Special Issue "Differential Geometry"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 November 2018)

Special Issue Editor

Guest Editor
Prof. Dr. Ion Mihai

Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest, Romania
Website | E-Mail
Interests: differentiable manifolds; Riemannian manifolds; distinguished vector fields; Riemannian invariants; sectional curvature; complex manifolds; contact manifolds; affine manifolds; statistical manifolds; submanifold theory

Special Issue Information

Dear Colleagues,

Differential geometry is the field of mathematics that studies geometrical structures on differentiable manifolds by using techniques of differential calculus, integral calculus, and linear algebra. Starting from some classical examples (open sets in Euclidean spaces, spheres, tori, projective spaces, Grassmannians, etc.) one may construct new manifolds by using algebraic tools: Product of manifolds, quotient spaces, pull-back of manifolds by smooth functions, tensor product of submanifolds, etc.

Differential geometry became a field of research in late 19th century, but it is very actual by its applications and new approaches. In order to determine the lengths of curves, the areas of surfaces, and the volumes of manifolds, the geometers have considered Riemannian manifolds, or, more generally, pseudo-Riemannian manifolds. On such manifolds, distinguished vector fields (Killing, conformal, concurrent, torse forming vector fields) have interesting applications in geometry and relativity. Curvature invariants are the most natural and most important Riemannian invariants as they play key roles in physics and biology. Among the Riemannian curvature invariants, the most investigated are the sectional curvature, scalar curvature, Ricci curvature, and Chen invariants. Mostly studied are the Riemannian manifolds endowed with certain endomorphisms of their tangent bundles: Almost complex, almost product, almost contact, almost paracontact manifolds, etc. More general manifolds, for instance, affine manifolds and statistical manifolds, are also considered. On the other hand, the geometry of submanifolds in Riemannian manifolds is an important topic of research in differential geometry. Its origins are in the theory of curves and surfaces in the three-dimensional Euclidean space. Obstructions to the existence of minimal, Lagrangian, slant submanifolds were obtained in terms of their Riemannian curvature invariants.

The purpose of this Special Issue is to provide a collection of papers that reflect modern topics of research and new developments in the field of differential geometry and explore applications in other areas.

Prof. Dr. Ion Mihai

Guest Editor

Keywords

  • differentiable manifolds
  • Riemannian manifolds
  • distinguished vector fields
  • Riemannian invariants
  • sectional curvature
  • complex manifolds
  • contact manifolds
  • affine manifolds
  • statistical manifolds
  • submanifold theory

Published Papers (14 papers)

