A Flat Nonmetrizable Connexion

H. G. Helfenstein

doi:10.4153/cmb-1972-037-2

A Flat Nonmetrizable Connexion

Canadian Mathematical Bulletin ◽

10.4153/cmb-1972-037-2 ◽

1972 ◽

Vol 15 (2) ◽

pp. 205-206

Author(s):

H. G. Helfenstein

Keyword(s):

Smooth Manifold ◽

Polar Coordinates ◽

Riemannian Structure

A linear connexion ∇ on a smooth manifold M with vanishing torsion and curvature is called flat. It is well known that such a connexion need not be the Levi-Civita connexion of a Riemannian (nor pseudo-Riemannian) structure (cf. [1]).The following example with M homeomorphic to a cylinder is particularly simple. Using polar coordinates the punctured plane M=R2\{0} is covered by a smooth atlas with two charts

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PARTIAL ISOMETRIES OF A SUB-RIEMANNIAN MANIFOLD

International Journal of Mathematics ◽

10.1142/s0129167x12500437 ◽

2012 ◽

Vol 23 (02) ◽

pp. 1250043

Author(s):

MAHUYA DATTA

Keyword(s):

Riemannian Manifold ◽

Tangent Bundle ◽

Smooth Manifold ◽

Partial Isometry ◽

Riemannian Metric ◽

Riemannian Structure ◽

Partial Isometries ◽

Smooth Map

In this article, we obtain the following generalization of isometric C1-immersion theorem of Nash and Kuiper. Let M be a smooth manifold of dimension m and H a rank k subbundle of the tangent bundle TM with a Riemannian metric gH. Then the pair (H, gH) defines a sub-Riemannian structure on M. We call a C1-map f : (M, H, gH) → (N, h) into a Riemannian manifold (N, h) a partial isometry if the derivative map df restricted to H is isometric, that is if f*h|H = gH. We prove that if f0 : M → N is a smooth map such that df0|H is a bundle monomorphism and [Formula: see text], then f0 can be homotoped to a C1-map f : M → N which is a partial isometry, provided dim N > k. As a consequence of this result, we obtain that every sub-Riemannian manifold (M, H, gH) admits a partial isometry in ℝn, provided n ≥ m + k.

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Non-Polar Variation of Latitude

International Astronomical Union Colloquium ◽

10.1017/s0252921100026890 ◽

1972 ◽

Vol 1 ◽

pp. 93-101 ◽

Cited By ~ 1

Author(s):

S. Yumi

Keyword(s):

Polar Motion ◽

Polar Coordinates ◽

Normal Variation ◽

Latitude Variation ◽

Local Trend

ABSTRACTAnalysing the residual latitude of the station, local trend in latitude variation other than by the polar motion was found.Residual latitude was calculated for each of 26 stations which gave the continuous records of observation during 6 years comprising — 1962 — 1967 as a difference between observed variation of latitude and – normal variation calculated by the polar coordinates Iderived from all the results of 26 stations.As far as the results during these six years are concerned, local trend at any station it seemed to be expressed in terms of 3λ.Assumed effect of local trend on the coordinates values of the instantaneous pole is also discussed.

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The solution problem of Dirihle for equation of Laplace in polar coordinates by method Monte Carlo

Applied Questions of Mathematical Modeling ◽

10.32782/2618-0340-2018-2-114-122 ◽

2018 ◽

Vol 2 ◽

pp. 114-122

Author(s):

Yu.I. Nikolayenko ◽

◽

V.G. Ilvovsky ◽

S.V. Moiseenko ◽

◽

...

Keyword(s):

Monte Carlo ◽

Polar Coordinates ◽

Solution Problem

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Banach Lie Algebroids

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/v10157-011-0035-y ◽

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.

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Gauge invariance, polar coordinates and inflation

Journal of High Energy Physics ◽

10.1007/jhep03(2021)096 ◽

2021 ◽

Vol 2021 (3) ◽

Cited By ~ 1

Author(s):

Ali Akil ◽

Xi Tong

Keyword(s):

Coordinate System ◽

Effective Potential ◽

Gauge Invariance ◽

Polar Coordinates ◽

Quantum Corrections ◽

Background Current ◽

Gauge Invariant ◽

Gauge Dependence ◽

Current Term ◽

Invariant Current

Abstract We point out the necessity of resolving the apparent gauge dependence in the quantum corrections of cosmological observables for Higgs-like inflation models. We highlight the fact that this gauge dependence is due to the use of an asymmetric background current which is specific to a choice of coordinate system in the scalar manifold. Favoring simplicity over complexity, we further propose a practical shortcut to gauge-independent inflationary observables by using effective potential obtained from a polar-like background current choice. We demonstrate this shortcut for several explicit examples and present a gauge-independent prediction of inflationary observables in the Abelian Higgs model. Furthermore, with Nielsen’s gauge dependence identities, we show that for any theory to all orders, a gauge-invariant current term gives a gauge-independent effective potential and thus gauge-invariant inflationary observables.

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Approach to hand posture recognition based on hand shape features for human–robot interaction

Complex & Intelligent Systems ◽

10.1007/s40747-021-00333-w ◽

2021 ◽

Author(s):

Jing Qi ◽

Kun Xu ◽

Xilun Ding

Keyword(s):

Gaussian Mixture ◽

Human Robot Interaction ◽

Polar Coordinates ◽

Support Vector ◽

Hand Posture ◽

Data Set ◽

Hand Shape ◽

Hand Posture Recognition ◽

Hand Segmentation ◽

Posture Recognition

AbstractHand segmentation is the initial step for hand posture recognition. To reduce the effect of variable illumination in hand segmentation step, a new CbCr-I component Gaussian mixture model (GMM) is proposed to detect the skin region. The hand region is selected as a region of interest from the image using the skin detection technique based on the presented CbCr-I component GMM and a new adaptive threshold. A new hand shape distribution feature described in polar coordinates is proposed to extract hand contour features to solve the false recognition problem in some shape-based methods and effectively recognize the hand posture in cases when different hand postures have the same number of outstretched fingers. A multiclass support vector machine classifier is utilized to recognize the hand posture. Experiments were carried out on our data set to verify the feasibility of the proposed method. The results showed the effectiveness of the proposed approach compared with other methods.

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Missing the point in noncommutative geometry

Synthese ◽

10.1007/s11229-020-02998-1 ◽

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.

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