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20 pages, 311 KiB

Open AccessArticle

Gauss–Bonnet Theorem Related to the Semi-Symmetric Metric Connection in the Heisenberg Group

by Haiming Liu and Song Peng

Symmetry 2024, 16(6), 762; https://doi.org/10.3390/sym16060762 - 18 Jun 2024

Viewed by 1332

In this paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a [...] Read more.

In this paper, we introduce the notion of the semi-symmetric metric connection in the Heisenberg group. Moreover, by using the method of Riemannian approximations, we define the notions of intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on a surface, and the intrinsic Gaussian curvature of the surface away from characteristic points in the Heisenberg group with the semi-symmetric metric connection. Finally, we derive the expressions of those curvatures and prove the Gauss–Bonnet theorem related to the semi-symmetric metric connection in the Heisenberg group. Full article

(This article belongs to the Section Mathematics)

25 pages, 359 KiB

Open AccessArticle

Sub-Riemannian Geometry of Curves and Surfaces in Roto-Translation Group Associated with Canonical Connection

by Han Zhang and Haiming Liu

Mathematics 2024, 12(11), 1683; https://doi.org/10.3390/math12111683 - 28 May 2024

Viewed by 1037

Abstract

The aim of this paper is to obtain the sub-Riemannian properties of the roto-translation group

R T

. At the same time, we compute the sub-Riemannian limits of Gaussian curvature associated with two kinds of canonical connections for a

C^{2}

-smooth surface [...] Read more.

The aim of this paper is to obtain the sub-Riemannian properties of the roto-translation group

R T

. At the same time, we compute the sub-Riemannian limits of Gaussian curvature associated with two kinds of canonical connections for a

C^{2}

-smooth surface in the roto-translation group away from characteristic points and signed geodesic curvature associated with two kinds of canonical connections for

C^{2}

-smooth curves on surfaces. Based on these results, we obtain a Gauss-Bonnet theorem in the

R T

. Full article

(This article belongs to the Section B: Geometry and Topology)

82 pages, 748 KiB

Open AccessArticle

C-R Immersions and Sub-Riemannian Geometry

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Axioms 2023, 12(4), 329; https://doi.org/10.3390/axioms12040329 - 28 Mar 2023

Cited by 1 | Viewed by 2220

Abstract

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form

θ

, we consider natural

ϵ

-contractions, i.e., contractions

g_{ϵ}^{M}

of the Levi form

G_{θ}

, such that the norm [...] Read more.

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form

θ

, we consider natural

ϵ

-contractions, i.e., contractions

g_{ϵ}^{M}

of the Levi form

G_{θ}

, such that the norm of the Reeb vector field T of

(M, θ)

is of order

O (ϵ^{- 1})

. We study isopseudohermitian (i.e.,

f^{*} Θ = θ

) Cauchy–Riemann immersions

f : M \to (A, Θ)

between strictly pseudoconvex CR manifolds M and A, where

Θ

is a contact form on A. For every contraction

g_{ϵ}^{A}

of the Levi form

G_{Θ}

, we write the embedding equations for the immersion

f : M \to (A, g_{ϵ}^{A})

. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as

ϵ \to 0^{+}

. For every isopseudohermitian immersion

f : M \to S^{2 N + 1}

into a sphere

S^{2 N + 1} \subset C^{N + 1}

, we show that Webster’s pseudohermitian scalar curvature R of

(M, θ)

satisfies the inequality

R \leq 2 n [(f^{*} g_{Θ}) (T, T) + n + 1] + \frac{1}{2} {∥ H (f) ∥_{g_{Θ}^{f}}^{2} + ∥ {trace}_{G_{θ}} Π_{H (M)} (\nabla^{⊤} - \nabla) ∥_{f^{*} g_{Θ}}^{2}}

with equality if and only if

B (f) = 0

and

\nabla^{⊤} = \nabla

on

H (M) \otimes H (M)

. This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)

20 pages, 380 KiB

Open AccessReview

Integral Formulas for Almost Product Manifolds and Foliations

by Vladimir Rovenski

Mathematics 2022, 10(19), 3645; https://doi.org/10.3390/math10193645 - 5 Oct 2022

Cited by 2 | Viewed by 1625

Abstract

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying [...] Read more.

Integral formulas are powerful tools used to obtain global results in geometry and analysis. The integral formulas for almost multi-product manifolds, foliations and multiply twisted products of Riemannian, metric-affine and sub-Riemannian manifolds, to which this review paper is devoted, are useful for studying such problems as (i) the existence and characterization of foliations with a given geometric property, such as being totally geodesic, minimal or totally umbilical; (ii) prescribing the generalized mean curvatures of the leaves of a foliation; (iii) minimizing volume-like functionals defined for tensors on foliated manifolds. We start from the series of integral formulas for codimension one foliations of Riemannian and metric-affine manifolds, and then we consider integral formulas for regular and singular foliations of arbitrary codimension. In the second part of the article, we represent integral formulas with the mixed scalar curvature of an almost multi-product structure on Riemannian and metric-affine manifolds, give applications to hypersurfaces of space forms with

k = 2, 3

distinct principal curvatures of constant multiplicities and then discuss integral formulas for foliations or distributions on sub-Riemannian manifolds. Full article

(This article belongs to the Section E: Applied Mathematics)

10 pages, 3143 KiB

Open AccessArticle

Liouville Integrability in a Four-Dimensional Model of the Visual Cortex

by Ivan Galyaev and Alexey Mashtakov

J. Imaging 2021, 7(12), 277; https://doi.org/10.3390/jimaging7120277 - 17 Dec 2021

Cited by 1 | Viewed by 2229

Abstract

We consider a natural extension of the Petitot–Citti–Sarti model of the primary visual cortex. In the extended model, the curvature of contours is taken into account. The occluded contours are completed via sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and [...] Read more.

