The Classical Elastic Curves in A 3-Dimensional Indefinite-Riemannian Manifold

2012 ◽  
Vol 12 (12) ◽  
pp. 1303-1307
Author(s):  
Nemat Abazari ◽  
Yusuf Yayli
Keyword(s):  
Riemannian Manifold ◽  
3 Dimensional ◽  
Elastic Curves

scholarly journals Eikonal slant helices and Eikonal Darboux helices in 3-dimensional Riemannian manifold

2014 ◽  
Vol 11 (05) ◽  
pp. 1450045 ◽  
Author(s):  
Mehmet Önder ◽  
Evren Zıplar ◽  
Onur Kaya
Keyword(s):  
Riemannian Manifold ◽  
3 Dimensional ◽  
Slant Helix ◽  
Curvature And Torsion

In this study, we give the definitions and characterizations of Eikonal slant helices, Eikonal Darboux helices and non-modified Eikonal Darboux helices in 3-dimensional Riemannian manifold M3. We show that every Eikonal slant helix is also an Eikonal Darboux helix. Furthermore, we obtain that if the curve α is a non-modified Eikonal Darboux helix, then α is an Eikonal slant helix if and only if κ2 + τ2 = constant, where κ and τ are curvature and torsion of α, respectively.

Riemannian manifolds whose curvature operator R(X, Y) has constant eigenvalues

2004 ◽  
Vol 70 (2) ◽  
pp. 301-319 ◽  
Author(s):  
Y. Nikolayevsky
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Riemannian Manifolds ◽  
Warped Product ◽  
Curvature Operator ◽  
Specific Function ◽  
3 Dimensional ◽  
Space Of Constant Curvature

A Riemannian manifold Mn is called IP, if, at every point x ∈ Mn, the eigenvalues of its skew-symmetric curvature operator R(X, Y) are the same, for every pair of orthonormal vectors X, Y ∈ TxMn. In [5, 6, 12] it was shown that for all n ≥ 4, except n = 7, an IP manifold either has constant curvature, or is a warped product, with some specific function, of an interval and a space of constant curvature. We prove that the same result is still valid in the last remaining case n = 7, and also study 3-dimensional IP manifolds.

Fiber structures of special (4 + 3 + 1) warped-like manifolds with Spin(7) holonomy

2014 ◽  
Vol 11 (08) ◽  
pp. 1450076
Author(s):  
Selman Uğuz ◽  
İbrahim Ünal
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Warped Product ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Simply Connected ◽  
Block Diagonal

A generalization of 8-dimensional multiply-warped product manifolds is considered as a special warped product, by allowing the fiber metric to be non-block diagonal. Motivating from the previous paper [S. Uğuz and A. H. Bilge, (3 + 3 + 2) warped-like product manifolds with Spin(7) holonomy, J. Geom. Phys.61 (2011) 1093–1103], we present a special warped product as a (4 + 3 + 1) warped-like manifold of the form M = F × B, where the base B is a 1-dimensional Riemannian manifold, and the fiber F is of the form F = F1 × F2 where Fi's (i = 1, 2) are Riemannian 4- and 3-manifolds, respectively. It is showed that the connection on M is entirely determined provided that the Bonan 4-form is closed. Assuming that the Fi's are complete, connected and simply connected, it is proved that the 3-dimensional fiber is isometric to S3 with constant curvature k > 0. Finally, the geometric properties of the 4-dimensional fiber of M are studied.

scholarly journals On a Riemannian manifold with a circulant structure whose third power is the identity

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3529-3539 ◽  
Author(s):  
Iva Dokuzova
Keyword(s):  
Riemannian Manifold ◽  
Lie Group ◽  
Covariant Derivative ◽  
Conformal Transformation ◽  
Circulant Matrix ◽  
Tensor Structure ◽  
Fundamental Tensor ◽  
3 Dimensional ◽  
Geometrical Characteristics

It is studied a 3-dimensional Riemannian manifold equipped with a tensor structure of type (1,1), whose third power is the identity. This structure has a circulant matrix with respect to some basis, i.e. the structure is circulant. On such a manifold a fundamental tensor by the metric and by the covariant derivative of the circulant structure is defined. An important characteristic identity for this tensor is obtained. It is established that the image of the fundamental tensor with respect to the usual conformal transformation satisfies the same identity. A Lie group as a manifold of the considered type is constructed and some of its geometrical characteristics are found.

scholarly journals Spherical Ruled Surfaces in S3 Characterized by the Spherical Gauss Map

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2106
Author(s):  
Young Ho Kim ◽  
Sun Mi Jung
Keyword(s):  
Riemannian Manifold ◽  
Finite Type ◽  
Eigenvalue Problems ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Dimensional Sphere ◽  
3 Dimensional ◽  
Smooth Maps ◽  
The Laplace Operator ◽  
Natural Way

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.

