Surfaces of the nearly Kähler S3×S3 preserved by the almost product structure

2021 ◽  
Author(s):  
Miroslava Antić ◽  
Zejun Hu ◽  
Marilena Moruz ◽  
Luc Vrancken
Keyword(s):  
Product Structure ◽  
Almost Product Structure ◽  
Nearly Kähler

scholarly journals Properties of the nearly kähler S3×S3

2018 ◽  
Vol 103 (117) ◽  
pp. 147-158 ◽  
Author(s):  
Marilena Moruz ◽  
Luc Vrancken
Keyword(s):  
Complex Structure ◽  
Product Structure ◽  
Almost Complex Structure ◽  
Almost Product Structure ◽  
Almost Complex ◽  
Nearly Kähler

We show how the metric, the almost complex structure and the almost product structure of the homogeneous nearly Kahler S3 ? S3 can be recovered from a submersion ? : S3 ? S3 ? S3 ? S3 ? S3. On S3 ? S3 ? S3 we have the maps obtained either by changing two coordinates, or by cyclic permutations. We show that these maps project to maps from S3 ? S3 to S3 ? S3 and we investigate their behavior.

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 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

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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