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Research

Open AccessArticle
The Characterization of Affine Symplectic Curves in ℝ4
Mathematics 2019, 7(1), 110; https://doi.org/10.3390/math7010110
Received: 29 November 2018 / Revised: 16 January 2019 / Accepted: 18 January 2019 / Published: 21 January 2019
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Abstract
Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic space. We also give the characterizations of the [...] Read more.
Symplectic geometry arises as the natural geometry of phase-space in the equations of classical mechanics. In this study, we obtain new characterizations of regular symplectic curves with respect to the Frenet frame in four-dimensional symplectic space. We also give the characterizations of the symplectic circular helices as the third- and fourth-order differential equations involving the symplectic curvatures. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
Completness of Statistical Structures
Mathematics 2019, 7(1), 104; https://doi.org/10.3390/math7010104
Received: 30 November 2018 / Revised: 8 January 2019 / Accepted: 11 January 2019 / Published: 19 January 2019
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Abstract
In this survey note, we discuss the notion of completeness for statistical structures. There are at least three connections whose completeness might be taken into account, namely, the Levi-Civita connection of the given metric, the statistical connection, and its conjugate. Especially little is [...] Read more.
In this survey note, we discuss the notion of completeness for statistical structures. There are at least three connections whose completeness might be taken into account, namely, the Levi-Civita connection of the given metric, the statistical connection, and its conjugate. Especially little is known on the completeness of statistical connections. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
Generic Properties of Framed Rectifying Curves
Mathematics 2019, 7(1), 37; https://doi.org/10.3390/math7010037
Received: 27 November 2018 / Revised: 25 December 2018 / Accepted: 26 December 2018 / Published: 3 January 2019
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Abstract
The position vectors of regular rectifying curves always lie in their rectifying planes. These curves were well investigated by B.Y.Chen. In this paper, the concept of framed rectifying curves is introduced, which may have singular points. We investigate the properties of framed rectifying [...] Read more.
The position vectors of regular rectifying curves always lie in their rectifying planes. These curves were well investigated by B.Y.Chen. In this paper, the concept of framed rectifying curves is introduced, which may have singular points. We investigate the properties of framed rectifying curves and give a method for constructing framed rectifying curves. In addition, we reveal the relationships between framed rectifying curves and some special curves. Full article
(This article belongs to the Special Issue Differential Geometry)
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Open AccessArticle
Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures
Mathematics 2018, 6(11), 259; https://doi.org/10.3390/math6110259
Received: 13 September 2018 / Revised: 14 November 2018 / Accepted: 15 November 2018 / Published: 17 November 2018
Cited by 1 | PDF Full-text (248 KB) | HTML Full-text | XML Full-text
Abstract
A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and * in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded [...] Read more.
A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and * in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant φ -sectional curvature. In addition, we give examples to show that the total space is a sphere. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
Trans-Sasakian 3-Manifolds with Reeb Flow Invariant Ricci Operator
Mathematics 2018, 6(11), 246; https://doi.org/10.3390/math6110246
Received: 19 September 2018 / Revised: 3 November 2018 / Accepted: 6 November 2018 / Published: 9 November 2018
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Abstract
Let M be a three-dimensional trans-Sasakian manifold of type (α,β). In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an α-Sasakian manifold, cosymplectic manifold [...] Read more.
Let M be a three-dimensional trans-Sasakian manifold of type ( α , β ) . In this paper, we obtain that the Ricci operator of M is invariant along Reeb flow if and only if M is an α -Sasakian manifold, cosymplectic manifold or a space of constant sectional curvature. Applying this, we give a new characterization of proper trans-Sasakian 3-manifolds. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
Pinching Theorems for a Vanishing C-Bochner Curvature Tensor
Mathematics 2018, 6(11), 231; https://doi.org/10.3390/math6110231
Received: 21 September 2018 / Revised: 23 October 2018 / Accepted: 26 October 2018 / Published: 30 October 2018
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Abstract
The main purpose of this article is to construct inequalities between a main intrinsic invariant (the normalized scalar curvature) and an extrinsic invariant (the Casorati curvature) for some submanifolds in a Sasakian manifold with a zero C-Bochner tensor. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
Inextensible Flows of Curves on Lightlike Surfaces
Mathematics 2018, 6(11), 224; https://doi.org/10.3390/math6110224
Received: 28 September 2018 / Revised: 26 October 2018 / Accepted: 26 October 2018 / Published: 29 October 2018
Cited by 1 | PDF Full-text (240 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we study inextensible flows of a curve on a lightlike surface in Minkowski three-space and give a necessary and sufficient condition for inextensible flows of the curve as a partial differential equation involving the curvatures of the curve on a [...] Read more.
In this paper, we study inextensible flows of a curve on a lightlike surface in Minkowski three-space and give a necessary and sufficient condition for inextensible flows of the curve as a partial differential equation involving the curvatures of the curve on a lightlike surface. Finally, we classify lightlike ruled surfaces in Minkowski three-space and characterize an inextensible evolution of a lightlike curve on a lightlike tangent developable surface. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
Hypersurfaces with Generalized 1-Type Gauss Maps
Mathematics 2018, 6(8), 130; https://doi.org/10.3390/math6080130
Received: 18 May 2018 / Revised: 18 July 2018 / Accepted: 23 July 2018 / Published: 26 July 2018
