Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

Chen, Bang‐Yen; Güler, Erhan; Yaylı, Yusuf; Hacisalihoğlu, H. Hilmi

doi:10.36890/iejg.1216024

Cited by 13 publications

(6 citation statements)

References 155 publications

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“…Moreover, an up-to-date comprehensive survey, up to 2015, on this topic was presented in the author's book [42]. For more results on this, we refer to [40,41,42,43,44].…”

Section: Topic Iii: Finite Type Submanifoldsmentioning

confidence: 99%

Differential Geometry of Position Vector Fields

Chen

2023

Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

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Differential geometry studies the geometry of curves, surfaces, and higher dimensional smooth manifolds. It uses the ideas and techniques of differential and integral calculus, linear and multilinear algebras, topology, and differential equations. This subject in mathematics is closely related to differential topology, which concerns itself with properties of smooth manifolds. Differential geometry also closely relates to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.Curvature is an important notion in mathematics, which has been investigated extensively in differential geometry. There are two types of curvatures: namely, "intrinsic" and "extrinsic"."Intrinsic curvature" describes the curvature at a point on a surface or a smooth manifold and is independent of how the surface or manifold is embedded in space. Borrow a term from biology, intrinsic invariants of a manifold are the DNA of the manifold. The Gauss curvature of a surface is the most commonly studied intrinsic measure of curvature. In higher dimensions, curvature is too complicated to be described by a single number. In this case, tensors are used to describe the curvature as pioneered by B. Riemann in his famous 1854 inaugural lecture at Gottingen: "Über die Hypothesen welche der Geometrie zu Grunde liegen."In Einstein's theory of general relativity, intrinsic curvature is key to understanding the shape of the universe."Extrinsic curvature" of a manifold depends on how it is embedded within a space. Examples of extrinsic measures of curvature include geodesic curvature, principal curvature, and mean curvature. The most important extrinsic invariant for a submanifolds in an ambient Riemannian manifold is the mean curvature vector, which is known as the tension field in physics.Differential geometry studies the geometry of curves, surfaces and higher dimensional smooth manifolds. For submanifolds in Euclidean spaces, the position vector is the most natural geometric object. Position vectors find applications throughout mathematics, engineering and natural sciences. The purpose of this survey article is to present six research topics in differential geometry in which the position vector plays a very important role. In addition to this, we explain the link between position vectors with mechanics, dynamics, and D'Arcy Thompson's law of natural growth in biology.

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“…Moreover, an up-to-date comprehensive survey, up to 2015, on this topic was presented in the author's book [42]. For more results on this, we refer to [40,41,42,43,44].…”

Section: Topic Iii: Finite Type Submanifoldsmentioning

confidence: 99%

Differential Geometry of Position Vector Fields

Chen

2023

Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

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“…In space forms, Chen et al [5] served the 40 years of one-type sub-manifolds and a one-type Gauss map.…”

Section: Introductionmentioning

confidence: 99%

“…Arvanitoyeorgos et al [33] introduced ∆H = αH (H denotes the mean curvature, α is a constant). Güler [34] introduced the helical hypersurface determined by a space-like axis in E 5 1 . Li and Güler [35,36] studied a hypersurfaces of revolution family in pseudo-Euclidean spaces E 5 2 and E 7 3 .…”

Section: Introductionmentioning

confidence: 99%

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Twisted Hypersurfaces in Euclidean 5-Space

Li,

Güler

2023

Mathematics

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The twisted hypersurfaces x with the (0,0,0,0,1) rotating axis in five-dimensional Euclidean space E5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In E5, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces x are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of x are determined. Additionally, the Laplace–Beltrami operator relation of x is given.

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“…Additionally, Chen and Piccinni [13] researched submanifolds in E m that possess a finitetype Gauss map. In the realm of space forms, Chen et al [14] dedicated four decades to the study of 1-type submanifolds and the 1-type Gauss map. These investigations have contributed significantly to our understanding of the properties and characteristics of these submanifolds.…”

Section: Introductionmentioning

confidence: 99%

Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime L4

Güler

2023

Universe

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We investigate the class of helical hypersurfaces parametrized by x=x(u,v,w), characterized by a light-like axis in Minkowski spacetime L4. We determine the matrices that represent the fundamental forms, Gauss map, and shape operator of x. Furthermore, employing the Cayley–Hamilton theorem, we compute the curvatures associated with x. We explore the conditions under which the curvatures of x possess the property of being umbilical. Moreover, we provide the Laplace–Beltrami operator for the family of helical hypersurfaces with a light-like axis in L4.

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Differential Geometry of 1-type Submanifolds and Submanifolds with 1-type Gauss Map

Cited by 13 publications

References 155 publications

Differential Geometry of Position Vector Fields

Differential Geometry of Position Vector Fields

Twisted Hypersurfaces in Euclidean 5-Space

Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime L4

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