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 Conformally flat hypersurfaces in Euclidean 4-space

2000 ◽  
Vol 158 ◽  
pp. 1-42 ◽  
Author(s):  
Yoshihiko Suyama
Keyword(s):  
Euclidean Space ◽  
Coordinate System ◽  
Fundamental Form ◽  
Conformally Flat ◽  
Curvature Line ◽  
Precise Representation ◽  
First Fundamental Form

AbstractWe study generic and conformally flat hypersurfaces in Euclidean four-space. What kind of conformally flat three manifolds are really immersed generically and conformally in Euclidean space as hypersurfaces? According to the theorem due to Cartan [1], there exists an orthogonal curvature-line coordinate system at each point of such hypersurfaces. This fact is the first step of our study. We classify such hypersurfaces in terms of the first fundamental form. In this paper, we consider hypersurfaces with the first fundamental forms of certain specific types. Then, we give a precise representation of the first and the second fundamental forms of such hypersurfaces, and give exact shapes in Euclidean space of them.

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 ON NONUNIQUENESS FOR THE ANISOTROPIC CALDERÓN PROBLEM WITH PARTIAL DATA

2020 ◽  
Vol 8 ◽  
Author(s):  
THIERRY DAUDÉ ◽  
NIKY KAMRAN ◽  
FRANÇOIS NICOLEAU
Keyword(s):  
Infinite Number ◽  
Riemannian Manifolds ◽  
Riemannian Metrics ◽  
Unique Continuation ◽  
Manifolds With Boundary ◽  
Conformal Class ◽  
Hölder Continuous ◽  
Continuation Principle ◽  
Partial Data ◽  
Calderón Problem

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order $\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M,g)$ and we show that there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric $g$ and do not satisfy the unique continuation principle.

scholarly journals On compactness of isospectral conformal, metrics of 4-manifolds

1995 ◽  
Vol 140 ◽  
pp. 77-99 ◽  
Author(s):  
Xingwang Xu
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifold ◽  
Riemannian Metrics ◽  
Conformal Class ◽  
Conformal Metrics ◽  
Laplace Operators ◽  
Definition Of ◽  
The Laplace Operator ◽  
Isometry Class

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.Let us recall the definition of the isospectral metrics. Two Riemannian metrics g, g′ on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operator Δ on the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).

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.

Sign in / Sign up

Export Citation Format

Share Document