Research

16 pages, 265 KiB

Open AccessArticle

by Sharief Deshmukh, Josef Mikeš, Nasser Bin Turki and Gabriel-Eduard Vîlcu

Mathematics 2020, 8(10), 1663; https://doi.org/10.3390/math8101663 - 27 Sep 2020

Cited by 6 | Viewed by 1994

The concircularity property of vector fields implies the geodesicity property, while the converse of this statement is not true. The main objective of this note is to find conditions under which the concircularity and geodesicity properties of vector fields are equivalent. Moreover, it [...] Read more.

The concircularity property of vector fields implies the geodesicity property, while the converse of this statement is not true. The main objective of this note is to find conditions under which the concircularity and geodesicity properties of vector fields are equivalent. Moreover, it is shown that the geodesicity property of vector fields is also useful in characterizing not only spheres, but also Euclidean spaces. Full article

(This article belongs to the Special Issue Geometric Structures and Interdisciplinary Applications)

14 pages, 805 KiB

Open AccessArticle

Classifications of Canal Surfaces with the Gauss Maps in Minkowski 3-Space

by Jinhua Qian, Xueqian Tian, Xueshan Fu and Young Ho Kim

Mathematics 2020, 8(9), 1453; https://doi.org/10.3390/math8091453 - 30 Aug 2020

Cited by 2 | Viewed by 1413

Abstract

In this work, we study the canal surfaces foliated by pseudo spheres

S_{1}^{2}

along a Frenet curve in terms of their Gauss maps in Minkowski 3-space. Such kind of surfaces with pointwise 1-type Gauss maps are classified completely. For example, the [...] Read more.

In this work, we study the canal surfaces foliated by pseudo spheres

S_{1}^{2}

along a Frenet curve in terms of their Gauss maps in Minkowski 3-space. Such kind of surfaces with pointwise 1-type Gauss maps are classified completely. For example, the canal surface with proper pointwise 1-type Gauss map of the first kind if and only if it is a part of a minimal surface of revolution. Full article

(This article belongs to the Special Issue Geometric Structures and Interdisciplinary Applications)

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6 pages, 248 KiB

Open AccessArticle

Measure-Expansive Homoclinic Classes for C¹ Generic Flows

by Manseob Lee

Mathematics 2020, 8(8), 1232; https://doi.org/10.3390/math8081232 - 27 Jul 2020

Cited by 2 | Viewed by 1382

Abstract

In this paper, we prove that for a generically

C^{1}

vector field X of a compact smooth manifold M, if a homoclinic class

H (γ, X)

which contains a hyperbolic closed orbit

γ

is measure expansive for X [...] Read more.

In this paper, we prove that for a generically

C^{1}

vector field X of a compact smooth manifold M, if a homoclinic class

H (γ, X)

which contains a hyperbolic closed orbit

γ

is measure expansive for X then

H (γ, X)

is hyperbolic. Full article

(This article belongs to the Special Issue Geometric Structures and Interdisciplinary Applications)

11 pages, 257 KiB

Open AccessArticle

The Curve Shortening Flow in the Metric-Affine Plane

by Vladimir Rovenski

Mathematics 2020, 8(5), 701; https://doi.org/10.3390/math8050701 - 02 May 2020

Viewed by 1604

Abstract

We investigated, for the first time, the curve shortening flow in the metric-affine plane and prove that under simple geometric condition (when the curvature of initial curve dominates the torsion term) it shrinks a closed convex curve to a “round point” in finite [...] Read more.

We investigated, for the first time, the curve shortening flow in the metric-affine plane and prove that under simple geometric condition (when the curvature of initial curve dominates the torsion term) it shrinks a closed convex curve to a “round point” in finite time. This generalizes the classical result by M. Gage and R.S. Hamilton about convex curves in a Euclidean plane. Full article

(This article belongs to the Special Issue Geometric Structures and Interdisciplinary Applications)

18 pages, 357 KiB

Open AccessArticle

The Weitzenböck Type Curvature Operator for Singular Distributions

by Paul Popescu, Vladimir Rovenski and Sergey Stepanov

Mathematics 2020, 8(3), 365; https://doi.org/10.3390/math8030365 - 06 Mar 2020

Cited by 1 | Viewed by 1618

Abstract

We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior [...] Read more.

We study geometry of a Riemannian manifold endowed with a singular (or regular) distribution, determined as an image of the tangent bundle under smooth endomorphisms. Following construction of an almost Lie algebroid on a vector bundle, we define the modified covariant and exterior derivatives and their

L^{2}

adjoint operators on tensors. Then, we introduce the Weitzenböck type curvature operator on tensors, prove the Weitzenböck type decomposition formula, and derive the Bochner–Weitzenböck type formula. These allow us to obtain vanishing theorems about the null space of the Hodge type Laplacian. The assumptions used in the results are reasonable, as illustrated by examples with f-manifolds, including almost Hermitian and almost contact ones. Full article

(This article belongs to the Special Issue Geometric Structures and Interdisciplinary Applications)

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