Cited by 2 | PDF Full-text (340 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, En, is said to be of generalized 1-type if, for the [...] Read more.
In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, En, is said to be of generalized 1-type if, for the Laplace operator, Δ, on the submanifold, it satisfies ΔG=fG+gC, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in E3. Second, we show that the Gauss map of any cylindrical surface in E3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E3, except planes. Finally, we show that cylindrical hypersurfaces in En+2 always have generalized 1-type Gauss maps. Full article
(This article belongs to the Special Issue Differential Geometry)
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Open AccessArticle
Comparison of Differential Operators with Lie Derivative of Three-Dimensional Real Hypersurfaces in Non-Flat Complex Space Forms
Mathematics 2018, 6(5), 84; https://doi.org/10.3390/math6050084
Received: 29 March 2018 / Revised: 11 May 2018 / Accepted: 14 May 2018 / Published: 20 May 2018
Cited by 1 | PDF Full-text (295 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field P=ϕA-Aϕ is given and three-dimensional real hypersurfaces in non-flat complex space forms whose [...] Read more.
In this paper, three-dimensional real hypersurfaces in non-flat complex space forms, whose shape operator satisfies a geometric condition, are studied. Moreover, the tensor field P = ϕ A - A ϕ is given and three-dimensional real hypersurfaces in non-flat complex space forms whose tensor field P satisfies geometric conditions are classified. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
L2-Harmonic Forms on Incomplete Riemannian Manifolds with Positive Ricci Curvature
Mathematics 2018, 6(5), 75; https://doi.org/10.3390/math6050075
Received: 28 February 2018 / Revised: 24 April 2018 / Accepted: 29 April 2018 / Published: 9 May 2018
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Abstract
We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L2 -harmonic forms and on which the L2 -Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds. [...] Read more.
We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L 2 -harmonic forms and on which the L 2 -Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessFeature PaperArticle
On Angles and Pseudo-Angles in Minkowskian Planes
Mathematics 2018, 6(4), 52; https://doi.org/10.3390/math6040052
Received: 1 March 2018 / Revised: 20 March 2018 / Accepted: 22 March 2018 / Published: 3 April 2018
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Abstract
The main purpose of the present paper is to well define Minkowskian angles and pseudo-angles between the two null directions and between a null direction and any non-null direction, respectively. Moreover, in a kind of way that will be tried to be made [...] Read more.
The main purpose of the present paper is to well define Minkowskian angles and pseudo-angles between the two null directions and between a null direction and any non-null direction, respectively. Moreover, in a kind of way that will be tried to be made clear at the end of the paper, these new sorts of angles and pseudo-angles can similarly to the previously known angles be seen as (combinations of) Minkowskian lengths of arcs on a Minkowskian unit circle together with Minkowskian pseudo-lengths of parts of the straight null lines. Full article
(This article belongs to the Special Issue Differential Geometry)
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Open AccessFeature PaperArticle
Curvature Invariants for Statistical Submanifolds of Hessian Manifolds of Constant Hessian Curvature
Mathematics 2018, 6(3), 44; https://doi.org/10.3390/math6030044
Received: 23 February 2018 / Revised: 11 March 2018 / Accepted: 12 March 2018 / Published: 15 March 2018
Cited by 5 | PDF Full-text (206 KB) | HTML Full-text | XML Full-text
Abstract
We consider statistical submanifolds of Hessian manifolds of constant Hessian curvature. For such submanifolds we establish a Euler inequality and a Chen-Ricci inequality with respect to a sectional curvature of the ambient Hessian manifold. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
A New Proof of a Conjecture on Nonpositive Ricci Curved Compact Kähler–Einstein Surfaces
Mathematics 2018, 6(2), 21; https://doi.org/10.3390/math6020021
Received: 26 December 2017 / Revised: 26 January 2018 / Accepted: 26 January 2018 / Published: 7 February 2018
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Abstract
In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of Hong et al. of 1988 and 2011. Moreover, we proved that any compact Kähler–Einstein surface M is a quotient [...] Read more.
In an earlier paper, we gave a proof of the conjecture of the pinching of the bisectional curvature mentioned in those two papers of Hong et al. of 1988 and 2011. Moreover, we proved that any compact Kähler–Einstein surface M is a quotient of the complex two-dimensional unit ball or the complex two-dimensional plane if (1) M has a nonpositive Einstein constant, and (2) at each point, the average holomorphic sectional curvature is closer to the minimal than to the maximal. Following Siu and Yang, we used a minimal holomorphic sectional curvature direction argument, which made it easier for the experts in this direction to understand our proof. On this note, we use a maximal holomorphic sectional curvature direction argument, which is shorter and easier for the readers who are new in this direction. Full article
(This article belongs to the Special Issue Differential Geometry)
Open AccessArticle
Euclidean Submanifolds via Tangential Components of Their Position Vector Fields
Mathematics 2017, 5(4), 51; https://doi.org/10.3390/math5040051
Received: 6 September 2017 / Revised: 6 September 2017 / Accepted: 10 October 2017 / Published: 16 October 2017
Cited by 3 | PDF Full-text (305 KB) | HTML Full-text | XML Full-text
Abstract
The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (t) [...] Read more.
The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (t) is usually the most sought-after quantity because the position vector field defines the motion of a particle (i.e., a point mass): its location relative to a given coordinate system at some time variable t. This article is a survey article. The purpose of this article is to survey recent results of Euclidean submanifolds associated with the tangential components of their position vector fields. In the last section, we present some interactions between torqued vector fields and Ricci solitons. Full article
(This article belongs to the Special Issue Differential Geometry)
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