Facta Universitatis, Series: Mathematics and Informatics, 2021
This paper deals with the Lagrange vertical structure on the vertical space TV (E) endowed with a... more 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
The results of f-structures manifold on a differentiable manifold was initiated and developed by ... more The results of f-structures manifold on a differentiable manifold was initiated and developed by Yano [1], Ishihara and Yano [2]. J. B. Kim [3] has defined and studied the structure of rank r and of degree k. The purpose of this paper is to study a differentiable manifold with F-structure of rank r using tensor as a vector valued linear function [4]. The case when k is odd and even have been considered in this paper.
In this paper, authors have shown that if an almost product structure P on the tangent space of a... more In this paper, authors have shown that if an almost product structure P on the tangent space of a 2n – dimensional Lagrangian manifold E is defined and the F (a1, a2, . . ., an) – 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 the tangent space T (E) of E. Linear connections on the Lagrangian F (a1, a2, . . .an) – structure manifold E are also discussed. Certain other interesting results like geodesics in E are also studied. M.S.C. 2000: 53D12, 53D35.
Let f be a nonzero tensor field of type (1,1) on an manifold V n such that (f 3 -λ 2 f)(f 3 -μ 2 ... more Let f be a nonzero tensor field of type (1,1) on an manifold V n such that (f 3 -λ 2 f)(f 3 -μ 2 f)=0 for λ,μ∈ℝ + such that λ≠μ and f 2 ≠λ 2 and μ 2 respectively. Such a structure is called an f λ,μ (3,1)-structure of rank r, where r=rank(f). Theorem 1: Let M n be an n-dimensional manifold with an f λ,μ (3,1)-structure of rank r. Then there exist complementary distributions L of dimension (2r-n) and M 1 and M 2 both of dimension (n-r) and a positive definite Riemannian metric g with respect to which L, M 1 and M 2 are orthogonal such that g(fX,fY)=μ 2 g(X,Y), ∀X,Y∈L and g(fX,fY)=λ 2 g(X,Y), ∀X,Y∈M i . Theorem 2: The necessary and sufficient condition for the n-dimensional manifold M n to admit an f λ,μ (3,1)-structure is that the group of the tangent bundle can be reduced to the group O(h)×O(2r-n-h)×O(n-r).Reviewer: Nicolai Konstantinovich Smolentsev (Kemerovo)
Cartesian products of two manifolds were defined and studied by A. Z. Petrov [Einstein spaces, Ox... more Cartesian products of two manifolds were defined and studied by A. Z. Petrov [Einstein spaces, Oxford: Pergamon Press (1969; Zbl 0174.28305)] among others. In this paper, we consider Cartesian product spaces of r-manifolds, where r is some finite integer. Some properties of such a product manifold are defined and studied. The curvature of the product manifold is also studied and it is shown that the product manifold is an Einstein space if the manifolds M 1 , M 2 , M r are Einstein spaces.
Journal of International Academy Of Physical Sciences, Sep 15, 2010
Almost r – contact structure was defined and studied by Vanzura 1 and several other geometers inc... more Almost r – contact structure was defined and studied by Vanzura 1 and several other geometers including Mishra, Pandey 2 and Imai 3 . Recently, Das, Ram Nivas, S. Ali and M. Ahmad 4 have studied quarter symmetric connections and have obtained some interesting results. In this paper, authors have studied submanifolds of an almost r – contact structure manifold. Quarter symmetric ( F, G ) – connection has also been defined and submanifolds of a manifolds with such connection have been studied. Study of ( F, G ) geodesic and ( F, G ) umbilical submanifolds is also the subject matter of this paper.
Facta Universitatis, Series: Mathematics and Informatics, 2021
This paper deals with the Lagrange vertical structure on the vertical space TV (E) endowed with a... more 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
The results of f-structures manifold on a differentiable manifold was initiated and developed by ... more The results of f-structures manifold on a differentiable manifold was initiated and developed by Yano [1], Ishihara and Yano [2]. J. B. Kim [3] has defined and studied the structure of rank r and of degree k. The purpose of this paper is to study a differentiable manifold with F-structure of rank r using tensor as a vector valued linear function [4]. The case when k is odd and even have been considered in this paper.
In this paper, authors have shown that if an almost product structure P on the tangent space of a... more In this paper, authors have shown that if an almost product structure P on the tangent space of a 2n – dimensional Lagrangian manifold E is defined and the F (a1, a2, . . ., an) – 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 the tangent space T (E) of E. Linear connections on the Lagrangian F (a1, a2, . . .an) – structure manifold E are also discussed. Certain other interesting results like geodesics in E are also studied. M.S.C. 2000: 53D12, 53D35.
Let f be a nonzero tensor field of type (1,1) on an manifold V n such that (f 3 -λ 2 f)(f 3 -μ 2 ... more Let f be a nonzero tensor field of type (1,1) on an manifold V n such that (f 3 -λ 2 f)(f 3 -μ 2 f)=0 for λ,μ∈ℝ + such that λ≠μ and f 2 ≠λ 2 and μ 2 respectively. Such a structure is called an f λ,μ (3,1)-structure of rank r, where r=rank(f). Theorem 1: Let M n be an n-dimensional manifold with an f λ,μ (3,1)-structure of rank r. Then there exist complementary distributions L of dimension (2r-n) and M 1 and M 2 both of dimension (n-r) and a positive definite Riemannian metric g with respect to which L, M 1 and M 2 are orthogonal such that g(fX,fY)=μ 2 g(X,Y), ∀X,Y∈L and g(fX,fY)=λ 2 g(X,Y), ∀X,Y∈M i . Theorem 2: The necessary and sufficient condition for the n-dimensional manifold M n to admit an f λ,μ (3,1)-structure is that the group of the tangent bundle can be reduced to the group O(h)×O(2r-n-h)×O(n-r).Reviewer: Nicolai Konstantinovich Smolentsev (Kemerovo)
Cartesian products of two manifolds were defined and studied by A. Z. Petrov [Einstein spaces, Ox... more Cartesian products of two manifolds were defined and studied by A. Z. Petrov [Einstein spaces, Oxford: Pergamon Press (1969; Zbl 0174.28305)] among others. In this paper, we consider Cartesian product spaces of r-manifolds, where r is some finite integer. Some properties of such a product manifold are defined and studied. The curvature of the product manifold is also studied and it is shown that the product manifold is an Einstein space if the manifolds M 1 , M 2 , M r are Einstein spaces.
Journal of International Academy Of Physical Sciences, Sep 15, 2010
Almost r – contact structure was defined and studied by Vanzura 1 and several other geometers inc... more Almost r – contact structure was defined and studied by Vanzura 1 and several other geometers including Mishra, Pandey 2 and Imai 3 . Recently, Das, Ram Nivas, S. Ali and M. Ahmad 4 have studied quarter symmetric connections and have obtained some interesting results. In this paper, authors have studied submanifolds of an almost r – contact structure manifold. Quarter symmetric ( F, G ) – connection has also been defined and submanifolds of a manifolds with such connection have been studied. Study of ( F, G ) geodesic and ( F, G ) umbilical submanifolds is also the subject matter of this paper.
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