scholarly journals Riemannian Geometry and Modern Developments

2019 ◽  
Vol 39 ◽  
pp. 71-85
Author(s):  
AKM Nazimuddin ◽  
Md Showkat Ali
Keyword(s):  
Differential Geometry ◽  
Scalar Curvature ◽  
Constant Curvature ◽  
Riemannian Geometry ◽  
Ricci Tensor ◽  
Software Tool ◽  
3 Dimensional ◽  
Christoffel Symbols ◽  
Implementation Method

In this paper, we compute the Christoffel Symbols of the first kind, Christoffel Symbols of the second kind, Geodesics, Riemann Christoffel tensor, Ricci tensor and Scalar curvature from a metric which plays a fundamental role in the Riemannian geometry and modern differential geometry, where we consider MATLAB as a software tool for this implementation method. Also we have shown that, locally, any Riemannian 3-dimensional metric can be deformed along a directioninto another metricthat is conformal to a metric of constant curvature GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 71-85

scholarly journals On static manifolds and related critical spaces with cyclic parallel Ricci tensor

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
H. Baltazar ◽  
A. Da Silva
Keyword(s):  
Scalar Curvature ◽  
Ricci Tensor ◽  
Critical Spaces ◽  
3 Dimensional ◽  
Critical Metrics ◽  
Volume Functional ◽  
Total Scalar Curvature ◽  
Cyclic Parallel Ricci Tensor ◽  
Scalar Curvature Functional ◽  
Parallel Ricci Tensor

Abstract We classify 3-dimensional compact Riemannian manifolds (M 3, g) that admit a non-constant solution to the equation −Δfg +Hess f − f Ric = μ Ric +λg for some special constants (μ, λ), under the assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we study here are: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We also classify n-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.

A 3-DIMENSIONAL SASAKIAN METRIC AS A YAMABE SOLITON

2012 ◽  
Vol 09 (04) ◽  
pp. 1220003 ◽  
Author(s):  
RAMESH SHARMA
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Curvature ◽  
Constant Scalar Curvature ◽  
Complete Manifold ◽  
3 Dimensional ◽  
Infinitesimal Automorphism ◽  
Contact Metric Structure ◽  
Sasakian Metric ◽  
Yamabe Soliton

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

2019 ◽  
Vol 16 (03) ◽  
pp. 1950039 ◽  
Author(s):  
V. Venkatesha ◽  
Devaraja Mallesha Naik
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Flow Vector ◽  
Contact Metric Manifolds ◽  
Yamabe Soliton

If [Formula: see text] is a 3-dimensional contact metric manifold such that [Formula: see text] which admits a Yamabe soliton [Formula: see text] with the flow vector field [Formula: see text] pointwise collinear with the Reeb vector field [Formula: see text], then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if [Formula: see text] is endowed with a Yamabe soliton [Formula: see text], then either [Formula: see text] is flat or it has constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, we show that if [Formula: see text] is non-flat, then either [Formula: see text] is a Sasakian manifold of constant curvature [Formula: see text] or [Formula: see text] is an infinitesimal automorphism of the contact metric structure on [Formula: see text].

Constant curvature models in sub-Riemannian geometry

2019 ◽  
Vol 138 ◽  
pp. 241-256 ◽  
Author(s):  
D. Alekseevsky ◽  
A. Medvedev ◽  
J. Slovak
Keyword(s):  
Constant Curvature ◽  
Riemannian Geometry

scholarly journals A modular platform for automated cryo-FIB workflows

eLife ◽  
2021 ◽  
Vol 10 ◽  
Author(s):  
Sven Klumpe ◽  
Herman K H Fung ◽  
Sara K Goetz ◽  
Ievgeniia Zagoriy ◽  
Bernhard Hampoelz ◽  
...  
Keyword(s):  
Focused Ion Beam ◽  
Structural Integrity ◽  
Electron Tomography ◽  
Ion Beam ◽  
Light And Electron Microscopy ◽  
Software Tool ◽  
3 Dimensional ◽  
Improved Stability ◽  
Modular Platform ◽  
Beam Currents

