Consistent Curvature Approximation on Riemannian Shape Spaces
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DISCRETE DIFFERENTIAL GEOMETRY: an APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE
DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 Curvature—Overview • Intuitively, describes “how much a shape bends” – Extrinsic: how quickly does the tangent plane/normal change? – Intrinsic: how much do quantities differ from flat case? N T B Curvature—Overview • Driving force behind wide variety of physical phenomena – Objects want to reduce—or restore—their curvature – Even space and time are driven by curvature… Curvature—Overview • Gives a coordinate-invariant description of shape – fundamental theorems of plane curves, space curves, surfaces, … • Amazing fact: curvature gives you information about global topology! – “local-global theorems”: turning number, Gauss-Bonnet, … Curvature—Overview • Geometric algorithms: shape analysis, local descriptors, smoothing, … • Numerical simulation: elastic rods/shells, surface tension, … • Image processing algorithms: denoising, feature/contour detection, … Thürey et al 2010 Gaser et al Kass et al 1987 Grinspun et al 2003 Curvature of Curves Review: Curvature of a Plane Curve • Informally, curvature describes “how much a curve bends” • More formally, the curvature of an arc-length parameterized plane curve can be expressed as the rate of change in the tangent Equivalently: Here the angle brackets denote the usual dot product, i.e., . Review: Curvature and Torsion of a Space Curve •For a plane curve, curvature captured the notion of “bending” •For a space curve we also have torsion, which captures “twisting” Intuition: torsion is “out of plane bending” increasing torsion Review: Fundamental Theorem of Space Curves •The fundamental theorem of space curves tells that given the curvature κ and torsion τ of an arc-length parameterized space curve, we can recover the curve (up to rigid motion) •Formally: integrate the Frenet-Serret equations; intuitively: start drawing a curve, bend & twist at prescribed rate. -
Curvatures of Smooth and Discrete Surfaces
Curvatures of Smooth and Discrete Surfaces John M. Sullivan The curvatures of a smooth curve or surface are local measures of its shape. Here we consider analogous quantities for discrete curves and surfaces, meaning polygonal curves and triangulated polyhedral surfaces. We find that the most useful analogs are those which preserve integral relations for curvature, like the Gauß/Bonnet theorem. For simplicity, we usually restrict our attention to curves and surfaces in euclidean three space E3, although many of the results would easily generalize to other ambient manifolds of arbitrary dimension. Most of the material here is not new; some is even quite old. Although some refer- ences are given, no attempt has been made to give a comprehensive bibliography or an accurate picture of the history of the ideas. 1. Smooth curves, Framings and Integral Curvature Relations A companion article [Sul06] investigated the class of FTC (finite total curvature) curves, which includes both smooth and polygonal curves, as a way of unifying the treatment of curvature. Here we briefly review the theory of smooth curves from the point of view we will later adopt for surfaces. The curvatures of a smooth curve γ (which we usually assume is parametrized by its arclength s) are the local properties of its shape invariant under Euclidean motions. The only first-order information is the tangent line; since all lines in space are equivalent, there are no first-order invariants. Second-order information is given by the osculating circle, and the invariant is its curvature κ = 1/r. For a plane curve given as a graph y = f(x) let us contrast the notions of curvature and second derivative. -
Differential Geometry: Curvature and Holonomy Austin Christian
University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics. -
Simulation of a Soap Film Catenoid
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Kanazawa University Repository for Academic Resources Simulation of A Soap Film Catenoid ! ! Pornchanit! Subvilaia,b aGraduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa 920-1192 Japan bDepartment of Mathematics, Faculty of Science, Chulalongkorn University, Pathumwan, Bangkok 10330 Thailand E-mail: [email protected] ! ! ! Abstract. There are many interesting phenomena concerning soap film. One of them is the soap film catenoid. The catenoid is the equilibrium shape of the soap film that is stretched between two circular rings. When the two rings move farther apart, the radius of the neck of the soap film will decrease until it reaches zero and the soap film is split. In our simulation, we show the