scholarly journals Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four

2006 ◽  
Vol 58 (2) ◽  
pp. 282-311 ◽  
Author(s):  
M. E. Fels ◽  
A. G. Renner
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Isotropy Subgroup ◽  
Algebraic Classification ◽  
Ricci Flat ◽  
Einstein Spaces ◽  
Simply Connected

AbstractA method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2, 2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to ℝ4. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.

TOPONOGOV-TYPE AREA COMPARISON THEOREM OF 2-DIMENSIONAL MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1350007
Author(s):  
XIAOLE SU ◽  
HONGWEI SUN ◽  
YUSHENG WANG
Keyword(s):  
Riemannian Manifold ◽  
Comparison Theorem ◽  
Constant Curvature ◽  
Dimensional Manifold ◽  
Geodesic Triangle ◽  
Type Area ◽  
Simply Connected

Let △p1p2p3 be a geodesic triangle on M, a complete 2-dimensional Riemannian manifold of curvature ≥ k, and let [Formula: see text] be its comparison triangle on [Formula: see text] (a complete and simply connected 2-dimensional manifold of constant curvature k). Our main result is that if △p1p2p3 is areable, then its area is not less than that of [Formula: see text].

scholarly journals The even Clifford structure of the fourth Severi variety

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Maurizio Parton ◽  
Paolo Piccinni
Keyword(s):  
Vector Bundle ◽  
Riemannian Manifolds ◽  
Severi Variety ◽  
Simply Connected

AbstractTheHermitian symmetric spaceM = EIII appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure [19]. This means the existence of a real oriented Euclidean vector bundle E over it together with an algebra bundle morphism φ : Cl

scholarly journals Classification of Ricci solitons on Euclidean hypersurfaces

2014 ◽  
Vol 25 (11) ◽  
pp. 1450104 ◽  
Author(s):  
Bang-Yen Chen ◽  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Riemannian Manifolds ◽  
Potential Field ◽  
Vector Fields ◽  
Ricci Soliton ◽  
Position Vector ◽  
Ricci Solitons ◽  
Euclidean Submanifolds

A Ricci soliton (M, g, v, λ) on a Riemannian manifold (M, g) is said to have concurrent potential field if its potential field v is a concurrent vector field. Ricci solitons arisen from concurrent vector fields on Riemannian manifolds were studied recently in [Ricci solitons and concurrent vector fields, preprint (2014), arXiv:1407.2790]. The most important concurrent vector field is the position vector field on Euclidean submanifolds. In this paper we completely classify Ricci solitons on Euclidean hypersurfaces arisen from the position vector field of the hypersurfaces.

scholarly journals Randers manifolds of positive constant curvature

2003 ◽  
Vol 2003 (18) ◽  
pp. 1155-1165 ◽  
Author(s):  
Aurel Bejancu ◽  
Hani Reda Farran
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Complete Riemannian Manifold ◽  
Flag Curvature ◽  
Dimensional Sphere ◽  
Randers Metric ◽  
Constant Flag Curvature ◽  
Simply Connected

We prove that any simply connected and complete Riemannian manifold, on which a Randers metric of positive constant flag curvature exists, must be diffeomorphic to an odd-dimensional sphere, provided a certain 1-form vanishes on it.

scholarly journals Riemannian manifolds with local symmetry

2014 ◽  
Vol 06 (02) ◽  
pp. 211-236 ◽  
Author(s):  
Wouter van Limbeek
Keyword(s):  
Riemannian Manifolds ◽  
Local Symmetry ◽  
Universal Cover ◽  
Connected Components ◽  
Curved Manifolds ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Positively Curved ◽  
Aspherical Manifolds

We give a classification of many closed Riemannian manifolds M whose universal cover [Formula: see text] possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that [Formula: see text] has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.

On semi-symmetric metric connection in sub-Riemannian manifold

2016 ◽  
Vol 47 (4) ◽  
pp. 373-384
Author(s):  
Yanling Han ◽  
Fengyun Fu ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Conformally Flat ◽  
Ricci Flat ◽  
Metric Connection

The authors firstly in this paper define a semi-symmetric metric non-holonomic connection (in briefly, SS-connection) on sub-Riemannian manifolds. An invariant under a SS-connection transformation is obtained. The authors then further give a result that a sub-Riemannian manifold $(M,V_{0},g,\bar{\nabla})$ is locally horizontally flat if and only if $M$ is horizontally conformally flat and horizontally Ricci flat.

Recurrent conformal 2-forms on pseudo-Riemannian manifolds

2014 ◽  
Vol 11 (06) ◽  
pp. 1450056 ◽  
Author(s):  
Carlo Alberto Mantica ◽  
Young Jin Suh
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Higher Dimensions ◽  
Topological Properties ◽  
Lorentzian Manifolds ◽  
Differential Structure ◽  
Arbitrary Signature

In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdziński–Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions.

scholarly journals On a construction of complete simply-connected riemannian manifolds with negative curvature

1989 ◽  
Vol 113 ◽  
pp. 7-13
Author(s):  
Haruo Kitahara ◽  
Hajime Kawakami ◽  
Jin Suk Pak
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Harmonic Forms ◽  
Simply Connected

Let M be a complete simply-connected riemannian manifold of even dimension m. J. Dodziuk and I.M. Singer ([D1]) have conjectured that H2p(M) = 0 if p ≠ m/2 and dim H2m/2(M) = ∞, where H2*(M) is the space of L2-harmonic forms on M.

scholarly journals A Note on the Characterization of Two-Dimensional Quasi-Einstein Manifolds

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2013
Author(s):  
Gabriel Bercu
Keyword(s):  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Product Type ◽  
Two Dimensional ◽  
Einstein Manifolds ◽  
Local Coordinates ◽  
Elementary Methods

In this article, we aim to introduce new classes of two-dimensional quasi-Einstein pseudo-Riemannian manifolds with constant curvature. We also give a classification of 2D quasi-Einstein manifolds of warped product type working in local coordinates. All the results are obtained by elementary methods.

scholarly journals Metric connections with parallel twistor-free torsion

2021 ◽  
pp. 2140011
Author(s):  
Andrei Moroianu ◽  
Mihaela Pilca
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Cyclic Component ◽  
Metric Connection ◽  
Simply Connected ◽  
Three Components

The torsion of every metric connection on a Riemannian manifold has three components: one totally skew-symmetric, one of vectorial type and one of twistorial type, which is also called the traceless cyclic component. In this paper we classify complete simply connected Riemannian manifolds carrying a metric connection whose torsion is parallel, has nonzero vectorial component and vanishing twistorial component.

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