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Young Jin Suh - One of the best experts on this subject based on the ideXlab platform.
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real hypersurfaces in the complex quadric with parallel Ricci Tensor
Advances in Mathematics, 2015Co-Authors: Young Jin SuhAbstract:We introduce the notion of parallel Ricci Tensor for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2. According to the A-principal or the A-isotropic unit normal vector field N, we give a complete classification of real hypersurfaces in Qm=SOm+2/SOmSO2 with parallel Ricci Tensor.
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real hypersurfaces in complex hyperbolic two plane grassmannians with commuting Ricci Tensor
International Journal of Mathematics, 2015Co-Authors: Young Jin SuhAbstract:In this paper we first introduce the full expression of the curvature Tensor of a real hypersurface M in complex hyperbolic two-plane Grassmannians SU2,m/S(U2 ⋅ Um), m ≥ 2 from the equation of Gauss. Next we derive a new formula for the Ricci Tensor of M in SU2,m/S(U2 ⋅ Um). Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m/S(U2 ⋅ Um) with commuting Ricci Tensor. Each can be described as a tube over a totally geodesic SU2,m-1/S(U2 ⋅ Um-1) in SU2,m/S(U2 ⋅ Um) or a horosphere whose center at infinity is singular.
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real hypersurfaces in complex hyperbolic two plane grassmannians with commuting Ricci Tensor
arXiv: Differential Geometry, 2014Co-Authors: Young Jin SuhAbstract:In this paper we first introduce the full expression of the curvature Tensor of a real hypersurface $M$ in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$, $m{\ge}2$ from the equation of Gauss. Next we derive a new formula for the Ricci Tensor of $M$ in $SU_{2,m}/S(U_2{\cdot}U_m)$. Finally we give a complete classification of Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians $SU_{2,m}/S(U_2{\cdot}U_m)$ with commuting Ricci Tensor. Each can be described as a tube over a totally geodesic $SU_{2,m-1}/S(U_2{\cdot}U_{m-1})$ in $SU_{2,m}/S(U_2{\cdot}U_m)$ or a horosphere whose center at infinity is singular.
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real hypersurfaces in complex hyperbolic two plane grassmannians with commuting Ricci Tensor
Mathematische Nachrichten, 2014Co-Authors: Young Jin Suh, Changhwa WooAbstract:In this paper we prove that there does not exist any Hopf real hypersurface in complex hyperbolic two-plane Grassmannians with parallel Ricci Tensor.
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real hypersurfaces in complex two plane grassmannians with harmonic curvature
Journal de Mathématiques Pures et Appliquées, 2013Co-Authors: Young Jin SuhAbstract:Abstract In this paper we first introduce the full expression of the curvature Tensor of a real hypersurface M in complex two-plane Grassmannians G 2 ( C m + 2 ) from the equation of Gauss and derive a new formula for the Ricci Tensor of M in G 2 ( C m + 2 ) . Finally we give a complete classification for Hopf real hypersurfaces in complex two-plane Grassmannians G 2 ( C m + 2 ) with harmonic curvature or harmonic Weyl Tensor.
Constantin Călin - One of the best experts on this subject based on the ideXlab platform.
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on the 1 3 threading of spacetime with respect to an arbitrary timelike vector field
European Physical Journal C, 2015Co-Authors: Aurel Bejancu, Constantin CălinAbstract:We develop a new approach on the (\(1+3\)) threading of spacetime \((M, g)\) with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial Tensor fields and on the Riemannian spatial connection \(\nabla ^{\star }\), which behave as 3D geometric objects. We obtain new formulas for local components of the Ricci Tensor field of \((M, g)\) with respect to the threading frame field, in terms of the Ricci Tensor field of \(\nabla ^{\star }\) and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri’s equation enables us to prove Lemma 6.3, which completes a well-known lemma used in the proof of the Penrose–Hawking singularity theorems. Finally, we apply the new \((1+3)\) formalism to the study of the dynamics of a Kerr–Newman black hole.
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on the 1 3 threading of spacetime with respect to an arbitrary timelike vector field
arXiv: Differential Geometry, 2015Co-Authors: Aurel Bejancu, Constantin CălinAbstract:We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial Tensor fields and on the Riemannian spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We obtain new formulas for local components of the Ricci Tensor field of $(M, g)$ with respect to the threading frame field, in terms of the Ricci Tensor field of $\nabla^{\star}$ and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri's equation enables us to prove Lemma 6.2, which completes a well known lemma used in the proof of Penrose-Hawking singularity theorems.Finally, we apply the new $(1+3)$ formalism to the study of the dynamics of a Kerr-Newman black hole.
Ana Bela Cruzeiro - One of the best experts on this subject based on the ideXlab platform.
