≧1. Let T(Mn) be the tangent bundle of M. T(Mn) is a 2n dimensional manifold of class C∞. Let xi be the local coordinates of a point P of Mn. The local coordinates of T(Mn) can be expressed by 2n variables (xi, yi) where xi are coordinates of the point P and yi are components of a tangent vector at P with respect to the natural frame constituted by the vectior ∂/∂xi at P.

scholarly journals The Geometry of Product Conjugate Connections

2013 ◽  
Vol 59 (1) ◽  
pp. 73-84 ◽  
Author(s):  
Adara M. Blaga ◽  
Mircea Crăşmăreanu
Keyword(s):  
Product Structure ◽  
Tensor Fields ◽  
Almost Product Structure ◽  
Conjugate Connections ◽  
Hermitian Geometry ◽  
The Given

Abstract Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of these tensors. Some examples from the geometry of a pair of complementary distributions are discussed and for this case the above structural and virtual tensors are expressed in terms of O’Neill-Gray tensor fields.

scholarly journals The Principal Fibre Bundle on Lorentzian Almost R-para Contact Structure

2021 ◽  
Vol 27_NS1 (1) ◽  
pp. 16-20
Author(s):  
Lovejoy S. Das ◽  
Mohammad Nazrul Islam Khan
Keyword(s):  
Lie Group ◽  
Tensor Field ◽  
Contact Structure ◽  
Principal Bundle ◽  
Fibre Bundle ◽  
Product Structure ◽  
Frame Bundle ◽  
Principal Fibre ◽  
Principal Fibre Bundle ◽  
Almost Paracontact Structure

The purpose of this paper is to study the principal fibre bundle ( P , M , G , π p ) with Lie group G , where M admits Lorentzian almost paracontact structure ( Ø , ξ p , η p , g ) satisfying certain condtions on (1, 1) tensor field J , indeed possesses an almost product structure on the principal fibre bundle. In the later sections, we have defined trilinear frame bundle and have proved that the trilinear frame bundle is the principal bundle and have proved in Theorem 5.1 that the Jacobian map π * is the isomorphism.

AN ALMOST-PRODUCT STRUCTURE ON THE FOLIATED MANIFOLD R.

2020 ◽  
Vol 9 (12) ◽  
pp. 11019-11041
Author(s):  
S. Herinantenaina ◽  
H. S. G. Ravelonirina
Keyword(s):  
Product Structure ◽  
Foliated Manifold ◽  
Almost Product Structure
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