The Geometry of Submanifolds

Aminov, Yu. A.

doi:10.1201/9781482296860

Cited by 42 publications

(45 citation statements)

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“…We observe that the normal connection D of M is flat if and only if K N = 0, and by a result of Cartan, this equivalent to the diagonalisability of all shape operators A Nα of M which means that M is a totally umbilical surface in E n [1].…”

Section: Basic Conceptsmentioning

confidence: 74%

On Generalized Rotational Surfaces in Euclidean Spaces

Arslan¹,

Bulca²,

Kosova³

2017

Journal of the Korean Mathematical Society

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Abstract. In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized tractrices in Euclidean (n + 1)-space E n+1 . Further, we introduce some kind of generalized rotational surfaces in Euclidean spaces E 3 and E 4 , respectively. We have also obtained some basic properties of generalized rotational surfaces in E 4 and some results of their curvatures. Finally, we give some examples of generalized Beltrami surfaces in E 3 and E 4 , respectively.

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Section: Basic Conceptsmentioning

confidence: 74%

On Generalized Rotational Surfaces in Euclidean Spaces

Arslan¹,

Bulca²,

Kosova³

2017

Journal of the Korean Mathematical Society

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“…which is equivalent to (45) due to (35). The existence of such curvature is guaranteed by the existence of a solution of the heat equation and so is (45).…”

Section: Proposition 2 a Linear First-order Equationmentioning

confidence: 99%

“…On the other hand, to see what happens with our representation if the viscosity is added, identity (35) for the Gaussian curvature tells us that viscous Burgers' equation represents a surface with curvature…”

Section: Proposition 2 a Linear First-order Equationmentioning

confidence: 99%

“…This representation for the curvature is worthwhile to word in some aspects: By means of expression (35), the inviscid and viscous Burgers' equations and also KdV equation can be seen as the equations for surfaces with certain Gaussian curvatures. Also, by virtue of our representation, the problem of the construction of Riemannian metrics associated with certain first-order ODEs turns out to finding a solution of certain Burgers' equations.…”

Section: Theorem 1 Gaussian Curvature Of the Metric Connection (29) mentioning

confidence: 99%

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Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

Bayrakdar

2018

Advances in Mathematical Physics

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We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers' equations describe surfaces attached to an ODE of the form / = ( , ) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

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“…Let us recall the classical notion of Bianchi transformation [1,2,9]. Let F 2 be a pseudo-spherical surface in E 3 , i.e.…”

Section: Introductionmentioning

confidence: 99%