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Sugato Mukhopadhyay - One of the best experts on this subject based on the ideXlab platform.
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Levi-Civita Connections for a class of spectral triples
Letters in Mathematical Physics, 2020Co-Authors: Jyotishman Bhowmick, Debashish Goswami, Sugato MukhopadhyayAbstract:We give a new definition of Levi-Civita Connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence–uniqueness result for a class of modules of one-forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes–Landi deformations of spectral triples on the Connes–Dubois- Violette–Rieffel deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in Frohlich et al. (Commun Math Phys 203:119–184, 1999) failed to ensure the existence of a unique Levi-Civita Connection. In the case of the matrix geometry, the Levi-Civita Connection that we get coincides with the unique real torsion-less unitary Connection obtained by Frohlich et al. (1999).
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Levi-Civita Connections for a class of spectral triples
Letters in Mathematical Physics, 2019Co-Authors: Jyotishman Bhowmick, Debashish Goswami, Sugato MukhopadhyayAbstract:We give a new definition of Levi-Civita Connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois Violette-Rieffel-deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in the paper by Frolich et al failed to ensure the existence of a unique Levi-Civita Connection. In the case of the matrix geometry, the Levi-Civita Connection that we get coincides with the unique real torsion-less unitary Connection obtained by Frolich et al.
Jyotishman Bhowmick - One of the best experts on this subject based on the ideXlab platform.
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A New look at Levi-Civita Connection in noncommutative geometry
International Journal of Geometric Methods in Modern Physics, 2021Co-Authors: Jyotishman Bhowmick, Debashish Goswami, Soumalya JoardarAbstract:We prove the existence and uniqueness of Levi-Civita Connections for strongly [Formula: see text]-compatible pseudo-Riemannian metrics on tame differential calculi. Such pseudo-Riemannian metrics properly contain the classes of bilinear metrics as well as their conformal deformations. This extends the previous results in [J. Bhowmick, D. Goswami and S. Mukhopadhyay, Levi-Civita Connections for a class of spectral triples, Lett. Math. Phys. 110 (2020) 835–884] and [J. Bhowmick, D. Goswami and G. Landi, On the Koszul formula in noncommutative geometry, Rev. Math. Phys. 32(10) (2020) 2050032].
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Levi-Civita Connections for a class of spectral triples
Letters in Mathematical Physics, 2020Co-Authors: Jyotishman Bhowmick, Debashish Goswami, Sugato MukhopadhyayAbstract:We give a new definition of Levi-Civita Connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence–uniqueness result for a class of modules of one-forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes–Landi deformations of spectral triples on the Connes–Dubois- Violette–Rieffel deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in Frohlich et al. (Commun Math Phys 203:119–184, 1999) failed to ensure the existence of a unique Levi-Civita Connection. In the case of the matrix geometry, the Levi-Civita Connection that we get coincides with the unique real torsion-less unitary Connection obtained by Frohlich et al. (1999).
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Levi-Civita Connections for a class of spectral triples
Letters in Mathematical Physics, 2019Co-Authors: Jyotishman Bhowmick, Debashish Goswami, Sugato MukhopadhyayAbstract:We give a new definition of Levi-Civita Connection for a noncommutative pseudo-Riemannian metric on a noncommutative manifold given by a spectral triple. We prove the existence-uniqueness result for a class of modules of one forms over a large class of noncommutative manifolds, including the matrix geometry of the fuzzy 3-sphere, the quantum Heisenberg manifolds and Connes-Landi deformations of spectral triples on the Connes-Dubois Violette-Rieffel-deformation of a compact manifold equipped with a free toral action. It is interesting to note that in the example of the quantum Heisenberg manifold, the definition of metric compatibility given in the paper by Frolich et al failed to ensure the existence of a unique Levi-Civita Connection. In the case of the matrix geometry, the Levi-Civita Connection that we get coincides with the unique real torsion-less unitary Connection obtained by Frolich et al.
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A new look at Levi-Civita Connection in noncommutative geometry
arXiv: Quantum Algebra, 2016Co-Authors: Jyotishman Bhowmick, Debashish Goswami, Soumalya JoardarAbstract:We prove the existence and uniqueness of Levi-Civita Connections for a noncommutative pseudo-Riemannian metric on a class of centered bimodule of one forms. As an application, we compute the Ricci and scalar curvature for a general conformal perturbation of the canonical metric on the noncommutative $2$-torus as well as for a natural metric on the quantum Heisenberg manifold. For the latter, the scalar curvature turns out to be a negative constant.