evolution of the soap film when the rings move apart before the !film splits. We use the BMO algorithm for the evolution of a surface accelerated by the mean curvature. !Keywords: soap film catenoid, minimal surface, hyperbolic mean curvature flow, BMO algorithm !1. Introduction The phenomena that concern soap bubble and soap films are very interesting. For example when soap bubbles are blown with any shape of bubble blowers, the soap bubbles will be round to be a minimal surface that is the minimized surface area. One of them that we are interested in is a soap film catenoid. The catenoid is the minimal surface and the equilibrium shape of the soap film stretched between two circular rings. In the observation of the behaviour of the soap bubble catenoid [3], if two rings move farther apart, the radius of the neck of the soap film will decrease until it reaches zero. -
Mean Curvature in Manifolds with Ricci Curvature Bounded from Below
Comment. Math. Helv. 93 (2018), 55–69 Commentarii Mathematici Helvetici DOI 10.4171/CMH/429 © Swiss Mathematical Society Mean curvature in manifolds with Ricci curvature bounded from below Jaigyoung Choe and Ailana Fraser Abstract. Let M be a compact Riemannian manifold of nonnegative Ricci curvature and † a compact embedded 2-sided minimal hypersurface in M . It is proved that there is a dichotomy: If † does not separate M then † is totally geodesic and M † is isometric to the Riemannian n product † .a; b/, and if † separates M then the map i 1.†/ 1.M / induced by W ! inclusion is surjective. This surjectivity is also proved for a compact 2-sided hypersurface with mean curvature H .n 1/pk in a manifold of Ricci curvature RicM .n 1/k, k > 0, and for a free boundary minimal hypersurface in an n-dimensional manifold of nonnegative Ricci curvature with nonempty strictly convex boundary. As an application it is shown that a compact .n 1/-dimensional manifold N with the number of generators of 1.N / < n 1 n cannot be minimally embedded in the flat torus T . Mathematics Subject Classification (2010). 53C20, 53C42. Keywords. Ricci curvature, minimal surface, fundamental group. 1. Introduction Euclid’s fifth postulate implies that there exist two nonintersecting lines on a plane. But the same is not true on a sphere, a non-Euclidean plane. Hadamard [11] generalized this to prove that every geodesic must meet every closed geodesic on a surface of positive curvature. Note that a k-dimensional minimal submanifold of a Riemannian manifold M is a critical point of the k-dimensional area functional. -
Examples of Constant Mean Curvature Immersions of the 3-Sphere Into
Proc. NatL Acad. Sci. USA Vol. 79, pp. 3931-3932, June 1982 Mathematics Examples of constant mean curvature immersions of the 3-sphere into euclidean 4-space (global differential geometry of submanifolds) Wu-yi HSIANG*, ZHEN-HUAN TENG*t, AND WEN-CI YU*t *Department of Mathematics, University of California, Berkeley, California 94720; tPeking University, Peking, China; and tFudan University, China Communicated by S. S. Chern, December 14, 1981 ABSTRACT Mean curvature is one of the simplest and most is probably the only possible shape of closed constant-mean- basic of local differential geometric invariants. Therefore, closed curvature hypersurface in euclidean spaces ofarbitrary dimen- hypersurfaces of constant mean curvature in euclidean spaces of sion or, at least, that Hopf's theorem should be generalizable high dimension are basic objects of fundamental importance in to higher dimensions. global differential geometry. Before the examples of this paper, We announce here the construction of the following family the only known example was the obvious one ofthe round sphere. of examples of constant mean curvature immersions of the 3- Indeed, the theorems ofH. Hopf(for immersion ofS2 into E3) and sphere into euclidean 4-space. A. D. Alexandrov (for imbedded hypersurfaces of E") have gone MAIN THEOREM: There exist infinitely many distinct differ- a long way toward characterizing the round sphere as the only entiable immersions ofthe 3-sphere into euclidean 4-space hav- example ofa closed hypersurface ofconstant mean curvature with ing a given positive constant mean curvature. The total surface some added assumptions. Examples ofthis paper seem surprising area ofsuch examples can be as large as one wants. -
Basics of the Differential Geometry of Surfaces
