scholarly journals Discriminating the Weyl type in higher dimensions using scalar curvature invariants

2011 ◽  
Vol 43 (8) ◽  
pp. 2199-2207 ◽  
Author(s):  
Alan Coley ◽  
Sigbjørn Hervik
Keyword(s):  
Scalar Curvature ◽  
Higher Dimensions ◽  
Curvature Invariants

Discs area-minimizing in mean convex Riemannian n-manifolds

2021 ◽  
pp. 1-22
Author(s):  
Ezequiel Barbosa ◽  
Franciele Conrado
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Higher Dimensions ◽  
Dimensional Torus ◽  
Rigidity Result ◽  
Equality Case ◽  
Totally Geodesic ◽  
Zero Degree ◽  
Area Minimizing ◽  
Mean Convex

In this work, we consider oriented compact manifolds which possess convex mean curvature boundary, positive scalar curvature and admit a map to $\mathbb {D}^{2}\times T^{n}$ with non-zero degree, where $\mathbb {D}^{2}$ is a disc and $T^{n}$ is an $n$ -dimensional torus. We prove the validity of an inequality involving a mean of the area and the length of the boundary of immersed discs whose boundaries are homotopically non-trivial curves. We also prove a rigidity result for the equality case when the boundary is strongly totally geodesic. This can be viewed as a partial generalization of a result due to Lucas Ambrózio in (2015, J. Geom. Anal., 25, 1001–1017) to higher dimensions.

scholarly journals Dynamical projective curvature in gravitation

2018 ◽  
Vol 33 (36) ◽  
pp. 1850223 ◽  
Author(s):  
Samuel Brensinger ◽  
Vincent G. J. Rodgers
Keyword(s):  
Ricci Tensor ◽  
Higher Dimensions ◽  
Projective Connection ◽  
Two Dimensional ◽  
Affine Connections ◽  
Curvature Invariants ◽  
Projective Curvature ◽  
Dimensional Gravity

By using a projective connection over the space of two-dimensional affine connections, we are able to show that the metric interaction of Polyakov two-dimensional gravity with a coadjoint element arises naturally through the projective Ricci tensor. Through the curvature invariants of Thomas and Whitehead, we are able to define an action that could describe dynamics to the projective connection. We discuss implications of the projective connection in higher dimensions as related to gravitation.

SCALAR AVERAGING IN COSMOLOGY

2010 ◽  
Vol 19 (14) ◽  
pp. 2361-2364
Author(s):  
A. A. COLEY
Keyword(s):  
Scalar Curvature ◽  
Cosmological Evolution ◽  
Considerable Importance ◽  
Correct Interpretation ◽  
Curvature Invariants ◽  
Averaging Problem ◽  
Cosmological Data

The averaging problem in cosmology is of considerable importance for the correct interpretation of cosmological data. In this essay an approach to averaging based on scalar curvature invariants is presented, which gives rise to significant effects on cosmological evolution.

scholarly journals Canonical formulation of scalar curvature squared action in higher dimensions

2014 ◽  
Vol 90 (4) ◽  
Author(s):  
Subhra Debnath ◽  
Soumendranath Ruz ◽  
Abhik Kumar Sanyal
Keyword(s):  
Scalar Curvature ◽  
Higher Dimensions ◽  
Canonical Formulation

An inequality between intrinsic and extrinsic scalar curvature invariants for codimension 2 embeddings

2004 ◽  
Vol 52 (2) ◽  
pp. 101-112 ◽  
Author(s):  
Franki Dillen ◽  
Stefan Haesen ◽  
Miroslava Petrović-Torgašev ◽  
Leopold Verstraelen
Keyword(s):  
Scalar Curvature ◽  
Curvature Invariants

scholarly journals Basis for scalar curvature invariants in three dimensions

2014 ◽  
Vol 31 (23) ◽  
pp. 235010 ◽  
Author(s):  
A A Coley ◽  
A MacDougall ◽  
D D McNutt
Keyword(s):  
Scalar Curvature ◽  
Three Dimensions ◽  
Curvature Invariants

scholarly journals Curvature operators and scalar curvature invariants

2010 ◽  
Vol 27 (9) ◽  
pp. 095014 ◽  
Author(s):  
Sigbjørn Hervik ◽  
Alan Coley
Keyword(s):  
Scalar Curvature ◽  
Curvature Invariants

scholarly journals The Faraday 2-form in Einstein-Weyl geometry

2001 ◽  
Vol 89 (1) ◽  
pp. 97 ◽  
Author(s):  
David M. J. Calderbank
Keyword(s):  
Scalar Curvature ◽  
Higher Dimensions ◽  
Riemannian Metric ◽  
Constant Sign ◽  
Weyl Curvature ◽  
Weyl Geometry ◽  
Four Dimensions ◽  
Bach Tensor ◽  
Levi Civita Connection ◽  
Torsion Free

On a conformal manifold, a compatible torsion free connection $D$ need not be the Levi-Civita connection of a compatible Riemannian metric. The local obstruction is a real $2$-form $F^D$, the Faraday curvature. It is shown that, except in four dimensions, $F^D$ necessarily vanishes if it is divergence free. In four dimensions another differential operator may be applied to $F^D$ to show that an Einstein-Weyl $4$-manifold with selfdual Weyl curvature also has selfdual Faraday curvature and so is either Einstein or locally hypercomplex. More generally, the Bach tensor and the scalar curvature are shown to control the selfduality of $F^D$. Finally, the constancy of the sign of the scalar curvature on compact Einstein-Weyl $4$-manifolds [24] is generalised to higher dimensions. The scalar curvature need not have constant sign in dimensions two and three.

scholarly journals KILLING VECTORS IN HIGHER-DIMENSIONAL SPACETIMES WITH CONSTANT SCALAR CURVATURE INVARIANTS

2010 ◽  
Vol 07 (08) ◽  
pp. 1349-1369 ◽  
Author(s):  
DAVID MCNUTT ◽  
NICOS PELAVAS ◽  
ALAN COLEY
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Null Vector ◽  
Killing Vectors ◽  
Curvature Invariants ◽  
Higher Dimensional ◽  
Metric Functions ◽  
Kundt Spacetimes

We study the existence of a non-spacelike isometry, ζ, in higher-dimensional Kundt spacetimes with constant scalar curvature invariants (CSI). We present the particular forms for the null or timelike Killing vectors and a set of constraints for the metric functions in each case. Within the class of N-dimensional CSI Kundt spacetimes, admitting a non-spacelike isometry, we determine which of these can admit a covariantly constant null vector that also satisfy ζ[a;b] = 0.

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