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14 pages, 318 KiB

Open AccessArticle

Spherical Ruled Surfaces in S³ Characterized by the Spherical Gauss Map

by Young Ho Kim and Sun Mi Jung

Mathematics 2020, 8(12), 2106; https://doi.org/10.3390/math8122106 - 25 Nov 2020

Cited by 1 | Viewed by 1366

Abstract

The Laplace operator on a Riemannian manifold plays an important role with eigenvalue problems and the spectral theory. Extending such an eigenvalue problem of smooth maps including the Gauss map, the notion of finite-type was introduced. The simplest finite-type is of 1-type. In [...] Read more.

The Laplace operator on a Riemannian manifold plays an important role with eigenvalue problems and the spectral theory. Extending such an eigenvalue problem of smooth maps including the Gauss map, the notion of finite-type was introduced. The simplest finite-type is of 1-type. In particular, the spherical Gauss map is defined in a very natural way on spherical submanifolds. In this paper, we study ruled surfaces of the 3-dimensional sphere with generalized 1-type spherical Gauss map which generalizes the notion of 1-type. The classification theorem of ruled surfaces of the sphere with the spherical Gauss map of generalized 1-type is completed. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

5 pages, 342 KiB

Open AccessArticle

When Will a Sequence of Points in a Riemannian Submanifold Converge?

by Tuyen Trung Truong

Mathematics 2020, 8(11), 1934; https://doi.org/10.3390/math8111934 - 3 Nov 2020

Viewed by 1328

Abstract

Let X be a Riemannian manifold and

x_{n}

a sequence of points in X. Assume that we know a priori some properties of the set A of cluster points of

x_{n}

. The question is under what conditions that

x_{n}

[...] Read more.

Let X be a Riemannian manifold and

x_{n}

a sequence of points in X. Assume that we know a priori some properties of the set A of cluster points of

x_{n}

. The question is under what conditions that

x_{n}

will converge. An answer to this question serves to understand the convergence behaviour for iterative algorithms for (constrained) optimisation problems, with many applications such as in Deep Learning. We will explore this question, and show by some examples that having X a submanifold (more generally, a metric subspace) of a good Riemannian manifold (even in infinite dimensions) can greatly help. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

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16 pages, 821 KiB

Open AccessArticle

A Pinching Theorem for Compact Minimal Submanifolds in Warped Products

I \times_{f} S^{m} (c)

by Xin Zhan and Zhonghua Hou

Mathematics 2020, 8(9), 1445; https://doi.org/10.3390/math8091445 - 28 Aug 2020

Viewed by 1337

Abstract

Let

S^{m} (c)

be a Euclidean sphere of curvature

c > 0

and

R

be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type [...] Read more.

Let

S^{m} (c)

be a Euclidean sphere of curvature

c > 0

and

R

be a Euclidean line. We prove a pinching theorem for compact minimal submanifolds immersed in Riemannian warped products of the type

I \times_{f} S^{m} (c)

, where

f : I \to R^{+}

is a smooth positive function on an open interval I of

R

. This allows us to generalize Chen-Cui’s pinching theorem from Riemannian products

S^{m} (c) \times R

to Riemannian warped products

I \times_{f} S^{m} (c)

. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

8 pages, 250 KiB

Open AccessArticle

H-Umbilical Lagrangian Submanifolds of the Nearly Kähler \( {\mathbb{S}^3\times\mathbb{S}^3} \)

by Miroslava Antić, Marilena Moruz and Joeri Van der Veken

Mathematics 2020, 8(9), 1427; https://doi.org/10.3390/math8091427 - 26 Aug 2020

Cited by 12 | Viewed by 1325

Abstract

H-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that, in the homogeneous nearly Kähler

S^{3} \times S^{3}

, [...] Read more.

H-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that, in the homogeneous nearly Kähler

S^{3} \times S^{3}

, also H-umbilical Lagrangian submanifolds are automatically totally geodesic. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

12 pages, 235 KiB

Open AccessArticle

Pseudo-Isotropic Centro-Affine Lorentzian Surfaces

by Olivier Birembaux

Mathematics 2020, 8(8), 1284; https://doi.org/10.3390/math8081284 - 4 Aug 2020

Cited by 5 | Viewed by 1322

Abstract

In this paper, we study centro-affine Lorentzian surfaces

M^{2}

in

ℝ^{3}

which have pseudo-isotropic or lightlike pseudo-isotropic difference tensor. We first show that

M^{2}

is pseudo-isotropic if and only if the Tchebychev form

T = 0

. In that case, [...] Read more.

In this paper, we study centro-affine Lorentzian surfaces

M^{2}

in

ℝ^{3}

which have pseudo-isotropic or lightlike pseudo-isotropic difference tensor. We first show that

M^{2}

is pseudo-isotropic if and only if the Tchebychev form

T = 0

. In that case,

M^{2}

is a an equi-affine sphere. Next, we will get a complete classification of centro-affine Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

24 pages, 376 KiB

Open AccessArticle

Five-Dimensional Contact CR-Submanifolds in S 7 ( 1 )

by Mirjana Djorić and Marian Ioan Munteanu

Mathematics 2020, 8(8), 1278; https://doi.org/10.3390/math8081278 - 3 Aug 2020

Cited by 1 | Viewed by 2083

Abstract

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure

(φ, ξ, η)

, we study its five-dimensional contact

C R

-submanifolds, which are the analogue of

C R

[...] Read more.

