Book ChapterDOI
Differential-Geometric Structures On Manifolds
Krishan L. Duggal,Aurel Bejancu +1 more
- pp 18-51
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In this paper, the authors provide a review of vector bundles and introduce the main differential operators: Lie derivative, exterior differential, linear connection, general connection, and general connection 1-forms.Abstract:
In the present chapter we provide most of the prerequisites for reading the rest of the book. In the first two sections we present a review of vector bundles and introduce the main differential operators: Lie derivative, exterior differential, linear connection, general connection. Distributions on manifolds (known as non-holonomic spaces in classical terminology) are then introduced and studied by using both methods of vector fields and of differential 1-forms. We give here the characterization for the existence of a transversal distribution to a foliation, which is found to be very useful in Chapters 4 and 5 for a general study of lightlike submanifolds. In the last two sections we deal with semi-Riemannian manifolds and lightlike manifolds. While the geometry of a semi-Riemannian manifold is fully developed by using the Levi-Civita connection we stress the role of the radical distribution in studying the geometry of a lightlike manifold. The main formulas and results are expressed by using both the invariant form and the index form.read more
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Invariant f-structures in generalized Hermitian geometry
TL;DR: In this article, the authors consider all invariant f-structures on the complex flag manifold SU(3)/Tmax and describe them in the sense of generalized Hermitian geometry.
On Almost Contact Metric Hypersurfaces of Nearly Kählerian 6-Sphere
TL;DR: In this article, it is proved that if the type number of an oriented hypersurface of the nearly Kahlerian six-dimensional sphere can be computed, then it can be shown that it is a polygonal polygon.
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Geometric Aspects of Gauge and Spacetime Symmetries
TL;DR: In this paper, the authors apply the theory of Lie group deformations to isometry groups of exact solutions in general relativity, relating the algebraic properties of these groups to physical properties of the spacetimes, and make group deformation local, generalising deformed special relativity by describing gravity as a gauge theory of the de Sitter group.
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A Study of New Class of Almost Contact Metric Manifolds of Kenmotsu Type
TL;DR: In this paper, the authors characterized a new class of almost contact metric manifolds and established the equivalent conditions of the characterization identity in terms of Kirichenko's tensors, and demonstrated that the Kenmotsu manifold provides the mentioned class; i.e., the new class can be decomposed into a direct sum of the KG and other classes.
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Identity of the connection curvature tensor of almost manifold C(λ)
TL;DR: In this article, the geometry of the projective curvature tensor is investigated, and three classes of nearly infinite C(λ) are distinguished and studied, and some identities for this tensor are obtained.
References
More filters
Posted Content
Invariant f-structures in generalized Hermitian geometry
TL;DR: In this article, the authors consider all invariant f-structures on the complex flag manifold SU(3)/Tmax and describe them in the sense of generalized Hermitian geometry.
On Almost Contact Metric Hypersurfaces of Nearly Kählerian 6-Sphere
TL;DR: In this article, it is proved that if the type number of an oriented hypersurface of the nearly Kahlerian six-dimensional sphere can be computed, then it can be shown that it is a polygonal polygon.
Journal ArticleDOI
Geometric Aspects of Gauge and Spacetime Symmetries
TL;DR: In this paper, the authors apply the theory of Lie group deformations to isometry groups of exact solutions in general relativity, relating the algebraic properties of these groups to physical properties of the spacetimes, and make group deformation local, generalising deformed special relativity by describing gravity as a gauge theory of the de Sitter group.
Journal ArticleDOI
A Study of New Class of Almost Contact Metric Manifolds of Kenmotsu Type
TL;DR: In this paper, the authors characterized a new class of almost contact metric manifolds and established the equivalent conditions of the characterization identity in terms of Kirichenko's tensors, and demonstrated that the Kenmotsu manifold provides the mentioned class; i.e., the new class can be decomposed into a direct sum of the KG and other classes.
Journal ArticleDOI
Identity of the connection curvature tensor of almost manifold C(λ)
TL;DR: In this article, the geometry of the projective curvature tensor is investigated, and three classes of nearly infinite C(λ) are distinguished and studied, and some identities for this tensor are obtained.