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Articles 1 - 10 of 10
Full-Text Articles in Geometry and Topology
Differential Topology And The Poincaré-Hopf Theorem, Tara Saini
Differential Topology And The Poincaré-Hopf Theorem, Tara Saini
Rose-Hulman Undergraduate Mathematics Journal
In this paper, we will develop the ideas needed to understand and prove the Poincaré–Hopf Theorem, which connects the local behavior of smooth vector fields to global topological properties. We will begin by introducing smooth manifolds and smooth maps, which are the basis of differential topology. We will then define derivatives of smooth maps through tangent spaces and use these to classify points. To build toward the theorem, we will introduce orientation, degree, and smooth vector fields. These concepts will culminate in a proof of the Poincaré–Hopf Theorem, aided by Brouwer’s Fixed Point Theorem. Finally, we will apply the result …
Classifying Surfaces With Handle Decomposition, Elizabeth Sipes
Classifying Surfaces With Handle Decomposition, Elizabeth Sipes
Murray State Theses and Dissertations
Among topological spaces, manifolds draw a lot of interest. An
n-manifold is a space that is locally like R^n. Manifolds of dimension 2
are called surfaces. Using handle decomposition, we decompose surfaces
into k-handles, where 0< =k< =2. Techniques such as handle sliding and
handle cancellation allow us to get a more favorable representation of
our surface. We use these tools and calculation of the fundamental group
to classify all compact surfaces.
An Exposition Of The Curvature Of Warped Product Manifolds, Angelina Bisson
An Exposition Of The Curvature Of Warped Product Manifolds, Angelina Bisson
Electronic Theses, Projects, and Dissertations
The field of differential geometry is brimming with compelling objects, among which are warped products. These objects hold a prominent place in differential geometry and have been widely studied, as is evident in the literature. Warped products are topologically the same as the Cartesian product of two manifolds, but with distances in one of the factors in skewed. Our goal is to introduce warped product manifolds and to compute their curvature at any point. We follow recent literature and present a previously known result that classifies all flat warped products to find that there are flat examples of warped products …
An Equivalence Between Contact Gluing Maps In Sutured Floer Homology: A Conjecture Of Zarev, Charles Ryan Leigon
An Equivalence Between Contact Gluing Maps In Sutured Floer Homology: A Conjecture Of Zarev, Charles Ryan Leigon
LSU Doctoral Dissertations
We show that the contact gluing map of Honda, Kazez, and Matic has a natural algebraic description in bordered sutured Floer homology. In particular, we establish Zarev's conjecture that his gluing map on sutured Floer homology is equivalent, in the appropriate sense, to the contact gluing map. This further solidifies the relationship between bordered Floer theory and contact geometry.
Dupin Submanifolds In Lie Sphere Geometry (Updated Version), Thomas E. Cecil, Shiing-Shen Chern
Dupin Submanifolds In Lie Sphere Geometry (Updated Version), Thomas E. Cecil, Shiing-Shen Chern
Mathematics and Computer Science Department Faculty Scholarship
A hypersurface Mn-1 in Euclidean space En is proper Dupin if the number of distinct principal curvatures is constant on Mn-1, and each principal curvature function is constant along each leaf of its principal foliation. This paper was originally published in 1989 (see Comments below), and it develops a method for the local study of proper Dupin hypersurfaces in the context of Lie sphere geometry using moving frames. This method has been effective in obtaining several classification theorems of proper Dupin hypersurfaces since that time. This updated version of the paper contains the original exposition together …
The Poincaré Duality Theorem And Its Applications, Natanael Alpay, Melissa Sugimoto, Mihaela Vajiac
The Poincaré Duality Theorem And Its Applications, Natanael Alpay, Melissa Sugimoto, Mihaela Vajiac
SURF Posters and Papers
In this talk I will explain the duality between the deRham cohomology of a manifold M and the compactly supported cohomology on the same space. This phenomenon is entitled “Poincaré duality” and it describes a general occurrence in differential topology, a duality between spaces of closed, exact differentiable forms on a manifold and their compactly supported counterparts. In order to define and prove this duality I will start with the simple definition of the dual space of a vector space, with the definition of a positive definite inner product on a vector space, then define the concept of a manifold. …
A Structure Theorem For Bad 3-Orbifolds, Rachel Julie Lehman
A Structure Theorem For Bad 3-Orbifolds, Rachel Julie Lehman
Graduate Theses and Dissertations
We explicitly construct 10 families of bad 3-orbifolds, X , having the following property: given any bad 3-orbifold, O, it admits an embedded suborbifold X ∈ X such that after removing this member from O, and capping the resulting boundary, and then iterating this process finitely many times, you obtain a good 3-orbifold. Reversing this process gives us a procedure to obtain any possible bad 3-orbifold starting with a good 3-orbifold. Each member of X has 1 or 2 spherical boundary components and has underlying topological space S2 × I or (S2 × S1)\B3.
Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews
Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews
Honors Theses
This survey will develop the theory of normal surfaces as they apply to the S3 recognition algorithm. Sections 2 and 3 provide necessary background on manifold theory. Section 4 presents the theory of normal surfaces in triangulations of 3-manifolds. Section 6 discusses issues related to implementing algorithms based on normal surfaces, as well as an overview of the Regina, a program that implements many 3-manifold algorithms. Finally section 7 presents the proof of the 3-sphere recognition algorithm and discusses how Regina implements the algorithm.
Hidden Symmetries And Commensurability Of 2-Bridge Link Complements, Christian Millichap, William Worden
Hidden Symmetries And Commensurability Of 2-Bridge Link Complements, Christian Millichap, William Worden
Faculty Publications
In this paper, we show that any nonarithmetic hyperbolic 2-bridge link complement admits no hidden symmetries. As a corollary, we conclude that a hyperbolic 2-bridge link complement cannot irregularly cover a hyperbolic 3-manifold. By combining this corollary with the work of Boileau and Weidmann, we obtain a characterization of 3-manifolds with nontrivial JSJ-decomposition and rank-two fundamental groups. We also show that the only commensurable hyperbolic 2-bridge link complements are the figure-eight knot complement and the 622 link complement. Our work requires a careful analysis of the tilings of R2 that come from lifting the canonical triangulations of …
Dupin Submanifolds In Lie Sphere Geometry, Thomas E. Cecil, Shiing-Shen Chern
Dupin Submanifolds In Lie Sphere Geometry, Thomas E. Cecil, Shiing-Shen Chern
Mathematics and Computer Science Department Faculty Scholarship
No abstract provided.