Chapter 20 Basics of the Differential Geometry of Surfaces 20.1 Introduction The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We simply want to introduce the concepts needed to understand the notion of Gaussian curvature, mean curvature, principal curvatures, and geodesic lines. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate differential geometry course offered in the fall of 1994. Most of the topics covered in this course have been included, except a presentation of the global Gauss–Bonnet–Hopf theorem, some material on special coordinate systems, and Hilbert’s theorem on surfaces of constant negative curvature. What is a surface? A precise answer cannot really be given without introducing the concept of a manifold. An informal answer is to say that a surface is a set of points in R3 such that for every point p on the surface there is a small (perhaps very small) neighborhood U of p that is continuously deformable into a little flat open disk. Thus, a surface should really have some topology. Also,locally,unlessthe point p is “singular,” the surface looks like a plane. Properties of surfaces can be classified into local properties and global prop- erties.Intheolderliterature,thestudyoflocalpropertieswascalled geometry in the small,andthestudyofglobalpropertieswascalledgeometry in the large.Lo- cal properties are the properties that hold in a small neighborhood of a point on a surface. Curvature is a local property. Local properties canbestudiedmoreconve- niently by assuming that the surface is parametrized locally. -
Lecture 8: the Sectional and Ricci Curvatures
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ⊗2(^2V ∗) the set of 4-tensors that is anti-symmetric with respect to the first two entries and with respect to the last two entries. Lemma 1.1. Suppose T 2 ⊗2(^2V ∗), X; Y 2 V . Let X0 = aX +bY; Y 0 = cX +dY , then T (X0;Y 0;X0;Y 0) = (ad − bc)2T (X; Y; X; Y ): Proof. This follows from a very simple computation: T (X0;Y 0;X0;Y 0) = T (aX + bY; cX + dY; aX + bY; cX + dY ) = (ad − bc)T (X; Y; aX + bY; cX + dY ) = (ad − bc)2T (X; Y; X; Y ): 1 Now suppose (M; g) is a Riemannian manifold. Recall that 2 g ^ g is a curvature- like tensor, such that 1 g ^ g(X ;Y ;X ;Y ) = hX ;X ihY ;Y i − hX ;Y i2: 2 p p p p p p p p p p 1 Applying the previous lemma to Rm and 2 g ^ g, we immediately get Proposition 1.2. The quantity Rm(Xp;Yp;Xp;Yp) K(Xp;Yp) := 2 hXp;XpihYp;Ypi − hXp;Ypi depends only on the two dimensional plane Πp = span(Xp;Yp) ⊂ TpM, i.e. it is independent of the choices of basis fXp;Ypg of Πp. Definition 1.3. We will call K(Πp) = K(Xp;Yp) the sectional curvature of (M; g) at p with respect to the plane Πp. Remark. The sectional curvature K is NOT a function on M (for dim M > 2), but a function on the Grassmann bundle Gm;2(M) of M. -
AN INTRODUCTION to the CURVATURE of SURFACES by PHILIP ANTHONY BARILE a Thesis Submitted to the Graduate School-Camden Rutgers
AN INTRODUCTION TO THE CURVATURE OF SURFACES By PHILIP ANTHONY BARILE A thesis submitted to the Graduate School-Camden Rutgers, The State University Of New Jersey in partial fulfillment of the requirements for the degree of Master of Science Graduate Program in Mathematics written under the direction of Haydee Herrera and approved by Camden, NJ January 2009 ABSTRACT OF THE THESIS An Introduction to the Curvature of Surfaces by PHILIP ANTHONY BARILE Thesis Director: Haydee Herrera Curvature is fundamental to the study of differential geometry. It describes different geometrical and topological properties of a surface in R3. Two types of curvature are discussed in this paper: intrinsic and extrinsic. Numerous examples are given which motivate definitions, properties and theorems concerning curvature. ii 1 1 Introduction For surfaces in R3, there are several different ways to measure curvature. Some curvature, like normal curvature, has the property such that it depends on how we embed the surface in R3. Normal curvature is extrinsic; that is, it could not be measured by being on the surface. On the other hand, another measurement of curvature, namely Gauss curvature, does not depend on how we embed the surface in R3. Gauss curvature is intrinsic; that is, it can be measured from on the surface. In order to engage in a discussion about curvature of surfaces, we must introduce some important concepts such as regular surfaces, the tangent plane, the first and second fundamental form, and the Gauss Map. Sections 2,3 and 4 introduce these preliminaries, however, their importance should not be understated as they lay the groundwork for more subtle and advanced topics in differential geometry. -
The Riemann Curvature Tensor