C. S. Bagewadi - One of the best experts on this subject based on the ideXlab platform.
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Invariant Submanifolds of Sasakian Manifolds Admitting Semi-symmetric Metric Connection
Communications in Mathematics and Applications, 2013Co-Authors: B. S. S. Anitha, C. S. BagewadiAbstract:The object of this paper is to study invariant submanifolds $M$ of Sasakian manifolds $\widetilde{M}$ admitting a semi-symmetric metric Connection and to show that $M$ admits semi-symmetric metric Connection. Further it is proved that the second fundamental forms $\sigma$ and $\overline{\sigma}$ with respect to Levi-Civita Connection and semi-symmetric metric Connection coincide. It is shown that if the second fundamental form $\sigma$ is recurrent, 2-recurrent, generalized 2-recurrent and $M$ has parallel third fundamental form with respect to semi-symmetric metric Connection, then $M$ is totally geodesic with respect to Levi-Civita Connection.
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A Study on Conservative C-Bochner Curvature Tensor in K-Contact and Kenmotsu Manifolds Admitting Semisymmetric Metric Connection
ISRN Geometry, 2012Co-Authors: Gurupadavva Ingalahalli, C. S. BagewadiAbstract:The paper deals with the study on conservative C-Bochner curvature tensor in K-contact and Kenmotsu manifolds admitting semisymmetric metric Connection, and it is shown that these manifolds are η-Einstein with respect to Levi-Civita Connection, and the results are illustrated with examples.
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Invariant Submanifolds of Sasakian Manifolds Admitting Semisymmetric Nonmetric Connection
International Journal of Mathematics and Mathematical Sciences, 2012Co-Authors: B. S. S. Anitha, C. S. BagewadiAbstract:The object of this paper is to study invariant submanifolds
Jose Alberto Orejuela - One of the best experts on this subject based on the ideXlab platform.
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a non trivial Connection for the metric affine gauss bonnet theory in d 4
Physics Letters B, 2019Co-Authors: Bert Janssen, Alejandro Jimenezcano, Jose Alberto OrejuelaAbstract:Abstract We study non-trivial (i.e. non-Levi-Civita) Connections in metric-affine Lovelock theories. First we study the projective invariance of general Lovelock actions and show that all Connections constructed by acting with a projective transformation of the Levi-Civita Connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl Connection is also a solution for the specific case of the four-dimensional metric-affine Gauss–Bonnet action, for arbitrary vector fields. The existence of this solution is related to a two-vector family of transformations, that leaves the Gauss–Bonnet action invariant when acting on metric-compatible Connections. We argue that this solution is physically inequivalent to the Levi-Civita Connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of Connections.
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A non-trivial Connection for the metric-affine Gauss–Bonnet theory in D = 4
Physics Letters B, 2019Co-Authors: Bert Janssen, Alejandro Jiménez-cano, Jose Alberto OrejuelaAbstract:Abstract We study non-trivial (i.e. non-Levi-Civita) Connections in metric-affine Lovelock theories. First we study the projective invariance of general Lovelock actions and show that all Connections constructed by acting with a projective transformation of the Levi-Civita Connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl Connection is also a solution for the specific case of the four-dimensional metric-affine Gauss–Bonnet action, for arbitrary vector fields. The existence of this solution is related to a two-vector family of transformations, that leaves the Gauss–Bonnet action invariant when acting on metric-compatible Connections. We argue that this solution is physically inequivalent to the Levi-Civita Connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of Connections.
Nicola Tamanini - One of the best experts on this subject based on the ideXlab platform.
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Ghosts in pure and hybrid formalisms of gravity theories: a unified analysis
Physical Review D, 2013Co-Authors: Tomi S. Koivisto, Nicola TamaniniAbstract:In the first order formalism of gravitational theories, the spacetime Connection is considered as an independent variable to vary together with the metric. However, the metric still generates its Levi-Civita Connection that turns out to determine the geodesics of matter. Recently, "hybrid" gravity theories have been introduced by constructing actions involving both the independent Palatini Connection and the metric Levi-Civita Connection. In this study a method is developed to analyse the field content of such theories, in particular to determine whether the propagating degrees of freedom are ghosts or tachyons. New types of second, fourth and sixth order derivative gravity theories are investigated and the so called f(X) theories are singled out as a viable class of "hybrid" extensions of General Relativity.