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure

(φ, ξ, η)

, we study its five-dimensional contact

C R

-submanifolds, which are the analogue of

C R

-submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field

ξ

is tangent to M, the tangent bundle of contact

C R

-submanifold M can be decomposed as

T (M) = H (M) \oplus E (M) \oplus R ξ,

where

H (M)

is invariant and

E (M)

is anti-invariant with respect to

φ

. On this occasion we obtain a complete classification of five-dimensional proper contact

C R

-submanifolds in

S^{7} (1)

whose second fundamental form restricted to

H (M)

and

E (M)

vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

18 pages, 284 KiB

Open AccessArticle

Almost Complex Surfaces in the Nearly Kähler SL(2,ℝ) × SL(2,ℝ)

by Elsa Ghandour and Luc Vrancken

Mathematics 2020, 8(7), 1160; https://doi.org/10.3390/math8071160 - 15 Jul 2020

Cited by 2 | Viewed by 1503

Abstract

The space

S L (2, R) \times S L (2, R)

admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally [...] Read more.

The space

S L (2, R) \times S L (2, R)

admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

15 pages, 1055 KiB

Open AccessArticle

Dual Associate Null Scrolls with Generalized 1-Type Gauss Maps

by Jinhua Qian, Xueshan Fu and Seoung Dal Jung

Mathematics 2020, 8(7), 1111; https://doi.org/10.3390/math8071111 - 6 Jul 2020

Cited by 1 | Viewed by 1479

Abstract

In this work, a pair of dual associate null scrolls are defined from the Cartan Frenet frame of a null curve in Minkowski 3-space. The fundamental geometric properties of the dual associate null scrolls are investigated and they are related in terms of [...] Read more.

In this work, a pair of dual associate null scrolls are defined from the Cartan Frenet frame of a null curve in Minkowski 3-space. The fundamental geometric properties of the dual associate null scrolls are investigated and they are related in terms of their Gauss maps, especially the generalized 1-type Gauss maps. At the same time, some representative examples are given and their graphs are plotted by the aid of a software programme. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

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10 pages, 249 KiB

Open AccessArticle

A Remarkable Property of Concircular Vector Fields on a Riemannian Manifold

by Ibrahim Al-Dayel, Sharief Deshmukh and Olga Belova

Mathematics 2020, 8(4), 469; https://doi.org/10.3390/math8040469 - 29 Mar 2020

Cited by 14 | Viewed by 2008

Abstract

In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold

(M, g)

with potential function f, there exists a unique smooth function

ρ

on M that connects u to the gradient of [...] Read more.

In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold

(M, g)

with potential function f, there exists a unique smooth function

ρ

on M that connects u to the gradient of potential function

\nabla f

. We call the connecting function of the concircular vector field u. This connecting function is shown to be a main ingredient in obtaining characterizations of n-sphere

S^{n} (c)

and the Euclidean space

E^{n}

. We also show that the connecting function influences on a topology of the Riemannian manifold. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

13 pages, 1004 KiB

Open AccessArticle

Darboux Associated Curves of a Null Curve on Pseudo-Riemannian Space Forms

by Jinhua Qian, Xueshan Fu and Seoung Dal Jung

Mathematics 2020, 8(3), 395; https://doi.org/10.3390/math8030395 - 11 Mar 2020

Cited by 1 | Viewed by 1898

Abstract

In this work, the Darboux associated curves of a null curve on pseudo-Riemannian space forms, i.e., de-Sitter space, hyperbolic space and a light-like cone in Minkowski 3-space are defined. The relationships of such partner curves are revealed including the relationship of their Frenet [...] Read more.

In this work, the Darboux associated curves of a null curve on pseudo-Riemannian space forms, i.e., de-Sitter space, hyperbolic space and a light-like cone in Minkowski 3-space are defined. The relationships of such partner curves are revealed including the relationship of their Frenet frames and the curvatures. Furthermore, the Darboux associated curves of k-type null helices are characterized and the conclusion that a null curve and its self-associated curve share the same Darboux associated curve is obtained. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

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14 pages, 975 KiB

Open AccessArticle

Some Characterizations of Generalized Null Scrolls

by Jinhua Qian, Xueshan Fu and Seoung Dal Jung

Mathematics 2019, 7(10), 931; https://doi.org/10.3390/math7100931 - 10 Oct 2019

Cited by 4 | Viewed by 1727

Abstract

In this work, a family of ruled surfaces named generalized null scrolls in Minkowski 3-space are investigated via the defined structure functions. The relations between the base curve and the ruling flow of the generalized null scroll are revealed. The Gaussian curvature, mean [...] Read more.

In this work, a family of ruled surfaces named generalized null scrolls in Minkowski 3-space are investigated via the defined structure functions. The relations between the base curve and the ruling flow of the generalized null scroll are revealed. The Gaussian curvature, mean curvature, second Gaussian curvature and the second mean curvature are given and related to each other. Last but not least, the generalized null scrolls whose base curves are k-type null helices are discussed and several examples are presented. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

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Review

Jump to: Research

32 pages, 436 KiB

Open AccessReview

Differential Geometry of Identity Maps: A Survey

by Bang-Yen Chen

Mathematics 2020, 8(8), 1264; https://doi.org/10.3390/math8081264 - 2 Aug 2020

Cited by 4 | Viewed by 2949

Abstract

An identity map

i d_{M} : M \to M

is a bijective map from a manifold M onto itself which carries each point of M return to the same point. To study the differential geometry of an identity map [...] Read more.

An identity map

i d_{M} : M \to M

is a bijective map from a manifold M onto itself which carries each point of M return to the same point. To study the differential geometry of an identity map

i d_{M} : M \to M

, we usually assume that the domain M and the range M admit metrics g and

g^{'}

, respectively. The main purpose of this paper is to provide a comprehensive survey on the differential geometry of identity maps from various differential geometric points of view. Full article

(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)

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