The Riemann Curvature Tensor Jennifer Cox May 6, 2019 Project Advisor: Dr. Jonathan Walters Abstract A tensor is a mathematical object that has applications in areas including physics, psychology, and artificial intelligence. The Riemann curvature tensor is a tool used to describe the curvature of n-dimensional spaces such as Riemannian manifolds in the field of differential geometry. The Riemann tensor plays an important role in the theories of general relativity and gravity as well as the curvature of spacetime. This paper will provide an overview of tensors and tensor operations. In particular, properties of the Riemann tensor will be examined. Calculations of the Riemann tensor for several two and three dimensional surfaces such as that of the sphere and torus will be demonstrated. The relationship between the Riemann tensor for the 2-sphere and 3-sphere will be studied, and it will be shown that these tensors satisfy the general equation of the Riemann tensor for an n-dimensional sphere. The connection between the Gaussian curvature and the Riemann curvature tensor will also be shown using Gauss's Theorem Egregium. Keywords: tensor, tensors, Riemann tensor, Riemann curvature tensor, curvature 1 Introduction Coordinate systems are the basis of analytic geometry and are necessary to solve geomet- ric problems using algebraic methods. The introduction of coordinate systems allowed for the blending of algebraic and geometric methods that eventually led to the development of calculus. Reliance on coordinate systems, however, can result in a loss of geometric insight and an unnecessary increase in the complexity of relevant expressions. Tensor calculus is an effective framework that will avoid the cons of relying on coordinate systems. -
Curvatures of Smooth and Discrete Surfaces
Curvatures of Smooth and Discrete Surfaces John M. Sullivan Abstract. We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation. Keywords. Discrete Gauss curvature, discrete mean curvature, integral curvature rela- tions. The curvatures of a smooth surface are local measures of its shape. Here we con- sider analogous quantities for discrete surfaces, meaning triangulated polyhedral surfaces. Often the most useful analogs are those which preserve integral relations for curvature, like the Gauss/Bonnet theorem or the force balance equation for mean curvature. For sim- plicity, we usually restrict our attention to surfaces in euclidean three-space E3, although some of the results generalize to other ambient manifolds of arbitrary dimension. This article is intended as background for some of the related contributions to this volume. Much of the material here is not new; some is even quite old. Although some references are given, no attempt has been made to give a comprehensive bibliography or a full picture of the historical development of the ideas. 1. Smooth curves, framings and integral curvature relations A companion article [Sul08] in this volume investigates curves of finite total curvature. This class includes both smooth and polygonal curves, and allows a unified treatment of curvature. Here we briefly review the theory of smooth curves from the point of view we arXiv:0710.4497v1 [math.DG] 24 Oct 2007 will later adopt for surfaces. The curvatures of a smooth curve γ (which we usually assume is parametrized by its arclength s) are the local properties of its shape, invariant under euclidean motions. -
Ricci Curvature-Based Semi-Supervised Learning on an Attributed Network
entropy Article Ricci Curvature-Based Semi-Supervised Learning on an Attributed Network Wei Wu , Guangmin Hu and Fucai Yu * School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China; [email protected] (W.W.); [email protected] (G.H.) * Correspondence: [email protected] Abstract: In recent years, on the basis of drawing lessons from traditional neural network models, people have been paying more and more attention to the design of neural network architectures for processing graph structure data, which are called graph neural networks (GNN). GCN, namely, graph convolution networks, are neural network models in GNN. GCN extends the convolution operation from traditional data (such as images) to graph data, and it is essentially a feature extractor, which aggregates the features of neighborhood nodes into those of target nodes. In the process of aggregating features, GCN uses the Laplacian matrix to assign different importance to the nodes in the neighborhood of the target nodes. Since graph-structured data are inherently non-Euclidean, we seek to use a non-Euclidean mathematical tool, namely, Riemannian geometry, to analyze graphs (networks). In this paper, we present a novel model for semi-supervised learning called the Ricci curvature-based graph convolutional neural network, i.e., RCGCN. The aggregation pattern of RCGCN is inspired by that of GCN. We regard the network as a discrete manifold, and then use Ricci curvature to assign different importance to the nodes within the neighborhood of the target nodes. Ricci curvature is related to the optimal transport distance, which can well reflect the geometric structure of the underlying space of the network.