scholarly journals Almost contact metric3-submersions

1984 ◽  
Vol 7 (4) ◽  
pp. 667-688 ◽  
Author(s):  
Bill Watson
Keyword(s):  
Vector Fields ◽  
Betti Numbers ◽  
Riemannian Submersion ◽  
Second Fundamental Form ◽  
Horizontal Distribution ◽  
Total Space ◽  
Field Equations ◽  
Contact Metric Manifold ◽  
Almost Contact Metric ◽  
Weak Structure

An almost contact metric3-submersion is a Riemannian submersion,πfrom an almost contact metric manifold(M4m+3,(φi,ξi,ηi)i=13,g)onto an almost quaternionic manifold(N4n,(Ji)i=13,h)which commutes with the structure tensors of type(1,1);i.e.,π*φi=Jiπ*, fori=1,2,3. For various restrictions on∇φi, (e.g.,Mis3-Sasakian), we show corresponding limitations on the second fundamental form of the fibres and on the complete integrability of the horizontal distribution. Concommitantly, relations are derived between the Betti numbers of a compact total space and the base space. For instance, ifMis3-quasi-Saskian(dΦ=0), thenb1(N)≤b1(M). The respectiveφi-holomorphic sectional and bisectional curvature tensors are studied and several unexpected results are obtained. As an example, ifXandYare orthogonal horizontal vector fields on the3-contact (a relatively weak structure) total space of such a submersion, then the respective holomorphic bisectional curvatures satisfy:Bφi(X,Y)=B′J′i(X*,Y*)−2. Applications to the real differential geometry of Yarg-Milis field equations are indicated based on the fact that a principalSU(2)-bundle over a compactified realized space-time can be given the structure of an almost contact metric3-submersion.

scholarly journals The differential geometry of almost Hermitian almost contact metric submersions

2004 ◽  
Vol 2004 (36) ◽  
pp. 1923-1935
Author(s):  
T. Tshikuna-Matamba
Keyword(s):  
Differential Geometry ◽  
Total Space ◽  
Riemannian Submersions ◽  
Contact Metric Manifold ◽  
Almost Contact Metric Manifold ◽  
Fundamental Properties ◽  
Almost Contact Metric

Three types of Riemannian submersions whose total space is an almost Hermitian almost contact metric manifold are studied. The study is focused on fundamental properties and the transference of structures.

scholarly journals Semi-invariant Riemannian submersions from nearly Kaehler manifolds

2020 ◽  
Vol 17 (07) ◽  
pp. 2050100
Author(s):  
Rupali Kaushal ◽  
Rashmi Sachdeva ◽  
Rakesh Kumar ◽  
Rakesh Kumar Nagaich
Keyword(s):  
Sufficient Conditions ◽  
Riemannian Submersion ◽  
Horizontal Distribution ◽  
Product Manifold ◽  
Riemannian Submersions ◽  
Kaehler Manifolds ◽  
Necessary And Sufficient ◽  
Nearly Kaehler Manifold ◽  
The Vertical Distribution ◽  
Usual Product

We study semi-invariant Riemannian submersions from a nearly Kaehler manifold to a Riemannian manifold. It is well known that the vertical distribution of a Riemannian submersion is always integrable therefore, we derive condition for the integrability of horizontal distribution of a semi-invariant Riemannian submersion and also investigate the geometry of the foliations. We discuss the existence and nonexistence of semi-invariant submersions such that the total manifold is a usual product manifold or a twisted product manifold. We establish necessary and sufficient conditions for a semi-invariant submersion to be a totally geodesic map. Finally, we study semi-invariant submersions with totally umbilical fibers.

scholarly journals Plane poloidal-toroidal decomposition of doubly periodic vector fields. Part 2. The Stokes equations

2005 ◽  
Vol 47 (1) ◽  
pp. 39-50 ◽  
Author(s):  
G. D. McBain
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Momentum Equation ◽  
Integral Representations ◽  
Navier Stokes ◽  
Field Equations ◽  
Navier Stokes Equations ◽  
Time Discretisation ◽  
Mass Constraint ◽  
Conservation Of Momentum

AbstractWe continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.