Soliton Resolution for Corotational Wave Maps on a Wormhole

2017 ◽  
Vol 2019 (15) ◽  
pp. 4603-4706 ◽  
Author(s):  
Casey Rodriguez
Keyword(s):  
General Relativity ◽  
Riemannian Manifold ◽  
Topological Degree ◽  
Harmonic Map ◽  
Finite Energy ◽  
Constant Time ◽  
Wave Maps ◽  
3 Dimensional ◽  
Dimensional Spacetime

Abstract In this article, we initiate the study of finite energy equivariant wave maps from the $(1+3)$-dimensional spacetime ${\mathbb R} \times ({\mathbb R} \times {\mathbb S}^2) {\rightarrow} {\mathbb S}^3$ where the metric on ${\mathbb R} \times ({\mathbb R} \times {\mathbb S}^2)$ is given by \begin{align*} {\rm d}s^2 = -{\rm d}t^2 + {\rm d}r^2 + (r^2 + 1) \left ( {\rm d} \theta^2 + \sin^2 \theta {\rm d} \varphi^2 \right)\!, \quad t,r \in {\mathbb R}, (\theta,\varphi) \in {\mathbb S}^2. \end{align*} The constant time slices are each given by the Riemannian manifold ${\mathbb R} \times {\mathbb S}^2$ with metric \begin{align*} {\rm d}s^2 = {\rm d}r^2 + (r^2 + 1) \left ( {\rm d} \theta^2 + \sin^2 \theta{\rm d} \varphi^2 \right)\!. \end{align*} This Riemannian manifold contains two asymptotically Euclidean ends at $r \rightarrow \pm \infty$ that are connected by a spherical throat of area $4 \pi^2$ at $r = 0$. The spacetime ${\mathbb R} \times ({\mathbb R} \times {\mathbb S}^2) {\rightarrow} {\mathbb S}^3$ is a simple example of a wormhole geometry in general relativity. In this work, we will consider 1-equivariant or corotational wave maps. Each corotational wave map can be indexed by its topological degree $n$. For each $n$, there exists a unique energy minimizing corotational harmonic map $Q_{n} : {\mathbb R} \times {\mathbb S}^2 \rightarrow {\mathbb S}^3$ of degree $n$. In this work, we show that modulo a free radiation term, every corotational wave map of degree $n$ converges strongly to $Q_{n}$. This resolves a conjecture made by Bizon and Kahl for the corotational case.

scholarly journals Integral Formulas for Submanifolds and their Applications

1970 ◽  
Vol 22 (2) ◽  
pp. 376-388 ◽  
Author(s):  
Kentaro Yano
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
Integral Formulas ◽  
Closed Hypersurfaces

Liebmann [12] proved that the only ovaloids with constant mean curvature in a 3-dimensional Euclidean space are spheres. This result has been generalized to the case of convex closed hypersurfaces in an m-dimensional Euclidean space by Alexandrov [1], Bonnesen and Fenchel [3], Hopf [4], Hsiung [5], and Süss [14].The result has been further generalized to the case of closed hypersurfaces in an m-dimensional Riemannian manifold by Alexandrov [2], Hsiung [6], Katsurada [7; 8; 9], Ōtsuki [13], and by myself [15; 16].The attempt to generalize the result to the case of closed submanifolds in an m-dimensional Riemannian manifold has been recently done by Katsurada [10; 11], Kôjyô [10], and Nagai [11].

scholarly journals Triharmonic Curves in 3-Dimensional Homogeneous Spaces

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
S. Montaldo ◽  
A. Pámpano
Keyword(s):  
Riemannian Manifold ◽  
Homogeneous Spaces ◽  
Arbitrary Dimension ◽  
Complete Classification ◽  
Constant Gaussian Curvature ◽  
Space Forms ◽  
3 Dimensional ◽  
The Family ◽  
Full Classification

AbstractWe first prove that, unlike the biharmonic case, there exist triharmonic curves with nonconstant curvature in a suitable Riemannian manifold of arbitrary dimension. We then give the complete classification of triharmonic curves in surfaces with constant Gaussian curvature. Next, restricting to curves in a 3-dimensional Riemannian manifold, we study the family of triharmonic curves with constant curvature, showing that they are Frenet helices. In the last part, we give the full classification of triharmonic Frenet helices in space forms and in Bianchi–Cartan–Vranceanu spaces.

Three-Dimensional Reconstruction Using Stereo Pairs from Serial Thick Sections

1977 ◽  
Vol 35 ◽  
pp. 562-563
Author(s):  
Robert Glaeser ◽  
Thomas Bauer ◽  
David Grano
Keyword(s):  
Three Dimensional ◽  
Dimensional Structure ◽  
Serial Sections ◽  
Thin Sections ◽  
3 Dimensional ◽  
Transmission Electron ◽  
Freeze Dried ◽  
Dimensional Reconstruction ◽  
Stereo Imagery ◽  
Whole Mounts

In transmission electron microscopy, the 3-dimensional structure of an object is usually obtained in one of two ways. For objects which can be included in one specimen, as for example with elements included in freeze- dried whole mounts and examined with a high voltage microscope, stereo pairs can be obtained which exhibit the 3-D structure of the element. For objects which can not be included in one specimen, the 3-D shape is obtained by reconstruction from serial sections. However, without stereo imagery, only detail which remains constant within the thickness of the section can be used in the reconstruction; consequently, the choice is between a low resolution reconstruction using a few thick sections and a better resolution reconstruction using many thin sections, generally a tedious chore. This paper describes an approach to 3-D reconstruction which uses stereo images of serial thick sections to reconstruct an object including detail which changes within the depth of an individual thick section.

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