scholarly journals Lifts of connections to the bundle of (1,1) type tensor frames

2021 ◽  
pp. 177-183
Author(s):  
Habil FATTAYEV
Keyword(s):  
Smooth Manifold ◽  
Linear Connection ◽  
Sasaki Metric ◽  
Geodesic Lines ◽  
Complete Lift

In this paper we consider the bundle of (1,1) type tensor frames over a smooth manifold, define the horizontal and complete lifts of symmetric linear connection into this bundle. Also we study the properties of the geodesic lines corresponding to the complete lift of the linear connection and investigate the relations between Sasaki metric and lifted connections on the bundle of (1,1) type tensor frames.

scholarly journals Prolongations of linear connections to the frame bundle

1983 ◽  
Vol 28 (3) ◽  
pp. 367-381
Author(s):  
Luis A. Cordero ◽  
Manuel de Leon
Keyword(s):  
Linear Connection ◽  
Frame Bundle ◽  
Linear Connections ◽  
Complete Lift

In this paper we construct the prolongation of a linear connection Γ on a manifold Μ to the bundle space of its frame bundle, and show that such prolongated connection coincides with the so-called complete lift of Γ to .

scholarly journals Parabolic conformally symplectic structures I; definition and distinguished connections

2018 ◽  
Vol 30 (3) ◽  
pp. 733-751 ◽  
Author(s):  
Andreas Čap ◽  
Tomáš Salač
Keyword(s):  
Lie Algebra ◽  
Geometric Structure ◽  
Tangent Bundle ◽  
Smooth Manifold ◽  
Symplectic Structure ◽  
Linear Connection ◽  
Underlying Structure ◽  
Normalization Condition ◽  
Lie Algebra Cohomology ◽  
First Order

AbstractWe introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type {C_{n}} and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle {\mathrm{TM}}. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

An acceleration condition for b-incomplete timelike curves

1981 ◽  
Vol 90 (1) ◽  
pp. 191-193 ◽  
Author(s):  
C. T. J. Dodson ◽  
L. J. Sulley ◽  
P. M. Williams
Keyword(s):  
Smooth Manifold ◽  
Linear Connection ◽  
Timelike Curve ◽  
Acceptable Method ◽  
Proper Length ◽  
Lorentz Norm

AbstractThe b-completion of a smooth manifold with linear connection was introduced by Schmidt (6). He pointed out that, in a spacetime, an inextensible timelike curve of finite proper length and bounded acceleration has an endpoint on the b-boundary. However, it is not clear in what sense boundedness was meant. Geroch(1) suggested that any acceptable method of completing a spacetime should have this property with boundedness interpreted through the Lorentz norm. This result for the b-completion was given by Rosso(4), based on his earlier paper (3), which, however, contains some arguments that we do not accept. We give another proof of the result and improve it by weakening the condition of boundedness of the acceleration.

Banach Lie Algebroids

2011 ◽  
Vol 57 (2) ◽  
pp. 409-416
Author(s):  
Mihai Anastasiei
Keyword(s):  
Differential Equations ◽  
Smooth Manifold ◽  
Vector Bundles ◽  
Second Order ◽  
Lie Algebroids ◽  
Second Order Differential Equations

Banach Lie AlgebroidsFirst, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.

Visualization of Enneper’s surface by line graphics

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 387-405 ◽  
Author(s):  
Vesna Velickovic
Keyword(s):  
Minimal Surface ◽  
Graphical Representations ◽  
Lines Of Curvature ◽  
Geodesic Lines

Here we study Enneper?s minimal surface and some of its properties. We compute and visualize the lines of self-intersection, lines of intersections with planes, lines of curvature, asymptotic and geodesic lines of Enneper?s surface. For the graphical representations of all the results we use our own software for line graphics.

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 Gauging the higher-spin-like symmetries by the Moyal product

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
M. Cvitan ◽  
P. Dominis Prester ◽  
S. Giaccari ◽  
M. Paulišić ◽  
I. Vuković
Keyword(s):  
Linear Connection ◽  
Covariant Formulation ◽  
Formal Similarity ◽  
Commutative Field ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Higher Derivative ◽  
Novel Approach ◽  
Emergent Geometry ◽  
Higher Spin

Abstract We analyze a novel approach to gauging rigid higher derivative (higher spin) symmetries of free relativistic actions defined on flat spacetime, building on the formalism originally developed by Bonora et al. and Bekaert et al. in their studies of linear coupling of matter fields to an infinite tower of higher spin fields. The off-shell definition is based on fields defined on a 2d-dimensional master space equipped with a symplectic structure, where the infinite dimensional Lie algebra of gauge transformations is given by the Moyal commutator. Using this algebra we construct well-defined weakly non-local actions, both in the gauge and the matter sector, by mimicking the Yang-Mills procedure. The theory allows for a description in terms of an infinite tower of higher spin spacetime fields only on-shell. Interestingly, Euclidean theory allows for such a description also off-shell. Owing to its formal similarity to non-commutative field theories, the formalism allows for the introduction of a covariant potential which plays the role of the generalised vielbein. This covariant formulation uncovers the existence of other phases and shows that the theory can be written in a matrix model form. The symmetries of the theory are analyzed and conserved currents are explicitly constructed. By studying the spin-2 sector we show that the emergent geometry is closely related to teleparallel geometry, in the sense that the induced linear connection is opposite to Weitzenböck’s.

scholarly journals Pseudoinversion of degenerate metrics

2003 ◽  
Vol 2003 (55) ◽  
pp. 3479-3501 ◽  
Author(s):  
C. Atindogbe ◽  
J.-P. Ezin ◽  
Joël Tossa
Keyword(s):  
Differential Geometry ◽  
Large Class ◽  
Smooth Manifold ◽  
Differential Operators ◽  
Type Operator ◽  
Lorentzian Space ◽  
Lightlike Hypersurface ◽  
Definition Of ◽  
Lightlike Hypersurfaces ◽  
Theoretical Results

Let(M,g)be a smooth manifoldMendowed with a metricg. A large class of differential operators in differential geometry is intrinsically defined by means of the dual metricg∗on the dual bundleTM∗of 1-forms onM. If the metricgis (semi)-Riemannian, the metricg∗is just the inverse ofg. This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for whichg∗is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian spaceℝ1n+2.

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