scholarly journals On the Koszul formula in noncommutative geometry

2020 ◽  
Vol 32 (10) ◽  
pp. 2050032 ◽  
Author(s):  
Jyotishman Bhowmick ◽  
Debashish Goswami ◽  
Giovanni Landi
Keyword(s):  
Scalar Curvature ◽  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Fuzzy Sphere ◽  
Civita Connection ◽  
Canonical Metric ◽  
Levi Civita Connection

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical metric.

scholarly journals Levi-Civita connections for conformally deformed metrics on tame differential calculi

2021 ◽  
pp. 2150088
Author(s):  
Jyotishman Bhowmick ◽  
Debashish Goswami ◽  
Soumalya Joardar
Keyword(s):  
Scalar Curvature ◽  
Differential Calculus ◽  
Invertible Element ◽  
Conformal Deformation ◽  
Noncommutative Algebra ◽  
Connection Formula ◽  
Civita Connection ◽  
Canonical Metric ◽  
Negative Constant ◽  
Levi Civita Connection

Given a tame differential calculus over a noncommutative algebra [Formula: see text] and an [Formula: see text]-bilinear metric [Formula: see text] consider the conformal deformation [Formula: see text] [Formula: see text] being an invertible element of [Formula: see text] We prove that there exists a unique connection [Formula: see text] on the bimodule of one-forms of the differential calculus which is torsionless and compatible with [Formula: see text] We derive a concrete formula connecting [Formula: see text] and the Levi-Civita connection for the metric [Formula: see text] As an application, we compute the Ricci and scalar curvatures for a general conformal perturbation of the canonical metric on the noncommutative [Formula: see text]-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.

Certain inequalities for the Casorati curvatures of submanifolds of generalized (k,μ)-space-forms

2018 ◽  
Vol 13 (02) ◽  
pp. 2050040
Author(s):  
Shyamal Kumar Hui ◽  
Pradip Mandal ◽  
Ali H. Alkhaldi ◽  
Tanumoy Pal
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Space Forms ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
Casorati Curvature ◽  
Optimal Inequalities

The paper deals with the study of Casorati curvature of submanifolds of generalized [Formula: see text]-space-form with respect to Levi-Civita connection as well as semisymmetric metric connection and derived two optimal inequalities between scalar curvature and Casorati curvature of such space forms. The equality cases are also considered.

scholarly journals Totally real submanifolds of (LCS)n-manifolds

2018 ◽  
Vol 33 (2) ◽  
pp. 141
Author(s):  
Shyamal Kumar Hui ◽  
Tanumoy Pal
Keyword(s):  
Scalar Curvature ◽  
Real Submanifolds ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
Totally Real

The present paper deals with the study of totally real submanifolds and C-totally real submanifolds of (LCS)n-manifolds withrespect to Levi-Civita connection as well as quarter symmetric metric connection. It is proved that scalar curvature of C-totally real submanifolds of (LCS)n-manifold with respect to both the said connections are same.

scholarly journals A quarter-symmetric metric connection on almost contact B-metric manifolds

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5181-5190
Author(s):  
Şenay Bulut
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection

The aim of this paper is to study the notion of a quarter-symmetric metric connection on an almost contact B-metric manifold (M,?,?,?,g). We obtain the relation between the Levi-Civita connection and the quarter-symmetric metric connection on (M,?,?,?,g).We investigate the curvature tensor, Ricci tensor and scalar curvature tensor with respect to the quarter-symmetric metric connection. In case the manifold (M,?,?,?,g) is a Sasaki-like almost contact B-metric manifold, we get some formulas. Finally, we give some examples of a quarter-symmetric metric connection.

scholarly journals Curvature Invariants of Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature

Entropy ◽  
2018 ◽  
Vol 20 (7) ◽  
pp. 529 ◽  
Author(s):  
Simona Decu ◽  
Stefan Haesen ◽  
Leopold Verstraelen ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Totally Geodesic ◽  
Curvature Invariants ◽  
Statistical Manifolds ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Curvature Tensors

In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. For such submanifold, we investigate curvature properties. We establish some inequalities involving the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). Moreover, we prove that the equality cases of the inequalities hold if and only if the imbedding curvature tensors h and h∗ of the submanifold (associated with the dual connections) satisfy h=−h∗, i.e., the submanifold is totally geodesic with respect to the Levi–Civita connection.

scholarly journals Missing the point in noncommutative geometry

Synthese ◽  
2021 ◽  
Author(s):  
Nick Huggett ◽  
Fedele Lizzi ◽  
Tushar Menon
Keyword(s):  
Scalar Field ◽  
Noncommutative Geometry ◽  
Smooth Manifold ◽  
Operational Definition ◽  
Spectral Triple ◽  
Fundamental Quantity

AbstractNoncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

scholarly journals Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

2021 ◽  
Author(s):  
V. Cortés ◽  
A. Saha ◽  
D. Thung
Keyword(s):  
Curvature Tensor ◽  
Decomposition Theorem ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Cohomogeneity One ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

ON THE GAUSS–BONNET FORMULA IN RIEMANN–FINSLER GEOMETRY

2002 ◽  
Vol 34 (3) ◽  
pp. 329-340 ◽  
Author(s):  
BRAD LACKEY
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Euler Class ◽  
Constant Volume ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Riemannian Case ◽  
Torsion Free

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

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