scholarly journals ISOMETRIC IMMERSIONS INTO LORENTZIAN PRODUCTS

2011 ◽  
Vol 08 (06) ◽  
pp. 1269-1290 ◽  
Author(s):  
JULIEN ROTH
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Shape Operator ◽  
Equivalent Condition ◽  
Sufficient Condition ◽  
Maximal Surface ◽  
Necessary And Sufficient Condition ◽  
Isometric Immersions ◽  
Vertical Vector ◽  
Necessary And Sufficient

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products 𝕊n × ℝ1 or ℍn × ℝ1. This condition is expressed in terms of its first and second fundamental forms, the tangent and normal projections of the vertical vector field. As applications, we give an equivalent condition in a spinorial way and we deduce the existence of a one-parameter family of isometric maximal deformation of a given maximal surface obtained by rotating the shape operator.

scholarly journals On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1941
Author(s):  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Haila Alodan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Rigidity Results ◽  
Necessary And Sufficient

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

scholarly journals Killing Vector Fields in Generalized Conformalβ-Change of Finsler Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Mallikarjun Yallappa Kumbar ◽  
Narasimhamurthy Senajji Kampalappa ◽  
Thippeswamy Komalobiah Rajanna ◽  
Kavyashree Ambale Rajegowda
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Finsler Space ◽  
Killing Vector ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finsler Spaces ◽  
Killing Vector Fields ◽  
Necessary And Sufficient

We consider a Finsler space equipped with a Generalized Conformalβ-change of metric and study the Killing vector fields that correspond between the original Finsler space and the Finsler space equipped with Generalized Conformalβ-change of metric. We obtain necessary and sufficient condition for a vector field Killing in the original Finsler space to be Killing in the Finsler space equipped with Generalized Conformalβ-change of metric.

scholarly journals A Characterization of Invariant Affine Connections

1960 ◽  
Vol 16 ◽  
pp. 35-50 ◽  
Author(s):  
Bertram Kostant
Keyword(s):  
Riemannian Manifold ◽  
Simple Proof ◽  
General Theorem ◽  
Transitive Group ◽  
Complete Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Simply Connected

1. Introduction and statement of theorem. 1. In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem — Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the Tx of [1] is just the ax of [6] when X is restricted to p0, see [6], p. 539).

TOTALLY GEODESIC SUBMANIFOLDS OF GL-MANIFOLDS

2012 ◽  
Vol 09 (01) ◽  
pp. 1250002
Author(s):  
ABOLGHASEM LALEH ◽  
MORTEZA M. REZAII ◽  
ATAABAK BAAGHERZADEH HUSHMANDI
Keyword(s):  
Riemannian Manifold ◽  
Finsler Manifold ◽  
Finsler Metric ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Necessary And Sufficient ◽  
Sasakian Metric

In this paper, for a Finsler manifold (M, F) with a Finsler metric gij(x, y) we shall consider a generalized Lagrange metrics (FGL-metrics) as the form *gij(x, y) = gij(x, y) + σ(x, y)Bi(x, y)Bj(x, y) on TM. Then we shall consider a Riemannian manifold (TM, *G) in which *G is a generalized Sasakian metric of *g on [Formula: see text]. Then we restrict the above FGL-metrics to a submanifold of [Formula: see text], and show that it admits a GL-metric structure. Then we shall find a necessary and sufficient condition for this submanifold to be totally geodesic.

scholarly journals Anti-invariant Riemanian submersions from nearly-k-cosymplectic manifolds

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1159-1174
Author(s):  
Ju Tan ◽  
Na Xu
Keyword(s):  
Vector Field ◽  
Riemannian Manifolds ◽  
Riemannian Submersion ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Cosymplectic Manifolds ◽  
Necessary And Sufficient ◽  
Characteristic Vector Field

In this paper, we introduce anti-invariant Riemannian submersions from nearly-K-cosymplectic manifolds onto Riemannian manifolds. We study the integrability of horizontal distributions. And we investigate the necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. Moreover, we give examples of anti-invariant Riemannian submersions such that characteristic vector field ? is vertical or horizontal.

scholarly journals Defining amplituhedra and Grassmann polytopes

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Steven N. Karp
Keyword(s):  
Dimensional Subspace ◽  
Equivalent Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positivity Condition ◽  
Yang Mills ◽  
Cyclic Polytopes ◽  
Plücker Coordinates ◽  
International Audience ◽  
Necessary And Sufficient

International audience The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r).

Geometry of semi-transversal lightlike warped product submanifolds of indefinite Kaehler manifolds

2019 ◽  
pp. 2150001
Author(s):  
Sangeet Kumar
Keyword(s):  
Warped Product ◽  
Shape Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Kaehler Manifold ◽  
Type Formula ◽  
Kaehler Manifolds ◽  
Necessary And Sufficient ◽  
Lightlike Submanifolds ◽  
Lightlike Submanifold

In this paper, we investigate warped product semi-transversal lightlike submanifolds of indefinite Kaehler manifolds. It is shown that there does not exist any warped product semi-transversal lightlike submanifold of the type [Formula: see text] in an indefinite Kaehler manifold. Moreover, a necessary and sufficient condition for an isometrically immersed semi-transversal lightlike submanifold of an indefinite Kaehler manifold to be a semi-transversal lightlike warped product of the type [Formula: see text] is obtained, in terms of the shape operator.

scholarly journals On the ME-manifold inn-*g-UFT and its conformal change

1994 ◽  
Vol 17 (1) ◽  
pp. 79-90
Author(s):  
Kyung Tae Chung ◽  
Gwang Sik Eun
Keyword(s):  
Riemannian Manifold ◽  
Geometric Structure ◽  
Tensor Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Unique Existence ◽  
Conformal Change ◽  
Necessary And Sufficient

An Einstein's connection which takes the form (3.1) is called an ME-connection. A generalizedn-dimensional Riemannian manifoldXnon which the differential geometric structure is imposed by a tensor field*gλνthrough a unique ME-connection subject to the conditions of Agreement (4.1) is called*g-ME-manifold and we denote it by*g-MEXn. The purpose of the present paper is to introduce this new concept of*g-MEXnand investigate its properties. In this paper, we first prove a necessary and sufficient condition for the unique existence of ME-connection inXn, and derive a surveyable tensorial representation of the ME-connection. In the second, we investigate the conformal change of*g-MEXnand present a useful tensorial representation of the conformal change of the ME-connection.

On generalized concircularly recurrent manifolds

2009 ◽  
Vol 46 (2) ◽  
pp. 287-296
Author(s):  
U. De ◽  
A. Kalam Gazi
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Einstein Manifold ◽  
Constant Scalar Curvature ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Recurrent Manifold ◽  
New Type ◽  
Necessary And Sufficient

In this paper we study a new type of Riemannian manifold called generalized concircularly recurrent manifold. We obtain a necessary and sufficient condition for the constant scalar curvature of such a manifold. Next we study Ricci symmetric generalized concircularly recurrent manifold and prove that such a manifold is an Einstein manifold. Finally, we obtain a sufficient condition for a generalized concircularly recurrent manifold to be a special quasi-Einstein manifold.

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