Lifting semi-invariant submanifolds to distribu­tion of almost contact metric manifolds

2020 ◽  
pp. 39-48
Author(s):  
A. V. Bukusheva
Keyword(s):  
Parallel Transport ◽  
Linear Connection ◽  
Contact Metric Manifold ◽  
Almost Contact Metric Structure ◽  
Almost Contact Metric Manifold ◽  
Contact Metric Manifolds ◽  
Almost Contact Metric ◽  
Invariant Submanifolds ◽  
Contact Metric Structure ◽  
Almost Contact Metric Manifolds

Let M be an almost contact metric manifold of dimension n = 2m + 1. The distribution D of the manifold M admits a natural structure of a smooth manifold of dimension n = 4m + 1. On the manifold M, is defined a linear connection that preserves the distribution D; this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves. The assigment of the linear connection is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution D. Certain tensor field of type (1,1) on D defines a so-called prolonged almost contact metric structure. Each section of the distribution D defines a morphism of smooth manifolds. It is proved that if a semi-invariant sub­manifold of the manifold M and is a covariantly constant vec­tor field with respect to the N-connection , then is a semi-invariant submanifold of the manifold D with respect to the prolonged almost contact metric structure.

IMAGE PROCESSING AND SELF-ORGANIZING CNN

2005 ◽  
Vol 15 (09) ◽  
pp. 2939-2958 ◽  
Author(s):  
MAKOTO ITOH ◽  
LEON O. CHUA
Keyword(s):  
Image Processing ◽  
Pattern Formation ◽  
Vector Fields ◽  
Reaction Diffusion ◽  
Neural Field ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Self Organization ◽  
Field Equations ◽  
Neural Field Equations

CNN templates for image processing and pattern formation are derived from neural field equations, advection equations and reaction–diffusion equations by discretizing spatial integrals and derivatives. Many useful CNN templates are derived by this approach. Furthermore, self-organization is investigated from the viewpoint of divergence of vector fields.

On ruled surface in 3-dimensional almost contact metric manifold

2017 ◽  
Vol 14 (05) ◽  
pp. 1750076 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Nural Yuksel ◽  
Hasibe Ikiz
Keyword(s):  
Cross Product ◽  
Ruled Surface ◽  
Contact Metric Manifold ◽  
Almost Contact Metric Manifold ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Special Cases ◽  
Distribution Parameters ◽  
Almost Contact Metric ◽  
Phd Thesis

In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcı. In this study, we obtain the distribution parameters of the ruled surface and then some results and theorems are presented with special cases. Moreover, some relationships among asymptotic curve and striction line of the base curve of the ruled surface have been found.

scholarly journals On a semi-symmetric non-metric connection

Filomat ◽  
2011 ◽  
Vol 25 (4) ◽  
pp. 19-27 ◽  
Author(s):  
S.K. Chaubey ◽  
R.H. Ojha
Keyword(s):  
Riemannian Manifold ◽  
Contact Metric Manifold ◽  
Kenmotsu Manifold ◽  
Almost Contact Metric Manifold ◽  
Metric Connection ◽  
Almost Contact Metric

Yano [1] defined and studied semi-symmetric metric connection in a Riemannian manifold and this was extended by De and Senguta [8] and many other geometers. Recently, the present authors [3], [5] defined semi-symmetric non-metric connections in an almost contact metric manifold. In this paper, we studied some properties of a semi-symmetric non-metric connection in a Kenmotsu manifold.

scholarly journals On a semi-symmetric non-metric connection

Filomat ◽  
2012 ◽  
Vol 26 (2) ◽  
pp. 269-275 ◽  
Author(s):  
S.K. Chaubey ◽  
R.H. Ojha
Keyword(s):  
Riemannian Manifold ◽  
Contact Metric Manifold ◽  
Kenmotsu Manifold ◽  
Almost Contact Metric Manifold ◽  
Metric Connection ◽  
Almost Contact Metric

Yano [10] defined and studied semi-symmetric metric connection in a Riemannian manifold and this was extended by De and Senguta [4] and many other geometers. Recently, the present authors [2], [3] defined semi-symmetric non-metric connections in an almost contact metric manifold. In this paper, we studied some properties of a semi-symmetric non-metric connection in a Kenmotsu manifold.

scholarly journals Anti-invariant Riemannian submersions from nearly Kaehler manifolds

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1219-1235 ◽  
Author(s):  
Shahid Ali ◽  
Tanveer Fatima
Keyword(s):  
Riemannian Submersion ◽  
Horizontal Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Kaehler Manifolds ◽  
Necessary And Sufficient ◽  
Decomposition Theorems ◽  
The Vertical Distribution

We extend the notion of anti-invariant and Langrangian Riemannian submersion to the case when the total manifold is nearly Kaehler. We obtain the integrability conditions for the horizontal distribution while it is noted that the vertical distribution is always integrable. We also investigate the geometry of the foliations of the two distributions and obtain the necessary and sufficient condition for a Langrangian submersion to be totally geodesic. The decomposition theorems for the total manifold of the submersion are obtained.

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