A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle

We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ? is vertical or horizontal. Moreover we give decomposition theorems by using the existence of anti-invariant Riemannian submersions.

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Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2018-0007 ◽

2018 ◽

Vol 6 (1) ◽

pp. 129-145 ◽

Cited By ~ 2

Author(s):

Shouhei Honda

Keyword(s):

Riemannian Manifolds ◽

Short Note ◽

Sufficient Condition ◽

Metric Measure Spaces ◽

Necessary And Sufficient Condition ◽

Local Dimension ◽

Base Points ◽

Smooth Metric Measure Spaces ◽

Measure Spaces ◽

Necessary And Sufficient

Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.

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Some properties of lightlike submanifolds of semi-Riemannian manifolds

Demonstratio Mathematica ◽

10.1515/dema-2013-0257 ◽

2010 ◽

Vol 43 (3) ◽

Author(s):

Rakesh Kumar ◽

Rachna Rani ◽

R. K. Nagaich

Keyword(s):

Riemannian Manifolds ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Totally Geodesic ◽

Screen Distribution ◽

Necessary And Sufficient ◽

Lightlike Submanifolds ◽

Lightlike Submanifold

AbstractWe initially obtain various relations and then establish necessary and sufficient condition for the integrability of screen distribution of a lightlike submanifold. We also establish necessary and sufficient condition for a lightlike submanifold to be totally geodesic.

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Pseudo B-symmetric manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887817501195 ◽

2017 ◽

Vol 14 (09) ◽

pp. 1750119

Author(s):

Young Jin Suh ◽

Carlo Alberto Mantica ◽

Uday Chand De ◽

Prajjwal Pal

Keyword(s):

Riemannian Manifold ◽

Explicit Form ◽

Riemannian Manifolds ◽

Ricci Tensor ◽

Sufficient Condition ◽

Conformally Flat ◽

Harmonic Curvature ◽

Codazzi Type ◽

Curvature Tensors

In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.

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

Symmetry ◽

10.3390/sym12121941 ◽

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.

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

Filomat ◽

10.2298/fil1804159t ◽

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.

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Almost pseudo-Q-symmetric semi-Riemannian manifolds

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887818501177 ◽

2018 ◽

Vol 15 (07) ◽

pp. 1850117

Author(s):

Ljubica Velimirović ◽

Pradip Majhi ◽

Uday Chand De

Keyword(s):

Perfect Fluid ◽

Curvature Tensor ◽

Riemannian Manifolds ◽

Einstein Manifold ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Geometric Properties ◽

Dust Fluid ◽

Necessary And Sufficient ◽

Fluid Spacetimes

The object of the present paper is to study almost pseudo-[Formula: see text]-symmetric manifolds [Formula: see text]. Some geometric properties have been studied which recover some known results of pseudo [Formula: see text]-symmetric manifolds. We obtain a necessary and sufficient condition for the [Formula: see text]-curvature tensor to be recurrent in [Formula: see text]. Also, we provide several interesting results. Among others, we prove that a Ricci symmetric [Formula: see text] is an Einstein manifold under certain condition. Moreover we deal with [Formula: see text]-flat perfect fluid, dust fluid and radiation era perfect fluid spacetimes respectively. As a consequence, we obtain some important results. Finally, we consider [Formula: see text]-spacetimes.

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Remarks on metallic warped product manifolds

Facta Universitatis Series Mathematics and Informatics ◽

10.22190/fumi1802269b ◽

2018 ◽

Vol 33 (2) ◽

pp. 269

Author(s):

Adara-Monica Blaga ◽

Cristina-Elena Hretcanu

Keyword(s):

Riemannian Manifold ◽

Riemannian Manifolds ◽

Warped Product ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Metallic Structure ◽

Necessary And Sufficient

We characterize the metallic structure on the product of two metallic manifolds in terms of metallic maps and provide a necessary and sufficient condition for the warped product of two locally metallic Riemannian manifolds to be locally metallic. The particular case of product manifolds is discussed and an example of metallic warped product Riemannian manifold is provided.

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A relation between biharmonic Green’s functions of simply supported and clamped bodies

Nagoya Mathematical Journal ◽

10.1017/s0027763000017293 ◽

1976 ◽

Vol 61 ◽

pp. 59-71 ◽

Cited By ~ 2

Author(s):

James Ralston ◽

Leo Sario

Keyword(s):

Riemannian Manifold ◽

Riemannian Manifolds ◽

Elastic Plate ◽

Boundary Data ◽

Arbitrary Dimension ◽

Point Load ◽

Classification Theory ◽

Thin Elastic Plate ◽

Sufficient Condition ◽

Simply Supported

The deflection, under a point load, of a thin elastic plate clamped at the edges is the biharmonic Green’s function β with the boundary data β = ∂β/∂n = 0. If the boundary of the region is reasonably smooth, the construction of β offers no difficulty. In contrast, nothing is known about the existence of β in the general case. The purpose of our study is to give a sufficient condition for the existence of β on a given Riemannian manifold of arbitrary dimension and to construct β. Our results will have, apart from their physical meaning in elasticity, consequences in the biharmonic classification theory of Riemannian manifolds.

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Para-Ricci-like Solitons on Almost Paracontact Almost Paracomplex Riemannian Manifolds

10.20944/preprints202106.0622.v1 ◽

2021 ◽

Author(s):

Hristo Manev ◽

Mancho Manev

Keyword(s):

Vector Field ◽

Riemannian Manifolds ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Reeb Vector ◽

Reeb Vector Field ◽

Special Cases ◽

Necessary And Sufficient

It is introduced and studied para-Ricci-like solitons with potential Reeb vector field on almost paracontact almost paracomplex Riemannian manifolds. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field have been considered. It is proved a necessary and sufficient condition the manifold to admit a para-Ricci-like soliton which is the structure to be para-Einstein-like. Explicit examples are provided in support of the proven statements.

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