Lamella micromachining by focused ion beam milling at cryogenic temperature (cryo-FIB) has matured into a preparation method widely used for cellular cryo-electron tomography. Due to the limited ablation rates of low Ga+ ion beam currents required to maintain the structural integrity of vitreous specimens, common preparation protocols are time-consuming and labor intensive. The improved stability of new generation cryo-FIB instruments now enables automated operations. Here, we present an open-source software tool, SerialFIB, for creating automated and customizable cryo-FIB preparation protocols. The software encompasses a graphical user interface for easy execution of routine lamellae preparations, a scripting module compatible with available Python packages, and interfaces with 3-dimensional correlative light and electron microscopy (CLEM) tools. SerialFIB enables the streamlining of advanced cryo-FIB protocols such as multi-modal imaging, CLEM-guided lamella preparation and in situ lamella lift-out procedures. Our software therefore provides a foundation for further development of advanced cryogenic imaging and sample preparation protocols.

A Generalization of Finsler Geometry

1962 ◽  
Vol 14 ◽  
pp. 87-112 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Elliptic Partial Differential Equations ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Metric Function

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

scholarly journals Arthur Geoffrey Walker. 17 July 1909 — 31 March 2001

2006 ◽  
Vol 52 ◽  
pp. 413-421
Author(s):  
Nigel J. Hitchin
Keyword(s):  
Differential Geometry ◽  
Quantum Theory ◽  
Riemannian Geometry ◽  
Psychiatric Illness ◽  
Grammar School ◽  
Atomic Spectra ◽  
King's College

Arthur Geoffrey Walker was born in Watford, Hertfordshire, on 17 July 1909, and attended Watford Grammar School, from where he won in 1928 an Open Mathematical Scholarship to Balliol College, Oxford. There, his tutor was John William Nicholson FRS, who had been a professor at King's College, London, and was one of the first mathematical physicists to relate quantum theory to atomic spectra. However, in the late 1920s he was suffering from a psychiatric illness and in 1930 was hospitalized, so that Walker had to study on his own a great deal. This perhaps influenced his subsequent method of working on mathematics, which he normally did in the privacy of his room rather than in active consultation with others. He obtained a Second in Moderations, but in 1930 won a Junior Mathematical Exhibition and in 1931 took a First with a Distinction in the special subject of differential geometry, which was to become his life's work. Eisenhart's book Riemannian geometry (Eisenhart 1926) became his bible and he continually referred to it in many of his papers.

scholarly journals On inextensible flows of tangent developable of biharmonic B-Slant helices according to Bishop frames in the special 3-dimensional Kenmotsu manifold

2011 ◽  
Vol 31 (1) ◽  
pp. 89 ◽  
Author(s):  
Vedat Asil ◽  
Talat Körpınar ◽  
Essin Turhan
Keyword(s):  
Ricci Tensor ◽  
Three Dimensional ◽  
Kenmotsu Manifold ◽  
3 Dimensional ◽  
Developable Surfaces ◽  
Tangent Developable ◽  
Inextensible Flows ◽  
Parallel Ricci Tensor

In this paper, we study inextensible flows of tangent developable surfaces of biharmonic B-slant helices in the special three-dimensional Kenmotsu manifold K with η-parallel ricci tensor. We express some interesting relations about inextensible flows of this surfaces.

scholarly journals The Long-Time Behavior of the Ricci Tensor under the Ricci Flow

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Christian Hilaire
Keyword(s):  
Ricci Flow ◽  
Ricci Tensor ◽  
Closed Manifold ◽  
Time Behavior ◽  
Dimensional Manifold ◽  
Long Time Behavior ◽  
Bounded Curvature ◽  
3 Dimensional ◽  
Long Time ◽  
Uniformly Bounded

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as . We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the curvature and the square of the diameter is uniformly bounded, then this solution must be of type III.

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