Tangent planes and the first fundamental form

2018 ◽  
pp. 50-81
Keyword(s):  
Fundamental Form ◽  
First Fundamental Form

scholarly journals Classification of Some Special Types Ruled Surfaces in Simply Isotropic 3-Space

2017 ◽  
Vol 55 (1) ◽  
pp. 87-98 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Dae Won Yoon ◽  
Nural Yuksel
Keyword(s):  
Real Number ◽  
Laplace Operator ◽  
Fundamental Form ◽  
Three Dimensional ◽  
Ruled Surfaces ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
The Laplace Operator ◽  
First Fundamental Form

AbstractIn this paper, we classify two types ruled surfaces in the three dimensional simply isotropic space I13under the condition ∆xi= λixiwhere ∆ is the Laplace operator with respect to the first fundamental form and λ is a real number. We also give explicit forms of these surfaces.

scholarly journals Cuspidal edges with the same first fundamental forms along a knot

2020 ◽  
Vol 29 (07) ◽  
pp. 2050047
Author(s):  
Atsufumi Honda ◽  
Kosuke Naokawa ◽  
Kentaro Saji ◽  
Masaaki Umehara ◽  
Kotaro Yamada
Keyword(s):  
Fundamental Form ◽  
Local Existence ◽  
Reasonable Assumption ◽  
Real Analyticity ◽  
Cuspidal Edge ◽  
First Fundamental Form

Letting [Formula: see text] be a compact [Formula: see text]-curve embedded in the Euclidean [Formula: see text]-space ([Formula: see text] means real analyticity), we consider a [Formula: see text]-cuspidal edge [Formula: see text] along [Formula: see text]. When [Formula: see text] is non-closed, in the authors’ previous works, the local existence of three distinct cuspidal edges along [Formula: see text] whose first fundamental forms coincide with that of [Formula: see text] was shown, under a certain reasonable assumption on [Formula: see text]. In this paper, if [Formula: see text] is closed, that is, [Formula: see text] is a knot, we show that there exist infinitely many cuspidal edges along [Formula: see text] having the same first fundamental form as that of [Formula: see text] such that their images are non-congruent to each other, in general.

scholarly journals Some Equal-area, Conformal and Conventional Map Projections: A Tutorial Review

2016 ◽  
Vol 10 (3) ◽  
Author(s):  
Ebrahim Ghaderpour
Keyword(s):  
Fundamental Form ◽  
Positioning Systems ◽  
Equal Area ◽  
Map Projections ◽  
Flat Sheet ◽  
The Earth ◽  
Global Positioning ◽  
Different Types ◽  
First Fundamental Form ◽  
Shape And Size

AbstractMap projections have been widely used in many areas such as geography, oceanography, meteorology, geology, geodesy, photogrammetry and global positioning systems. Understanding different types of map projections is very crucial in these areas. This paper presents a tutorial review of various types of current map projections such as equal-area, conformal and conventional. We present these map projections from a model of the Earth to a flat sheet of paper or map and derive the plotting equations for them in detail. The first fundamental form and the Gaussian fundamental quantities are defined and applied to obtain the plotting equations and distortions in length, shape and size for some of these map projections.

A CLASSIFICATION AND A NON-EXISTENCE THEOREM FOR CONFORMALLY FLAT HYPERSURFACES IN EUCLIDEAN 4-SPACE

2005 ◽  
Vol 16 (01) ◽  
pp. 53-85 ◽  
Author(s):  
YOSHIHIKO SUYAMA
Keyword(s):  
Existence Theorem ◽  
Fundamental Form ◽  
Riemannian Metrics ◽  
Equivalence Classes ◽  
Classification Theorem ◽  
Conformally Flat ◽  
Conformal Class ◽  
First Fundamental Form ◽  
Conformal Equivalence ◽  
Conformally Flat Hypersurface

We study generic conformally flat hypersurfaces in the Euclidean 4-space satisfying a certain condition on the conformal class of the first fundamental form. We first classify such hypersurfaces by determining all conformal-equivalence classes of generic conformally flat hypersurfaces satisfying the condition. Next, as an application of the classification theorem, we give some examples of flat Riemannian metrics which are not conformal to the first fundamental form of any generic conformally flat hypersurface. These flat Riemannian metrics seem to provide counter-examples to Hertrich–Jeromin's claim [3, 5].

scholarly journals Classification of space-like translation surfaces in the 3-dimensional Lorentz Heisenberg group H3

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Rafik Medjati ◽  
Hanifi Zoubir ◽  
Brahim Medjahdi
Keyword(s):  
Heisenberg Group ◽  
Finite Type ◽  
Fundamental Form ◽  
Natural Extension ◽  
Laplacian Operator ◽  
Translation Surfaces ◽  
Content Type ◽  
Flat Metric ◽  
First Fundamental Form

PurposeIn the Lorentz Heisenberg space H3 endowed with flat metric g3, a translation surface is parametrized by r(x, y) = γ1(x)*γ2(y), where γ1 and γ2 are two planar curves lying in planes, which are not orthogonal. In this article, we classify translation surfaces in H3, which satisfy some algebraic equations in terms of the coordinate functions and the Laplacian operator with respect to the first fundamental form of the surface.Design/methodology/approachIn this paper, we classify some type of space-like translation surfaces of H3 endowed with flat metric g3 under the conditionΔri = λiri. We will develop the system which describes surfaces of type finite in H3. For solve the system thus obtained, we will use the calculation variational. Finally, we will try to give performances geometric surfaces that meet the condition imposed.FindingsClassification of six types of translation surfaces of finite type in the three-dimensional Lorentz Heisenberg group H3.Originality/valueThe subject of this paper lies at the border of geometry differential and spectral analysis on manifolds. Historically, the first research on the study of sub-finite type varieties began around the 1970 by B.Y.Chen. The idea was to find a better estimate of the mean total curvature of a compact subvariety of a Euclidean space. In fact, the notion of finite type subvariety is a natural extension of the notion of a minimal subvariety or surface, a notion directly linked to the calculation of variations. The goal of this work is the classification of surfaces in H3, in other words the surfaces which satisfy the condition/Delta (ri) = /Lambda (ri), such that the Laplacian is associated with the first, fundamental form.

scholarly journals A fluid generalization of membranes

Open Physics ◽  
2011 ◽  
Vol 9 (4) ◽  
Author(s):  
Mark Roberts
Keyword(s):  
Perfect Fluid ◽  
Fundamental Form ◽  
Point Particle ◽  
Lagrangian Formulation ◽  
Quantum Theories ◽  
Higher Dimensional ◽  
First Fundamental Form

AbstractIn a certain sense a perfect fluid is a generalization of a point particle. This leads to the question as to what is the corresponding generalization for extended objects. Here the lagrangian formulation of a perfect fluid is much generalized by replacing the product of the co-moving vector which is a first fundamental form by higher dimensional first fundamental forms; this has as a particular example a fluid which is a classical generalization of a membrane; however there is as yet no indication of any relationship between their quantum theories.

scholarly journals Classification of totally umbilical submanifolds in symmetric spaces

1980 ◽  
Vol 30 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Symmetric Space ◽  
Fundamental Form ◽  
Symmetric Spaces ◽  
Second Fundamental Form ◽  
Totally Umbilical ◽  
The Second Fundamental Form ◽  
Totally Umbilical Submanifold ◽  
Totally Umbilical Submanifolds ◽  
First Fundamental Form

AbstractA submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.

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