We consider a natural extension of the Petitot–Citti–Sarti model of the primary visual cortex. In the extended model, the curvature of contours is taken into account. The occluded contours are completed via sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and curvatures. Here,

M = R^{2} \times SO (2) \times R

models the configuration space of neurons of the visual cortex. We study the problem of sub-Riemannian geodesics on M via methods of geometric control theory. We prove complete controllability of the system and the existence of optimal controls. By application of the Pontryagin maximum principle, we derive a Hamiltonian system that describes the geodesics. We obtain the explicit parametrization of abnormal extremals. In the normal case, we provide three functionally independent first integrals. Numerical simulations indicate the existence of one more first integral that results in Liouville integrability of the system. Full article

(This article belongs to the Special Issue Mathematical Modeling of Human Vision and Its Application to Image Processing)

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21 pages, 2435 KiB

Open AccessArticle

Multi-Frequency Image Completion via a Biologically-Inspired Sub-Riemannian Model with Frequency and Phase

by Emre Baspinar

J. Imaging 2021, 7(12), 271; https://doi.org/10.3390/jimaging7120271 - 9 Dec 2021

Cited by 2 | Viewed by 2591

Abstract

We present a novel cortically-inspired image completion algorithm. It uses five-dimensional sub-Riemannian cortical geometry, modeling the orientation, spatial frequency and phase-selective behavior of the cells in the visual cortex. The algorithm extracts the orientation, frequency and phase information existing in a given two-dimensional [...] Read more.

We present a novel cortically-inspired image completion algorithm. It uses five-dimensional sub-Riemannian cortical geometry, modeling the orientation, spatial frequency and phase-selective behavior of the cells in the visual cortex. The algorithm extracts the orientation, frequency and phase information existing in a given two-dimensional corrupted input image via a Gabor transform and represents those values in terms of cortical cell output responses in the model geometry. Then, it performs completion via a diffusion concentrated in a neighborhood along the neural connections within the model geometry. The diffusion models the activity propagation integrating orientation, frequency and phase features along the neural connections. Finally, the algorithm transforms the diffused and completed output responses back to the two-dimensional image plane. Full article

(This article belongs to the Special Issue Mathematical Modeling of Human Vision and Its Application to Image Processing)

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11 pages, 280 KiB

Open AccessArticle

Integral Formulas for a Foliation with a Unit Normal Vector Field

by Vladimir Rovenski

Mathematics 2021, 9(15), 1764; https://doi.org/10.3390/math9151764 - 26 Jul 2021

Cited by 3 | Viewed by 1976

Abstract

In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation

F

and a unit vector field N orthogonal to

F

, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral [...] Read more.

In this article, we prove integral formulas for a Riemannian manifold equipped with a foliation

F

and a unit vector field N orthogonal to

F

, and generalize known integral formulas (due to Brito-Langevin-Rosenberg and Andrzejewski-Walczak) for foliations of codimension one. Our integral formulas involve Newton transformations of the shape operator of

F

with respect to N and the curvature tensor of the induced connection on the distribution

D = T F \oplus span (N)

, and this decomposition of

D

can be regarded as a codimension-one foliation of a sub-Riemannian manifold. We apply our formulas to foliated (sub-)Riemannian manifolds with restrictions on the curvature and extrinsic geometry of the foliation. Full article

(This article belongs to the Section E: Applied Mathematics)

6 pages, 533 KiB

Open AccessEditorial

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

by Frédéric Barbaresco and Jean-Pierre Gazeau

Entropy 2019, 21(3), 250; https://doi.org/10.3390/e21030250 - 6 Mar 2019

Cited by 2 | Viewed by 4349

Abstract

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern [...] Read more.

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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21 pages, 2983 KiB

Open AccessArticle

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

by Stefan Sommer

Entropy 2016, 18(12), 425; https://doi.org/10.3390/e18120425 - 26 Nov 2016

Cited by 10 | Viewed by 6470

Abstract

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be [...] Read more.

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic development is mapped to the manifold. We discuss how the paths are projections of geodesics for a sub-Riemannian metric on the frame bundle of the manifold, and how the curvature of the underlying connection appears in the sub-Riemannian Hamilton–Jacobi equations. Evolution equations for both metric and cometric formulations of the sub-Riemannian metric are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we relate the paths to geodesics and polynomials in Riemannian geometry. Examples from the family of paths are visualized on embedded surfaces, and we explore computational representations on finite dimensional landmark manifolds with geometry induced from right-invariant metrics on diffeomorphism groups. Full article

(This article belongs to the Special Issue Differential Geometrical Theory of Statistics)

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