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weak levi civita connection for the damped metric on the riemannian path space and vanishing of Ricci Tensor in adapted differential geometry
Journal of Functional Analysis, 2001Co-Authors: Ana Bela Cruzeiro, Shizan FangAbstract:Abstract We shall establish in the context of adapted differential geometry on the path space P m o ( M ) a Weitzenbock formula which generalizes that in (A. B. Cruzeiro and P. Malliavin, J. Funct. Anal . 177 (2000), 219–253), without hypothesis on the Ricci Tensor. The renormalized Ricci Tensor will be vanished. The connection introduced in (A. B. Cruzeiro and S. Fang, 1997, J. Funct. Anal. 143 , 400–414) will play a central role.
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frame bundle of riemannian path space and Ricci Tensor in adapted differential geometry
Journal of Functional Analysis, 2000Co-Authors: Ana Bela Cruzeiro, P MalliavinAbstract:Abstract The vanishing of the renormalized Ricci Tensor of the path space above a Ricci flat Riemannian manifold is discussed.
Aurel Bejancu - One of the best experts on this subject based on the ideXlab platform.
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on the 1 3 threading of spacetime with respect to an arbitrary timelike vector field
European Physical Journal C, 2015Co-Authors: Aurel Bejancu, Constantin CălinAbstract:We develop a new approach on the (\(1+3\)) threading of spacetime \((M, g)\) with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial Tensor fields and on the Riemannian spatial connection \(\nabla ^{\star }\), which behave as 3D geometric objects. We obtain new formulas for local components of the Ricci Tensor field of \((M, g)\) with respect to the threading frame field, in terms of the Ricci Tensor field of \(\nabla ^{\star }\) and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri’s equation enables us to prove Lemma 6.3, which completes a well-known lemma used in the proof of the Penrose–Hawking singularity theorems. Finally, we apply the new \((1+3)\) formalism to the study of the dynamics of a Kerr–Newman black hole.
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on the 1 3 threading of spacetime with respect to an arbitrary timelike vector field
arXiv: Differential Geometry, 2015Co-Authors: Aurel Bejancu, Constantin CălinAbstract:We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial Tensor fields and on the Riemannian spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We obtain new formulas for local components of the Ricci Tensor field of $(M, g)$ with respect to the threading frame field, in terms of the Ricci Tensor field of $\nabla^{\star}$ and of kinematic quantities. Also, new expressions for time covariant derivatives of kinematic quantities are stated. In particular, a new form of Raychaudhuri's equation enables us to prove Lemma 6.2, which completes a well known lemma used in the proof of Penrose-Hawking singularity theorems.Finally, we apply the new $(1+3)$ formalism to the study of the dynamics of a Kerr-Newman black hole.
Alan Barnes - One of the best experts on this subject based on the ideXlab platform.
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Ricci collineations in friedmann robertson walker spacetimes
Classical and Quantum Gravity, 2002Co-Authors: Ugur Camci, Alan BarnesAbstract:Ricci collineations and Ricci inheritance collineations of Friedmann–Robertson–Walker spacetimes are considered. When the Ricci Tensor is non-degenerate, it is shown that the spacetime always admits a 15-parameter group of Ricci inheritance collineations; this is the maximal possible dimension for spacetime manifolds. The general form of the vector generating the symmetry is exhibited. It is also shown, in the generic case, that the group of Ricci collineations is six-dimensional and coincides with the isometry group. In special cases the spacetime may admit either one or four proper Ricci collineations in addition to the six isometries. These special cases are classified and the general form of the vector fields generating the Ricci collineations is exhibited. When the Ricci Tensor is degenerate, the groups of Ricci inheritance collineations and Ricci collineations are infinite-dimensional. General forms for the generating vectors are obtained. Similar results are obtained for matter collineations and matter inheritance collineations.
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Ricci collineations in friedmann robertson walker spacetimes
arXiv: General Relativity and Quantum Cosmology, 2001Co-Authors: Ugur Camci, Alan BarnesAbstract:Ricci collineations and Ricci inheritance collineations of Friedmann-Robertson-Walker spacetimes are considered. When the Ricci Tensor is non-degenerate, it is shown that the spacetime always admits a fifteen parameter group of Ricci inheritance collineations; this is the maximal possible dimension for spacetime manifolds. The general form of the vector generating the symmetry is exhibited. It is also shown, in the generic case, that the group of Ricci collineations is six-dimensional and coincides with the isometry group. In special cases the spacetime may admit either one or four proper Ricci collineations in addition to the six isometries. These special cases are classified and the general form of the vector fields generating the Ricci collineations is exhibited. When the Ricci Tensor is degenerate, the groups of Ricci inheritance collineations and Ricci collineations are infinite-dimensional. General forms for the generating vectors are obtained. Similar results are obtained for matter collineations and matter inheritance collineations.
