Note on affine manifold with parallel volume element

1980 ◽  
Vol 9 (2) ◽  
Author(s):  
Huei-Shyong Lue
Keyword(s):  
Volume Element ◽  
Affine Manifold ◽  
Parallel Volume

scholarly journals THE UNIVERSAL COVER OF AN AFFINE THREE-MANIFOLD WITH HOLONOMY OF SHRINKABLE DIMENSION ≤ TWO

2000 ◽  
Vol 11 (03) ◽  
pp. 305-365 ◽  
Author(s):  
SUHYOUNG CHOI
Keyword(s):  
Affine Connection ◽  
Holonomy Group ◽  
Singular Locus ◽  
Universal Cover ◽  
Affine Structure ◽  
Affine Manifold ◽  
Cone Type ◽  
Parallel Volume ◽  
A Cell ◽  
Simply Connected

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold M with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to R3. Hence, M is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to R3, whose proof using the Morse theory takes most of this paper; and (ii) a closed affine manifold of holonomy of shrinkable dimension less or equal to d is d-convex. To prove (i), we show that 2-convexity is a geometric form of topological incompressibility of level sets. As a consequence, we show that the universal cover of a closed affine three-manifold with parallel volume form is diffeomorphic to R3, a part of the weak Markus conjecture. As applications, we show that the universal cover of a hyperbolic 3-manifold with cone-type singularity of arbitrarily assigned cone-angles along a link removed with the singular locus is diffeomorphic to R3. A fake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.

scholarly journals Affine analogues of the Sasaki-Shchepetilov connection

2016 ◽  
Vol 15 (1) ◽  
pp. 37-49
Author(s):  
Maria Robaszewska
Keyword(s):  
Volume Element ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Higher Dimension ◽  
Flat Connection ◽  
Trivial Bundle ◽  
Flat Connections ◽  
Parallel Volume ◽  
Symmetric Connection ◽  
Affine Embedding

AbstractFor two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.

The Theoretical Contrast in Scanning and Conventional High Resolution Microscopes

1969 ◽  
Vol 27 ◽  
pp. 170-171
Author(s):  
E. Zeitler ◽  
M. G. R. Thomson
Keyword(s):  
Small Volume ◽  
Volume Element ◽  
Optical Processing ◽  
Image Construction ◽  
Object Interaction ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Optical Treatment ◽  
Electron Microscopes ◽  
Intensity Contrast

In the formation of an image each small volume element of the object is correlated to an areal element in the image. The structure or detail of the object is represented by changes in intensity from element to element, and this variation of intensity (contrast) is determined by the interaction of the electrons with the specimen, and by the optical processing of the information-carrying electrons. Both conventional and scanning transmission electron microscopes form images which may be considered in this way, but the mechanism of image construction is very different in the two cases. Although the electron-object interaction is the same, the optical treatment differs.

Trace nanoanalysis

1994 ◽  
Vol 52 ◽  
pp. 940-941
Author(s):  
D. E. Newbury ◽  
R. D. Leapman
Keyword(s):  
High Performance ◽  
Secondary Ion Mass Spectrometry ◽  
Detection Efficiency ◽  
Volume Element ◽  
And Performance ◽  
Trace Constituents ◽  
Secondary Ion ◽  
Concentration Levels ◽  
Structure Properties

Trace constituents, which can be very loosely defined as those present at concentration levels below 1 percent, often exert influence on structure, properties, and performance far greater than what might be estimated from their proportion alone. Defining the role of trace constituents in the microstructure, or indeed even determining their location, makes great demands on the available array of microanalytical tools. These demands become increasingly more challenging as the dimensions of the volume element to be probed become smaller. For example, a cubic volume element of silicon with an edge dimension of 1 micrometer contains approximately 5×1010 atoms. High performance secondary ion mass spectrometry (SIMS) can be used to measure trace constituents to levels of hundreds of parts per billion from such a volume element (e. g., detection of at least 100 atoms to give 10% reproducibility with an overall detection efficiency of 1%, considering ionization, transmission, and counting).

scholarly journals Representative volume element based micromechanical modelling of rod shaped glass filled epoxy composites

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Aanchna Sharma ◽  
Yashwant Munde ◽  
Vinod Kushvaha
Keyword(s):  
Representative Volume Element ◽  
Poisson’S Ratio ◽  
Volume Fraction ◽  
Tensile Modulus ◽  
Volume Element ◽  
Poisson's Ratio ◽  
Epoxy Composites ◽  
Micromechanical Modeling ◽  
Representative Volume ◽  
Shaped Glass

AbstractIn this study, Representative Volume Element based micromechanical modeling technique has been implemented to assess the mechanical properties of glass filled epoxy composites. Rod shaped glass fillers having an aspect ratio of 80 were used for preparing the epoxy composite. The three-dimensional unit cell model of representative volume element was prepared with finite element analysis tool ANSYS 19 using the periodic square and hexagonal array with an assumption that there is a perfect bonding between the filler and the epoxy matrix. Results revealed that the tensile modulus increases and Poisson’s ratio decreases with increase in the volume fraction of the filler. To study the effect of filler volume fraction, the pulse echo techniques were used to experimentally measure the tensile modulus and Poisson’s ratio for 5% to 15% volume fraction of the filler. A good agreement was found between the RVE based predicted values and the experimental results.

Localization simulation of forming-induced residual stresses of composite using prescribed boundary

2021 ◽  
pp. 073168442094118
Author(s):  
Qi Wu ◽  
Hongzhou Zhai ◽  
Nobuhiro Yoshikawa ◽  
Tomotaka Ogasawara ◽  
Naoki Morita
Keyword(s):  
Residual Stresses ◽  
Representative Volume Element ◽  
Computational Cost ◽  
Volume Element ◽  
Thermoplastic Composite ◽  
Structural Deformation ◽  
Element Analysis ◽  
Micro Scale ◽  
Representative Volume ◽  
Localization Approach

A novel localization approach that seamlessly bridges the macro- and micro-scale models is proposed and used to model the forming-induced residual stresses within a representative volume element of a fiber reinforced composite. The approach uses a prescribed boundary that is theoretically deduced by integrating the asymptotic expansion of a composite and the equal strain transfer, thus rendering the simulation setting to be easier than conventional approaches. When the localization approach is used for the finite element analysis, the temperature and residual stresses within an ideal cubic representative volume element are precisely simulated, given a sandwiched thermoplastic composite is formed under one-side cooling condition. The simulation results, after being validated, show that the temperature gradient has an impact on the local residual stresses, especially on the in-plane normal stress transverse to the fiber, and consequently, influences the structural deformation. This newly designed localization approach demonstrates the advantages of enhanced precision and reduced computational cost owing to the fast modeling of the finely meshed representative volume element. This is beneficial for a detailed understanding of the actual residual stresses at the micro-scale.

Sign in / Sign up

Export Citation Format

Share Document